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Metaverse-as-a-Service +The three pillars to watch: Privacy and Security, +Edge Computing, and Blockchain +Vesal Ahsani +Department of Electrical Engineering, +Sharif University of Technology, +Tehran, Iran +vesal.ahsani@ee.sharif.edu +Ali Rahimi +Department of Electrical Engineering, +Sharif University of Technology, +Tehran, Iran +a.rahimi@ee.sharif.edu +Mehdi Letafati +Department of Electrical Engineering, +Sharif University of Technology, +Tehran, Iran +mletafati@ee.sharif.edu +Babak Hossein Khalaj +Department of Electrical Engineering, +Sharif University of Technology, +Tehran, Iran +khalaj@sharif.edu +Abstract +In this article, the authors provide a comprehensive overview on three core pillars of metaverse-as-a-service (MaaS) platforms; +privacy and security, edge computing, and blockchain technology. The article starts by investigating security aspects for the +wireless access to the metaverse. Then it goes through the privacy and security issues inside the metaverse from data-centric, +learning-centric, and human-centric points-of-view. The authors address private and secure mechanisms for privatizing sensitive +data attributes and securing machine learning algorithms running in a distributed manner within the metaverse platforms. Novel +visions and less-investigated methods are reviewed to help mobile network operators and metaverse service providers facilitate +the realization of secure and private MaaS through different layers of the metaverse, ranging from the access layer to the social +interactions among clients. Later in the article, it has been explained how the paradigm of edge computing can strengthen different +aspects of the metaverse. Along with that, the challenges of using edge computing in the metaverse have been comprehensively +investigated. Additionally, the paper has comprehensively investigated and analyzed 10 main challenges of MaaS platforms and +thoroughly discussed how blockchain technology provides solutions for these constraints. At the final, future vision and directions, +such as content-centric security and zero-trust metaverse, some blockchain’s unsolved challenges are also discussed to bring further +insights for the network designers in the metaverse era. +Index Terms +Metaverse-as-a-service (MaaS), privacy and security, edge computing, blockchain +I. INTRODUCTION +What is the metaverse, exactly? The metaverse is a concept in the tech world that refers to a digital living environment +where conventional social structures are changed. It is a term that combines the concepts of the Greek1 prefix “meta,” which +means “more complete” or “transcending,” and the acronym “Verse” for “universe,” which signifies a space-and-time container2. +The idea of the metaverse was introduced in Neal Stephenson’s science fiction novel Snow Crash nearly 30 years ago [1]. +The rapid advancements of technologies like blockchain, virtual and augmented reality, gaming, artificial intelligence, and the +Internet of Things have made the metaverse one of the most buzzworthy terms in the tech world. Solutions and services are +being developed for virtual worlds to allow users to have fun, intelligently engage with their surroundings, and form deeper +connections with others [2]. Investment in the metaverse has grown significantly, with technology giants investing billions of +dollars in its development and many businesses putting together their own plans for the metaverse. A McKinsey & Company +report predicts that the metaverse will be valued at over $5 trillion by 2030 [3]. +What is Metaverse-as-a-Service (MaaS)? The phrase “as-a-Service” originally appeared in a 1985 file with the United States +Patent and Trademark Office (USPTO), and it gained popularity throughout the cloud computing era [4]. Everything that +may be considered as a service through a network can be referred to as XaaS [5]. Everything-as-a-service (XaaS) is a recent +development in the information and communication technology sector that enables the provision of scalable computing resources +on demand. Accordingly, the Metaverse can profit from “as-a-service” models, in which the key elements and technologies of +the Metaverse, such as platforms, infrastructures, software, and artificial intelligence (AI), could be provided as service models, +1https://en.wikipedia.org/wiki/Meta +2https://alldimensions.fandom.com/wiki/Category:Verse +arXiv:2301.01221v1 [cs.CR] 1 Jan 2023 + +i.e., Metaverse-as-a-service (MaaS). As tech giants entering the Metaverse space, including Microsoft, Samsung, NVIDIA, and +others, it won’t be long until there are many marketplaces with a Metaverse-as-a-Service (MaaS) offering allowing businesses +to profit from the technology with lowered entry barriers. Even though MaaS is still a relatively new technology area, there +are currently a number of vendors making headway there; such as Lovelace World, Propel MaaS, Touchcast, and MetaVerse +Books. +MaaS is characterized as an on-demand subscription solution that enables companies and/or operators to create and implement +different forms (such as existence, management, coordination, and implementation) in the Metaverse to support the processing +of Metaverse services, collaboration, company operations, and products, among other related scenarios. Everything in Metaverse +can be thought of as a delivery model that can be simply generated and/or modified as function modules, similar to XaaS in +cloud computing systems. See Figure 1. +The main benefits of using MaaS can be summarized as follows: +• Products for the Metaverse may be created by businesses without substantial digital experience or knowledge. Without +prohibitive capital requirements, even small to mid-sized firms can engage in the Metaverse economy. +• It promotes financial investment in a still-developing technology. The majority of systems are currently only designed +for consumer usage, and solutions like Microsoft Mesh and the Horizons app suite from Meta have not yet been widely +released. In this setting, MaaS enables businesses to make low-risk investments and profit from technology. This model +also reduces the time of programming, setting up and installing systems and brings the investor a profit sooner. +• MaaS may ultimately lead to industry standardization, with selected few businesses serving as Metaverse “brokers” to aid +in infrastructure creation. +• MaaS model is a win-win situation. The seller of this service does research, programming and implementation only once, +but can sell this service many times. On the other hand, the buyer of the service can also use Metaverse at a competitive +cost without getting into the technical, management and implementation complexities of Metaverse. +Physical world +Perception Layer +Users +IoT, sensors, … +Devices +Human-computer +interaction +AR/VR/… +Holography comm. +Service providers +Edge computing +layer +Edge servers +Computing +Storage +Communication +Shared, Scalable +5G/6G +Network Core +Cloud +infrastructure +Semantic comm. +Virtual world +Metaverse service +models +AIaaS, NSaaS, +BCaaS, … +SaaS, IaaS, PaaS, +DaaS, … +Application layer +Avatars +Digital twins +Virtual environment +Metaverse engine +Security & privacy +Modeling, +simulation, … +Human-centric +Learning-centric +Data-centric +Real-time +Interoperable +Management +Virtualization +Machine learning +Resource +integration +AI controller +Metaverse service +instantiation +resource pool +Distributed +management +Blockchain +Consensus +Smart contracts +Trading +Data management +Policies +Rules, SLAs, … +Regulation +Fig. 1. Structure and important modules in the Metaverse network. +Despite the outstanding advances in today’s existing MaaS platforms, the Metaverse would still need a more comprehensive +and robust collection of standards and protocols to embrace interoperability, much more comprehensively than what the +current internet includes, based on a set of rules and policies for communication, visuals, graphical demonstration, and data. +For instance, Fortnite runs on practically all popular platforms (such as iOS, Android, PlayStation, and Xbox) and supports +numerous identity/account systems and payment options, which forces rivals to cooperate (i.e., engage in interoperability). Web2 + +goliaths like Apple and Google employ similar technology today, but they are not built to integrate with one another. Building +a scalable Metaverse will depend more on interoperability than anything else. Companies might create their own Metaverse +campuses using established protocols with the support of Metaverse-as-a-Service (MaaS), and then start providing immersive +experiences that promote social interaction. The metaverse infrastructure should be supported by extensive accessibility to +mediate various aspects of human-beings’ lives, to be able to provide immersive experiences. This opens up new venues for +privacy and security risks. +The Internet-of-Things (IoT) plays an important role in digitalizing the physical world by utilizing pervasive sensors, cameras, +wearables, etc. Networking connectivity is then provided via wireless networks, while the computing and storage are provisioned +through cloud and edge computing. The IoT networks act as “bridges” between the physical world and its digital counterpart +[6]. The information flow between the two worlds, can help facilitate the decision making in both physical and digital worlds. +Users, who are mainly represented as avatars, can also produce and exchange digital contents across various platforms in the +metaverse. Mobile users interact with the digital world via their smartphones, wearable devices, and augmented reality/virtual +reality (AR/VR) helmets, to create and share contents and gain knowledge. +Information is the core resource of the metaverse. The data flow within the employed networks of metaverse has the key role +in realizing the integration of physical and digital worlds. To better understand the security and privacy needs in the metaverse, +we indicate that there exist two main sources of information in this era: The first one is the data gathered from and exchanged +by the real world that might be utilized and visualized digitally in the virtual space. The second source of information is the +output of the virtual worlds, e.g., the information generated by digital assets and services in a MaaS platform. At the same +time, artificial intelligence and machine learning (AI/ML) algorithms, performed in the computation layer or the digital twin +layer, help facilitate rendering and offering various services. Accordingly, it is crucial to safeguard the privacy and security of +data flows within the MaaS platforms, as well as the learning algorithms. +On the other hand, performing all metaverse computations on cloud servers is not necessarily the best option. In some +situations, using edge servers close to users can facilitate low-latency services, reduce network traffic, help manage data, +and improve user experience. In addition, the paradigm of edge computing is closely related to other technologies such as +artificial intelligence and IoT. For example, with the help of edge servers, it is possible to train neural networks locally with +the participation of end devices without having any internet connection. Also, edge servers enable secure management and +access control of IoT devices and sensors on-premise. +Last but not least, blockchain is regarded as one of the metaverse’s core infrastructures and helps supply the metaverse +with laws that are clear, open, effective, and trustworthy [7] because it can connect disparate minor sectors and create a solid +economic system,. It will be challenging to determine the worth of the commodities and resources traded in the metaverse +without the assistance of blockchain technology, especially when those digital components interact economically with the +real-world economy. Therefore, it would be wise to investigate blockchain technology for MaaS platforms along with the other +two pillars; privacy and security and edge computing. +In the following, we provide a comprehensive review on guidelines to safeguard the privacy and security of MaaS platforms +from different perspectives. Additionally, in order to help metaverse operators to identify an appropriate approach for using +edge computing in the metaverse, we have focused on the advantages and challenges of using the edge computing paradigm +in MaaS in two separate subsections. Moreover, we thoroughly discuss the requirments of using blockchain technology and +the actions MaaS developers should take to solve many technical challeneges by using blockcahin. Finally, we conclude the +paper with future visions and directions. +II. PRIVACY AND SECURITY +A. An Overview of Privacy and Security Challenges for the metaverse +Despite the ever-increasing advances in developing numerous services for the metaverse, privacy and security challenges are +considered as the main concerns that need to be properly understood. Considering the fact that realizing the concept of MaaS +requires an extremely wide variety of computation, communication, and networking, a wide range of security and privacy +threats also arise in the metaverse. See Figure 2. +A variety of recent technologies are integrated into the metaverse as its basis, hence, the intrinsic vulnerabilities of them +may be inherited by the corresponding MaaS platforms. Recently, numerous privacy and security flaws have been identified +for the emerging technologies, including the vulnerabilities of cryptography-based key management schemes against quantum +computers [8], [9], the privacy leakage of distributed learning networks against honest-but-curious servers or malicious clients +[10], and the misuse of massive private data by service providers [11], just to name a few. Such threats can be intensified in +the virtual world, while simultaneously other threats can also happen, which does not exist in the physical world, e.g., virtual +spying [11], [12]. Notably, massive amount of sensitive data are utilized to create a digital replica of the real world. This +opens up new challenges in terms of threats to privacy and security. Individuals use wearables or AR/VR devices with built-in +sensors collecting biometric data, brain patterns, users’ speech and facial expressions and poses, and also the surrounding +environment in order to render a high-quality immersive experience for the MaaS users. In addition, clients are under the threat + +Privacy and security +challenges for the +Metaverse +Data fusion of +multimodal +private data +Distinguishing +the realness +Highly +heterogeneous +virtual worlds +Securing the 6G- +enabled access to the +Metaverse +Secure wireless +secret key +generation +PHY +authentication +for the metaverse +access +Adversarial +learning for +wireless access +Data-centric privacy +and security +Data collection +Incognito mode +Authenticating +synthetic data +Secure and private +AI/ML for the +Metaverse +Privacy for +AI/ML +algorithms in the +Metaverse +Secure AI/ML +for the +Metaverse +Secure +aggregation +framework +Human-centric privacy +and security: privatized +social interactions in the +Metaverse +Social +applications +Biometric data +Privacy- +enhancing +protocols +Fig. 2. Privacy and security for the Metaverse. +of being illegally tracked through their VR headsets or wearables. Such a miscellaneous set of sensitive information flow in +the context of metaverse services provides adversaries with a broad attacking surface which requires proposing novel secrecy- +and privacy-preserving mechanisms for mobile network operators and service providers. +The main privacy and security challenges in the era of metaverse can be identified as follows. +• Data fusion of multimodal private data corresponding to interactions between users, avatars, and environments imposes +serious challenges for the MaaS platforms. Collection, communication, and processing of such sensitive information must +be taken into consideration in terms of privacy and security risks. +• The blurring boundary between the real world and its virtual counterpart causes serious confusions in terms of distin- +guishing the realness. Therefore, we should focus on providing stringent authenticity guarantees from the MaaS providers’ +perspective. +• Virtual worlds in the metaverse are highly heterogeneous in terms of hardware implementation, communication, and +learning interfaces. The beyond-the-fifth and the sixth generation (B5G/6G) of wireless communication networks, as +the main fabric to facilitate the communications within the metaverse, pose critical challenges for conventional security +solutions. This raises urgent needs for proposing context-aware and flexible security and privacy mechanisms [9], [13]. +We aim to carefully review these items in this paper. We provide readers with useful insights through a comprehensive overview +of security solutions across different layers of the MaaS platforms, ranging from the access layer to the social aspects. We +also review different solutions in the contexts of communication, computation, and learning. +B. Securing the 6G-enabled Access to the Metaverse +For the deployment of access layer to the metaverse, the sixth generation (6G) of wireless networking will play a pivotal +role. 6G technologies, such as pencil-sharp beams, edge AI, and integrated sensing and communications (ISAC), provides +users with perceived understanding of their surroundings. These technologies are the key enablers for implementing novel +PHY layer security (PLS) schemes as promising approaches to safeguard the security of the metaverse access, while offering +information-theoretic security guarantees [14]. Leveraging PLS, intelligent, flexible, and context-aware mechanisms are feasible +for detection and mitigation of privacy and security vulnerabilities [9], [13], [15], [16]. Considering the new era of truly end-to- +end (E2E) quality-of-experience provisioning for the MaaS products, service level agreements (SLAs) are expected to include +guarantees about the quality of security (QoSec) as well [13]. This will include addressing the required security level, adaptive, +risk-aware, and adjustable security solutions, just to name a few [9]. The evolution of the metaverse systems is expected to +introduce new means to harvest and interpret the “context” of the communication, where PLS techniques can be considered +as a promising QoSec-assuring approach. +In order to secure the metaverse access, we introduce the following security mechanisms, which can provide lightweight +and scalable solutions for securing the access to the MaaS platforms. + +1) Wireless Secret Key Generation and the Role of PHY Security: Smart wearable devices and cameras, together with +head-mounted displays, e.g., Oculus helmet and Vive Pro headsets, and handheld controllers are considered as the main +terminals for entering the metaverse. In this regard, key generation and management is essentially required for smart devices +to establish secure connections for accessing the metaverse, transmit sensory data, and receive immersive experiences. In [17] +and [18], intrinsic features of specific wearable devices, such as gestures and motions, are taken into account to propose key +agreement schemes. Researchers in [19] propose a heartbeat-based key generation scheme based on the measurements from +electrocardiography (ECG) and photoplethysmography (PPG) sensors as the source of common randomness. +While the security mechanisms of the 5G standard rely on cryptography-based keys, such as the elliptic curve cryptography +(ECC) to fulfill the confidentiality and authentication requirements, 6G networks with many peer-to-peer communications +undermine the performance of conventional solutions. Besides, assumptions on the security of cryptographic methods based +on the mathematical hardness of solving a certain problem will no longer hold with the advent of quantum computers. To +efficiently secure the metaverse era, data communications should to be proactively secured, where PLS has such capabilities +for quantum-resilient security [9]. As a promising framework to migrate from the conventional complexity-based solutions +towards lightweight security techniques, PHY layer secret key generation (PHY-SKG) has been envisioned to be employed +for 6G networks [15], [16]. PHY-SKG is realized through implementing lightweight mechanisms with minimum required +changes at the control plane. This can actually provide the access networks with different advantages. In particular, PHY- +SKG is thoroughly decentralized without relying on any infrastructure of a particular entity, which can substantially reduce +the required time for key agreement, making it suitable for extremely low latency applications in the metaverse. Moreover, +continuous update of the key is realizable thanks to the dynamic variations of the PHY attributes. The PHY-SKG protocol +under realistic assumptions of hardware impairments and adversarial attacks is developed in [15]. Notably, the PHY-SKG has +shown to be resilient against man-in-the-middle adversarial attacks in [16]. Furthermore, for establishing secure communication +between digital healthcare devices and the access point, a lightweight learning-based key agreement protocol is proposed by +the authors in [20], which can guide designers to bring device-level intelligence for securing the metaverse access. +2) Adversarial Learning for Wireless Access: Due to the proliferation of virtualized services in the metaverse era, malicious +actors may have access to the network functions, such as the open-source software of virtualized access networks, and can +exploit vulnerabilities of the metaverse access through adversarial machine learning (AML) [21]. Adversaries can poison the +wireless access of the metaverse, by manipulating the inputs to the learning algorithms employed for authorizing users [22], +leading to unauthorized access to the metaverse network due to the mislead access model. Malicious users can also infer +sensitive information about edge devices, users, and applications which have access to the network, through membership +attribute inference attacks [23] and model inversion attacks [24]. As a privacy threat to the MaaS access, data characteristics +such as device-level information may leak to adversaries. Malicious users can exploit this leaked information using membership +inference attacks [23] by building an inference model to determine whether a sample of interest (associated with a particular +device) has been used in the training data of the MaaS provider. To mitigate such attacks, generative adversarial networks +(GANs) are shown to be capable of detecting anomalies and mitigating wireless attacks for the next generation of communication +and networking services [25]. +Connection requests for the metaverse services come with quality of experience (QoE) requirements such as throughput and +latency, which can be fulfilled by assigning radio resource blocks and processing power to the requests. Although network +slicing can manage the MaaS QoE requirements, AML techniques have the potential to attack network resource management +and disrupt B5G network slicing, which can impose huge losses for the network operators, if not properly secured [22]. To +combat such attacks, different reactive and proactive defense mechanisms are proposed in [26], including: the induction of +randomness to the decision processes to reinforce robustness against malicious nodes. +So far, the security challenges for accessing the metaverse were addressed. In the following sections, we go deeper into the +privacy and security aspects inside the metaverse, in a data-centric manner. We also address synthetic and authentic realness, +envisioning the privacy enhancement within the metaverse platforms. +C. Data-centric Privacy and Security +Inside the metaverse, the scope of data collection and processing far exceed what is traditionally performed in mobile and +web applications. The technologies employed for the metaverse do not just track where users click, but where they go, what +they do, whom they interact with, what they look at, etc. Therefore, it is crucial to the users’ privacy and security to implement +data-centric privacy-preserving mechanisms and push for meaningful, sensible, and aggressive regulations in terms of data +privacy. +Recent studies demonstrate how easily the metaverse users’ privacy can be compromised [27]. It is shown that an adversarial +program can accurately infer over 25 personal data attributes. To deal with the unprecedented data-centric privacy risks in the +metaverse platforms, the idea of “incognito mode” has recently been developed for the metaverse as a novel and promising +solution for safeguarding VR users’ privacy [28]. To realize the incognito mode in the metaverse, client-level differential privacy +framework is proposed which has been shown to be capable of protecting a wide variety of sensitive data attributes in VR + +applications. It is show in [28] that by employing randomized mechanisms (depending on the format of the private attributes), +sensitive user data attributes can be obscured effectively [28]. +We are facing a new trend of intelligent systems that are empowered by synthetic realness, in which AI-generated data is +vastly exploited to reflect various aspects of the physical world [29], [30]. Inside the metaverse, consumers are not simply +targeted via pop-up ads, rather, they are provided with “immersive contents” in the form of virtual people and activities that +look realistic [31]. Nevertheless, we should concern about the authenticity of synthetic data at different levels. Authenticating +synthetic data can make AI algorithms more fair and secure through correcting data biases and safeguarding data privacy +[32]. Synthetic data is utilized for training AI models in ways that real-world data practically cannot or should not, while +protecting privacy and confidentiality. It is critically important for the network operators and MetaaS providers to leverage +generative AI in an authentic way to maintain trust for their customers [33]. To realize authentic realness, the provenance, the +policy, and the purpose of utilizing synthetic data should be taken into account by companies and service providers [34]. To +this end, distributed ledger technology (DLT) can be employed to verify the provenance of digital content, hence addressing +the authenticity. Moreover, network operators should clarify the purpose behind the exploitation of synthetic content and the +resulting advantage over non-synthetic content. +D. Secure and Private AI/ML for the Metaverse +In this section, we focus on challenges and solutions for the privacy and security of AI/ML implementations in the metaverse. +Although metaverse can bring amazing AI/ML-based services to individuals and enterprises, the utilized learning algorithms +are still vulnerable to privacy and security risks. There exist challenging bottlenecks for efficient deployment of AI/ML +techniques. Among these challenges, the big concern is that the learning algorithms might leak users’ private data [35]. +Besides, users might not be willing to share their private data in the metaverses, making it challenging for AI/ML algorithms +to perform a comprehensive data analysis for enhancing the learning-based services of the metaverse platforms. To address +the privacy risks of AI/ML algorithms, the integration of federated learning (FL) [36] and cross-chain technologies can be +considered as a promising solution to design and implement privacy-aware frameworks for the MaaS platforms [37]–[39]. For +instance, [39] proposes a hierarchical blockchain-based framework for decentralized FL. A main chain is employed to mimic +the parameter server (aggregator server) of the FL algorithm, and multiple subchains are implemented to manage local model +updates generated by smart devices (or their digital counterpart) acting as workers. +Despite the fact that FL algorithms leverage locally-trained models instead of the raw data of metaverse participants, sensitive +information can still be inferred by analyzing the model parameters uploaded by clients [24]. If the model updates are inspected +by bad actors, participant users’ privacy and security would be threatened. In addition, there might exist some malicious users, +who proactively upload adversarial data to the aggregator server(s) to mislead the training process, known as backdoor attacks +[35]. FL still has limitations in terms of security and privacy: First, malicious clients may inject mislabeled or manipulated +data to mislead the global model, which is known as data poisoning attacks [21], [40]; Second, during model update of AI/ML +algorithms, adversarial participants can save the model parameters and infer sensitive information, e.g., private attributes of +participants’ data. They might also be able to recover the original training samples of users [24], [39]. In the following, we +discuss the possible countermeasures to the abovementioned threats in more details. +We take into account the fundamental fact that local model updates in distributed AI/ML algorithms carry extensive +information about the local datasets owned by the metaverse clients. Hence, secure aggregation of local models are necessary +for the distributed learning entities in the metaverse. Moreover, there might exist some adversarial participants in the metaverse +platform who actively upload “poisoned data” to the servers (as aggregator entities) to mislead the training processes. Such +attacking venues can provide the surface for the so called “backdoor attacks” on the collaborative model training procedures +within the MaaS platforms [35]. As an example, virtual service providers and mobile network operators employ learning +algorithms as a service for their users [41]. Malicious users have the capability to eavesdrop on the models exchanged to infer +the membership of specific clients or undermine the performance of AI/ML services [23]. In addition, attackers can hack edge +devices in the physical world or impersonate benign avatars as a man-in-the-middle. They can then inject adversarial training +sample to spoof the learning services within the MaaS products. +In the following, we address secure aggregation framework to protect learning tasks from being eavesdropped by malicious +participants. +Secure Model Aggregation: Generally speaking, state-of-the-art secure aggregation (SA) protocols rely on two main +principles: i) pairwise random-seed agreement between clients to generate “masks” for hiding the metaverse users’ models; +and ii) secret sharing of the random seeds. A SA protocol enables secure computation of distributed learning models, while +ensuring that the aggregator entities do not infer information about the local models [42]. Taking into account the fact that +the MaaS users might be “honest-but-curious” during the learning process, a SA protocol must guarantee that nothing can +be learned beyond the aggregated learning model, even if up to T users cooperate with each other. This is called T-privacy +property of SA algorithms. In [42], a modular system design for SA is proposed, which can help develop lightweight SA +protocols for the learning infrastructure of the MaaS platforms. + +Backdoor Attacks: Malicious clients may inject mislabeled or manipulated data to mislead the learning models within +the highly-distributed and heterogeneous infrastructure of the metaverse platforms, which is known as data poisoning attacks +[21], [40]. In this context, backdoor attack is a type of “data poisoning” attacks that aims to tamper a subset of training +data via injecting adversarial triggers, such that AI/ML models trained on the manipulated dataset make (targeted) incorrect +prediction/decisions during the inference [40]. +Backdoor attack is aimed to mislead the trained model to infer a target label on any input data that has an adversarially- +chosen embedded pattern (a.k.a trigger). Distributed version of backdoor attacks, known as DBA, decomposes a global trigger +pattern into separate local patterns and “embed” them into the training set of different adversarial parties respectively. This +is in contrast to the conventional attacks on the surface of distributed AI/ML algorithms [43], where a unified global trigger +pattern is embedded for all malicious parties. The experiments in [44] show that the DBA method outperforms centralized +attack significantly when evaluated with the global trigger. +To overcome backdoor attacks, adversarial training (AT) is currently considered as an empirically strong defense mechanism +[40]. AT corresponds to the mechanism of training a model on adversarial examples, with the aim of making the learning +service more robust to attacks and reducing the inference error on benign samples. AT is comprehensively reviewed in a wide +variety of researches, including [40], [45], [46]. Recently, a generic framework to defend against the general type of data +poisoning attacks is proposed in [40] by desensitizing networks to the effects of poisoning attacks. To mention an application +of AT within the metaverse, an AT-based protection against facial recognition systems is proposed in [47], which can be +effectively utilized for the MaaS platforms. +E. Human-centric Privacy and Security: Privatized Social Interactions in the Metaverse +In the metaverse, human users socially interact with others through their avatars and experience MaaS platforms by leveraging +ubiquitous smart devices and network access in a real-time manner [48]. The metaverse as a “human-centric” virtual environment +is supported by massive human-centric social applications, such as holography AR, immersive VR gaming, and the tactile +Internet. Such immersive services can be realized through the interdisciplinary communication-computation-storage co-design +techniques, integrated with the concepts of human perception, cognition, and physiology for haptic communication [49]. Hence, +the most sensitive and personalized information will be exchanged among different parties, which significantly highlights the +essence of employing novel intelligent privacy- and secrecy-preserving algorithms for the information security and social +privacy of MaaS users [50]. +The biometric data plays a significant role in terms of social interactions inside the metaverse. To elaborate, human-driven +agents and avatars, with whom the users interact, are being developed based on users’ personal data [51]. Such sensitive data +can be fed into AI/ML algorithms to create personalized “interaction partners” and influence social interaction behaviors of the +metaverse users. By exploiting physical and mental traits inferred from biometric data, the interaction partners can maliciously +lead users into undesired situations and behaviors. Hence, the metaverse users can be manipulated not only by businesses and +institutions, but by other individuals in a human-centric manner. +To mitigate such human-centric threats, attention should be paid to the users themselves. Metaverse service providers should +make users aware of the implications of biometric data extracted by AR/VR devices. Developers can implement privacy- +enhancing protocols such as differentially private mechanisms as proposed in [52] and [53]. Users can also be provided with +secondary avatars, e.g. clone, to hide their actions in the metaverse [11]. The secondary avatars help users hide their real +behavior within the metaverse. Furthermore, users must be able to have an adjustable level of privacy [28] to dynamically +manage their personal space in the virtual world by choosing their desired privacy configuration. From a governance layer, +a modular bottom-up governance approach is proposed in [54], which can be adapted to different platforms and use cases. +Decentralized autonomous organizations (DAOs) are examples of governance systems that allow users to be involved in +decision-making processes for the MaaS platforms [55]. In this regard, organizations such as XR Safety Initiative (XRSI)3 +will play an important role in encouraging MaaS providers and network operators to help facilitate enhancing trust within the +MaaS implementations in terms of human-centric privacy and regulations. +III. EDGE COMPUTING +To unlock the full potential of the metaverse, edge computing represents a promising paradigm for technical development of +the metaverse. Edge computing is a well-known technology that allows operators to perform processing, storage, and network +functions close to the edge of the network, where users are located. By using this paradigm in the Metaverse, network efficiency +is improved, costs are reduced, and new features are added. +As an illustrative example, consider a smart hospital in the Metaverse. In this center, telemedicine facilities are implemented. +Artificial intelligence is used to diagnose diseases and suggest treatment, hospitalized patients are monitored in real time with +IoT sensors, and remote surgery can be done in this center. In this scenario, if all this data is to be sent directly to the +3https://xrsi.org + +cloud server for processing and storage in a traditional way significant network bandwidth will be occupied. In addition, with +this method, there will be a long delay in sending data, and it is possible that the confidentiality of patients’ data will be +compromised. This motivates the need for the edge computing paradigm that can process and store data near the edge of the +network where data is generated. In short, this design can be such that several servers are installed in the hospital that can +quickly collect and process large amounts of data, feed it to neural network models, then send the extracted information to the +cloud server. +Edge computing can be a lucrative opportunity for network operators. One-third of network operators’ costs are spent on +real estate to house infrastructure [56]. Converting these locations into edge computing sites is a huge revenue potential for +network operators. They can also use their extra processing and storage resources to empower network edge fascilities. On +the other hand, operators can have a successful entry in the development of metaverse. They can play key roles by using +the services they provide in the 5G and 6G mobile networks [57]. Consequently, providing MaaS is a smart way to boost +operator income and develop diverse applications. In the MaaS model, organizations can flexibly create their own virtual world +according to their needs and with the most recent versions of the tools. This is made possible by using operator edge facilities +and networks. In addition to the network operator that provides a lot of hardware and software infrastructure, vendors such +as application developers, edge computing equipment vendors, system integrators, and edge computing solution providers are +also active in this market and new revenue lines are open to them. Furthermore, various vertical markets such as automotive +vehicles, virtual content productions, robotics, eHealth, entertainment, smart manufacturing, smart grids, and many others are +added to this market. For a survey on the edge computing market, see [58]. +An edge computing system includes the internet gateway, edge servers, and perception layer [59]. Internet gateway is +responsible for connecting the edge system with the network backbone and cloud servers. Edge servers process, store, and +aggregate data. Perception layer includes edge devices such as IoT sensors and actuators, AR glasses, VR headsets, haptic +gloves, smartphones, wearable devices, personal computers, and holographic displays. The metaverse can benefit from edge +computing in different ways. See Figure 3. In the following, we explain some of them. In addition, edge computing also brings +challenges. After explaining the advantages, we explain the challenges of using edge computing in the metaverse. It should be +noted that we provide a high-level review of these items and delving into their technical details is beyond the scope of this +paper. +Latency reduction +• Physics emulation +• Graphic rendering +• Real-time straming +Performance optimization +• Resolving computation bottlenecks +• Network traffic control +• Higher variety of services +Data control +• Access control +• Scalability +• Ubiquitous access +Capacity enhancement: +• Network traffic reduction +• Free up cloud resources +• Orchestrate resource sharing +Computation offloading +• Enrichment of sensors and actuators +• Expanding into new markets +• Reducing energy consumption +Privacy preserving +• Strengthen data ownership and trust +• Leveraging encryption +• Training neural networks on local data +Monitoring & management +• Edge-assisted communication +• Dynamic efficient network +• Remote management +Reliability & availability +• Tactile Internet +• Hazardous environments +• Anomaly detection +Edge intelligence +• Edge computing for AI +• Edge AI for network management +• Distributed machine learning +Cost reduction +• Profitable edge computing +• Local solutions for local problems +• Resource sharing +Modern networking techniques +• Semantic communication +• Virtualization of real-world edge networks +• Taking advantage of multi-user scenarios +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +Fig. 3. Metaverse requirements to be met by edge computing. +A. Metaverse requirements to be met by edge computing +Many applications in the metaverse need seamless synchronization of the real and virtual worlds. Some of these applications +are critical, such as remote control of a robotic assisted surgery system, and some of them, such as guiding an avatar in a +virtual environment, are related to the quality of the user experience (QoE). Therefore, it is necessary to provide a metaverse + +infrastructure to provide low-latency services. In traditional solutions, processing and storage are outsourced to cloud servers. +These servers can easily be located thousands of kilometers away from users, and this distance causes a lot of delays in +telecommunications. Edge computing, as an alternative solution, brings processing and storage closer to users and the place +where data is generated. Hence, in the literature, some metaverse processes that are very apt to run on edge servers and edge +devices such as physics emulation [60], [61], graphic rendering [62]–[65], and real-time streaming protocols [66] have been +investigated. +When edge servers are set up near users, it does not mean that the operator transfers some calculations from the cloud +server to the edge server forever. Instead, it flexibly uses edge facilities to optimize network performance depending on +different conditions. In other words, edge computing and edge storage capabilities add new dimensions to network performance +optimization problems and enlarge their feasible region. Therefore, the network has several more degrees of freedom. The +operator can dynamically decide, depending on the situation, how the edge servers come into play and bring advantages to +the metaverse network. For example, edge servers schedule computing tasks [67], provide better coverage [68], run load- +balancing algorithms [69], resolve computation bottlenecks [70], and provide heterogeneous services well addressed in the +MEC (Multi-access Edge Computing) standard [56]. +To make the metaverse network scalable, it is necessary to support issue tracking, data quality management, data protection, +and risk control functions. On the one hand, we know that the metaverse uses distributed systems. It also captures sensitive +data such as biometric data, eye movements, and financial data. In addition, the metaverse interacts with a wide range of +technologies from blockchain to artificial intelligence. For this reason, the impact of data control functions in the metaverse +is significant. On the other hand, we know that edge computing enables better and faster data control by keeping data in the +local network and close to the user. Edge computing makes it easier for organizations to control data access to a universal +metaverse [60] whilst manages data exchange in various ways such as W-LAN, cellular networks, and satellite communication +systems which are supported by 5G and 6G [71]. It is also possible to support distributed computing and use the processing +and storage resources of end-devices, which can help network scalability [72]. +Furthermore, small devices may not be able to run many heavy tasks in the metaverse like the inference of large neural +networks and real-time renderings. In this case, they should be able to outsource these tasks to powerful servers or perform the +tasks with the help of other devices in a distributed scheme [73]. This reduces telecommunications and thus reduces energy +consumption [74]. Edge servers can provide more performance, flexibility, and storage than edge devices. In addition, it is +usually easier to upgrade edge servers than end devices. This is because we can equip the edge server with more GPUs, RAM +cards, and storage devices, or change the previous components to increase the power of the edge server. But we may not be +able to disassemble all edge devices, including IoT modules, and change their features. So far, a lot of research has been done +on the outsourcing of computing of IoT devices to edge servers, in which its advantages, challenges, architectures, and various +approaches have been explained [75], [76]. Further, some IoT devices have limited batteries, and energy costs can be high. In +any case, optimizing energy consumption in the Metaverse network is crucial. A practical case study is presented in [77] that +explains the impact of edge computing on energy consumption in IoT devices. +Even with modest Metaverse assumptions, data usage could easily expand more than 20x during this decade [78]. If edge +servers perform storage and processing tasks for users, there is no need to send large amounts of raw data deep into the +network. For this reason, the telecommunication capacity of the network is less occupied. Hence, edge computing can replace +cloud computing at a lower cost in many applications. Additionally, since there are a lot of powerful edge devices in the +metaverse network, it is possible to encourage resource sharing among edge computing applications. Thus, edge devices can +serve as microcenters for storage and processing. In this case, the capacity of the metaverse network will increase without +occupying the resources of the cloud server [79]. The authors in [80] have presented an efficient method for sharing resources +at the edge of the network in the framework of the resource pool. For a survey of incentive mechanisms for sharing resources +at the network edge, see [81]. +Privacy protection is another challenge in the metaverse. Storing data on a local edge server (for example, at a university, +hospital, or airplane) can help keep users’ data confidential [82]. Edge servers can monitor the passage of sensitive data and +be equipped with hardware and software protection tools. Many IoT sensors and devices are not smart enough to not be +compromised. Sometimes their communication is not secure enough. Having an edge server near these devices and possibly +owned by the user can maintain confidentiality and bring more trust. Users connected to an edge server can use local area +network policies such as device and file sharing. On the other hand, authentication mechanisms can be implemented on edge +servers to secure the network against various types of hacker attacks [83]. Edge computing can come to the aid of end devices +and participate in cryptographic algorithm calculations [84] and training neural networks on local data [85]. +Edge servers make the network more self-aware and easy to manage and monitor. Spending on creating sites for edge servers +can be multi-purpose and simultaneously address several different needs. Edge servers provide monitoring of local network +information and its devices and help with higher-level configuration and customization. Edge computing is a key technology in +the achievement of 5G goals. It is also one of the enablers of 6G networks [86]. It enables distributed scenarios and low-latency +telecommunications which can help improve network traffic and be part of upcoming solutions in telecommunication issues + +such as channel selection. Furthermore, to choose the most appropriate coding and communication protocol, edge servers can +send network conditions to end devices and vice versa [87], [88]. Edge computing tools help end devices in many ways. In +addition to performing calculations, storage, caching, prefetching, and data traffic prediction of end devices, they can improve +network efficiency and dynamics. For example, they can have different communication channels and direct the information +packets of connected devices from the most efficient path [89]. Additionally, solutions have been offered to avoid the need for +physical presence to calibrate end devices with the help of edge computing [90]. Environmental threats such as temperature +rise, humidity, dust, and power outages are just as serious as cyber threats and can put sensitive edge devices out of service +[91]. The presence of processing and storage facilities at edge computing sites enables edge remote management solutions. +With the help of these solutions, we can take care of the critical facilities of the Metaverse network at the edge and reduce +the energy required for troubleshooting and repair. +Additionally, we want the metaverse network to be more reliable and available. Although there is no universal definition +of reliability and availability, and there is no standard way to measure them, we can talk about strengthening the reliability +and availability of some services compared to others. In some applications that are offered under MaaS, these two features +are prominent and should be considered in design, programming, and warranties given in agreements. In the event of a +network failure, edge computing infrastructure can still maintain the connection of the devices and perform some processing. +Additionally, it is possible to add redundancy to the system at edge computing sites for specific applications. By bringing +the processing tasks to the edge servers, the connection of the devices to the core network is reduced. This reduces the level +of vulnerability of the system to cyber threats. In [92], to increase the reliability of the Metaverse network, the authors have +presented a method based on distributed coded computing algorithms that can be implemented at the edge. Also, a growing +number of solutions are being offered for edge computing reliability in hazardous environments [93]. System reliability is +significantly affected by the detection and prediction of anomalies. Artificial intelligence can train machine learning models +on the large amount of local data that edge servers have access to and use them in real-time [94]. +3D reconstruction, character animation, detection of harmful content, machine vision, gesture recognition, facial expressions, +understanding emotions, body movement recognition, physical interactions, eye tracking, brain-computer interface, and many +more are activated by artificial intelligence in the Metaverse. Some strategies for using machine learning to strengthen security +in the Metaverse and to create recommendation systems are proposed in [95]. For various reasons, there is a rising desire to train +and execute machine learning models on edge servers, which is called Edge AI or Edge Intelligence. Many applications in the +metaverse such as autonomous vehicles, intelligent interaction of avatars with the environment in the virtual world, health care, +smart city, and industrial complexes need machine learning algorithms [96]. Having computing servers on the edge equipped +with artificial intelligence allows better network and telecommunication management. For example, in situations where we +have many sensors in an environment and we don’t want their massive data to leave the local network raw, we can reduce the +traffic volume with the help of an edge server that can process their data and reduce their size. In addition, in the edge server, +algorithms can be implemented to predict traffic and cache popular data, which increases the service quality. Today, machine +learning has been used in different telecommunication layers, from the physical layer and coding of telecommunication signals +[97] to the detection of cyberattacks in the application layer [98]. Designing distributed machine learning algorithms such as +federated learning [99], split learning [100] and gossip learning [101] is a hot research topic because it adds new capabilities to +the network. For example, with distributed learning, the neural network is trained without the data leaving the device and the +processing load is spread across multiple devices. In addition, in a local area network where Internet traffic is limited, devices +can continue to train neural networks with the help of each other without having a permanent connection to the Internet [102]. +As previously mentioned, edge computing is not necessarily a substitute for cloud computing. Rather, it is an option with its +advantages and challenges. The costs and desired quality of the various services provided in the metaverse should be examined +and the appropriate way to provide them should be chosen. In addition, attention should be paid to the interaction of cloud +servers and edge servers and how to divide their work. Due to the existence of various end devices and network equipment +that can be used in the metaverse, the importance of this review increases. Edge computing reduces network traffic load and +reduces costs. Additionally, it may be cheaper for some businesses to build edge servers than to outsource computing. For +example, in some regions, the cost of electricity makes it cost-effective to install edge servers. Cost-effective solutions for +edge servers are also provided. For example, the study [103] shows that micro data centers at the edge cost 42% less than +traditional centralized data centers. +In telecommunications, there are several techniques to reduce costs and increase capabilities. Some of them are in the +common domain of edge computing and the metaverse. Network designers can use these ideas. For example, semantic and +goal-oriented communication is provided to increase stability and efficiency [104], which can be used in the 6G network and +contribute to the metaverse. Edge servers can be equipped with various neural networks and perform semantic communication +tasks. On the other hand, the metaverse can help itself is by creating digital twins of edge devices. By monitoring the digital +twins of edge devices and edge infrastructures such as edge servers, I/O ports, RIS (re-configurable intelligent surface), UAV +(unmanned aerial vehicles), SAGIN (space-air-ground integrated network), and network equipment, the MaaS operator can +instantly control and calibrate these physical entities. + +B. Edge computing challenges +An operator who wants to use the edge computing paradigm in MaaS design must consider several challenges. Limitations of +edge servers is one of these challenges. Due to the limitations of end devices (power, bandwidth, computing, storage, etc.), edge +servers are proposed to assist end devices in tasks such as artificial intelligence processing and high-quality image rendering. +But it should be noted that the power of edge servers can also be limited due to hardware limitations or server room space. +In this case, the edge server cannot provide all the desired functions for a large number of users at the same time. +Another challenge to be considered is the resource allocation problem. Users in a region will likely make many requests +to the edge server at the same time and we will see network spikes. Managing these requests is a challenging issue. Another +issue is the management of faulty hardware. Cloud servers have more infrastructure compared to edge servers. The reason +why it is beneficial to have spare hardware on hand is so that in the event of a hardware failure, it is possible to replace it +quickly. But on edge servers, this issue is not economical. Suppose you own one device, and you buy a spare. The cost will +double. But suppose you have 100 pieces of equipment. If you buy a spare, your cost will be 101 instead of 100, which is +relatively not much. In addition, fairness is also essential in resource management. Because there is a lot of resource sharing +in the services provided at the edge, the metaverse operator at the edge needs to be very careful about the optimal scenarios. +For more information on resource management at the edge, see [105]. +Stragglers and malicious node are serious problems in outsourcing scenarios at the edge. When a computing load is distributed +among several servers or data is stored in several servers, it is possible that some of these servers, i.e., stragglers, will take +more time than usual to respond. This issue causes our entire work to be delayed on those servers. Malicious nodes in the +network are entities that seek to disrupt the functioning of the metaverse. There should be pre-prepared scenarios to deal with +stragglers and malicious nodes. There are various solutions to mitigate stragglers and malicious nodes in the network. For +example, in coded computing algorithms, we can divide a task into several parts and outsource it to several servers. If some +of these servers are stragglers or malignant, the computation’s final answer can still be recovered [106]. +There are also a couple of communication and networking challenges in using edge computing for the metaverse. Metaverse +services have diverse and strict requirements. For example, touch Internet requires high-reliability telecommunications, AR/VR +headsets require high bandwidth, and IoT devices require high coverage. A large amount of data in the network and its dynamic +nature should not cause friction and decrease the efficiency of the system in terms of reliability, delay, and rate. Increasing the +variety of services can make MaaS faster and more stable. For example, if different service modes and qualities are available, +for example, the video size can be changed or artificial intelligence can be adjusted with different accuracy levels, some of +the network challenges in overload conditions will be solved. +The distributed and heterogeneous nature of the metaverse network at the edge complicates its management too. In addition, +edge servers must be synchronized with end devices and cloud servers. In addition, various functions must be executed +simultaneously to maintain network performance, such as anomaly detection algorithms, resource provisioning, workload +prediction, and traffic routing. In places where edge servers are installed, such as a hospital, a network specialist is not always +available. For this reason, another challenge of network management at the edge is the high cost of admin intervention for +configuration, updates, and equipment maintenance. +Ownership considerations are another critical issue in this matter. Equipment, data, and software on edge servers can have +different owners. For example, in the case of equipment, some edge servers are privately owned, some are cooperatively owned, +some are rented out to generate revenue, and some are created with specific access requirements. This issue can make it difficult +for the Metaverse operator to enter into contracts and launch new services. +As the number of end devices, edge servers, and IoT modules increases, the metaverse network becomes challenged in various +sectors like wireless communication, queue management, and user authentication. For example, in wireless communication, +interference and congestion can occur. In queue management, there is a possibility of data being thrown away, and user +authentication can take a long time. We also know that edge servers hold sensitive data generated in places like hospitals, +financial institutions, and homes. But keeping data close to the user is not enough. Users and businesses should be given access +and data confidentiality guarantees. For more information on edge privacy solutions, see [107]. +In some applications, it is necessary to know who is responsible for each operation at the edge of the network. It should also +be possible to receive complaints and prevent manipulation, corruption, and abuse. Decision-making processes and functions +should be expanded to the extent that the privacy and property rights of individuals are not compromised. The participation of +different people and communities in monitoring the network not only improves the performance of edge servers in one place +but also creates knowledge that can be used throughout the metaverse network. +The needs of users are not the same in all regions. In addition, the data formats available in the network and devices connected +to the network are diverse. Therefore, the MaaS operator must design and install edge server equipment and software in a +flexible and compatible structure and configuration. Another issue is changes and developments in the network. The network +can change over time in terms of users or cloud servers. Accordingly, edge equipment should be able to meet these requirements +at a low cost. + +IV. BLOCKCHAIN TECHNOLOGY +To provide users with the best experiences possible, the metaverse gathers enormous amounts of private data. This information +is required by the businesses or programs in order to construct targeted systems successfully. With its authentication, access +control, and consensus methods, blockchain gives consumers total control over their data, protecting their personal information +[108]. +The blockchain utilizes hash algorithms and asymmetric-key encryption to protect data in the metaverse. Additionally, the +quality of the real-world data that users exchange is crucial to the construction of objects in the metaverse. This data is +collected from numerous applications, including those in entertainment, healthcare, and more. Blockchain enables individuals +and businesses to validate all transactions by providing comprehensive audit trails of all transactions. The metaverse’s data +quality will improve as a result [109]. +Blockchain +solutions +for MaaS +Data acquisition +Data storage +Data sharing +Data +interoperability +Data privacy +preservation +IoT +Digital twins +AI +Big data +Interactivity +Fig. 4. Blockchain solutions for MaaS. +On the other hand, the successful exchange of AR and VR data is essential to Metaverse. The smooth and safe data sharing +within the metaverse is made possible by the blockchain’s cutting-edge encoding information system. Moreover, stakeholders +in the metaverse must have access to and control over resources in various virtual environments. Due to the many settings +in which these virtual worlds are created, data interchange is constrained. A cross-chain protocol enables data interchange +between two or more blockchains that are present in different virtual worlds. In terms of integrity, data in the metaverse must +be constantly and accurately updated. Due to the immutability offered by the blockchain, the metaverse data is kept as a copy +in each block along the chain and cannot be changed or withdrawn without the agreement of a majority of the participants. +A number of vendors have already begun to enter the MaaS market; according to an announcement made by Propel, a +blockchain solutions platform, it would provide MaaS solutions for smart contracts, NFT utilities, and decentralized finance +(DeFi). Moreover, Lovelace, a different blockchain-based platform, offers a MaaS toolkit that gives users and developers +the technology required to build and trade NFTs, run smart contracts, monetize VR gaming, interact with other metaverse +platforms, and much more [110]. In the following, the authors concentrate on the steps that a MaaS developer should do and +utilize blockchain technology to address various difficulties associated with creating and developing the Metaverse platform. +See Figure 4. + +A. Open challenges for MaaS +Data Acquisition: The metaverse will generate large amounts of unstructured, real-time data through decentralized appli- +cations, but acquiring this data can be difficult. Building applications in the metaverse, such as recommender systems, will +require high levels of data integrity. The use of virtual reality and increased streaming in the metaverse will further strain data +acquisition systems [111], [112]. The quality of the data may also be impacted by the acquisition of duplicative or incorrect +data [113]. +Data Storage: Once the metaverse is fully functional, the physical world’s ability to store data may be strained to its +breaking point. This could create significant challenges for the metaverse [114]. If the metaverse relies on a central storage +system, there is a risk of data leakage, manipulation, or loss. The potential of the metaverse to offer biometric data, voice +inflections, and vital signs that depend on sensitive data is also jeopardized by the likelihood of data loss and corruption in +centralized applications [115], [116]. +Data Sharing: Data sharing on centralized platforms carries the risk of exposing private and sensitive information [117], +[118]. Additionally, due to data mutability, there is a risk of high latency and reduced data availability [119]. This is particularly +relevant in the metaverse, where many applications will generate large amounts of real-time data. When demand for real-time +data increases, data flexibility can become a problem. +Data Interoperability: The metaverse will be created through the merger of many digital domains, but these domains are +currently fragmented and disorganized. This can make it difficult for users to engage with multiple virtual worlds, as they must +set up separate accounts, avatars, hardware, and payment infrastructure for each one [114]. There are also few methods for +users to transfer their digital assets between different digital environments. In order for the metaverse to be truly interoperable, +digital world apps must be able to easily exchange information with one another, regardless of their location or the technology +being used. The conventional approach to interoperability is inadequate for the metaverse, so new solutions are needed [120]. +Data Privacy Preservation: In the early stages of the metaverse, attackers may be able to deceive users and steal important +data. This could be particularly dangerous if attackers use artificial intelligence bots, as users may not be aware that they are +not speaking with a real person. The metaverse also raises concerns about the confidentiality of personal data, particularly +personally identifiable information (PII) [121]. Finally, as the metaverse grows and more validity information is included, +managing the large amounts of data will become increasingly difficult. +IoT: The metaverse will have a large number of interconnected IoT sensors, which raises concerns about IoT security +and storage. Real-time analysis of unstructured IoT data is also challenging [122]. When storing data across virtual worlds, +a centralized solution is not ideal, as tampering with even one piece of data could compromise the entire set of findings. +Additionally, data sharing across virtual worlds will depend on the cross-platform capabilities of IoT devices [123]. Finally, +IoT data tracking is necessary for safety and legal compliance. +Digital Twins: The quality of the data used to build digital twin models is important for their accuracy, so the information +provided by the source must be accurate and of high quality [124]. Digital twins from different sectors, such as healthcare and +finance, must be able to communicate and connect with one another. To improve the accuracy and consistency of communication, +digital twins should be able to detect and correct faults. However, data security can be a challenge when using a range of +devices and sensors to create digital twin models that utilize real-time data. This can be particularly vulnerable to botnets and +other viruses [125]. +AI: The ownership of AI-powered content in the metaverse is difficult to determine, as users have no way to distinguish +between communicating with a real person and an avatar created by a computer. This could lead to users using AI technology +to exploit other users or resources in the metaverse, such as by cheating at games or stealing from other users’ accounts [126]. +Additionally, AI may make mistakes, which could lead to people losing trust in the metaverse. Another challenge is the use +of a similar blockchain across different AI applications in the metaverse. +Big Data: One of the main challenges is the sheer volume and rate of data production in the metaverse, which can be +difficult to keep up with, even with advances in data storage technology. Another challenge is the variety of data produced +by metaverse apps, which can make it difficult and time-consuming to collect and organize the necessary data for consumers. +Additionally, the rapid development of big data technology [127]–[132] can make it challenging to stay up to date with technical +advancements in the metaverse. +Interactivity: The metaverse, which is a virtual world created through the use of technology like holographic telepresence and +augmented reality, offers immersive, realistic experiences by combining audio, video, cognition, and other elements. However, +the use of XR technology in the metaverse also creates challenges related to data storage, data sharing, and data interoperability. +For example, the businesses can create recommendation systems using data from XR technology, but this data must be stored +securely and shared transparently with stakeholders. Additionally, the metaverse must be able to handle the exchange of data +between virtual worlds in an interoperable way in order to provide users with a seamless experience [133]. + +B. Blockchain solutions for MaaS +Data Acquisition: With the use of blockchain technology, it will be simpler to gather reliable data in the metaverse for +uses like social networking. Blockchain’s distributed ledger will make it possible to trace data in the metaverse and validate +transaction records [134], [135]. Because the majority of nodes in the ledger must consent before any modifications to the +data in the metaverse can be made, data collection is therefore resistant to attacks [136]. A blockchain-specific validation +process that is driven by consensus mechanisms is applied to all data collected in the metaverse [137], [138]. Every action is +documented as a transaction on a blockchain, and each block includes a cryptographic hash of the one before it, as well as the +metadata, a date, and the activity [139]. As a result, changing the data in one block will change the data in all the other blocks +as well. Any block’s data is impervious to manipulation [140]. There will be no repetition in the data collecting process since +the likelihood of producing a duplicate block is almost zero. Data obtained by blockchain enabled acquisition mechanisms in +the metaverse will be trustworthy since every block is approved on the blockchain [141]. +Data Storage: The metaverse storage is impermeable to hacking since a new block is generated for each transaction [142]. +As a result, data is stored across the chain as a copy of the original blocks, increasing data dependability and transparency +in the metaverse [143]. If the centralized data store is hacked, the metaverse applications, which include anything from real +estate to digital things, would be very vulnerable [144]. Utilizing blockchain technology will lead to a large number of blocks +contributing to data distribution, enhancing data accessibility in applications like vital monitoring and life support alerts in +the metaverse. Blockchain technology’s decentralized nature enables data scientists in the metaverse to work together and on +data cleaning, which will greatly minimize the time and expenses involved with labeling data and getting datasets ready for +analytics [145]. +Data Sharing: Blockchain technology has the potential to increase the accuracy and transparency of transactions in the +metaverse for applications such as education and cryptocurrency trading [118]. Stakeholders would be able to access a +decentralized, unchangeable record of all transactions created by applications like governance and finance. Therefore, increased +data openness will be advantageous to the metaverse’s stakeholders [146]. Users’ confidence will increase as a result of being +able to comprehend how third-party programs like Thunderbird, the Bat, and Pegasus manage data thanks to blockchain +technology, which can also reduce grey market transactions [147]. The owner of the data will also have total control over +the data. Distributed ledger technology can also be useful for data audits. Blockchain thus saves time and money by reducing +the need for data validation [148]. The flexibility of data sharing will be increased through smart contracts. Usually, they are +used to automate the execution of a contract so that all parties may be sure of the result right away, without the need for an +intermediary or a waste of time. The diverse programming of smart contracts is made possible by blockchain. As a result, +programs like Nmusik, Ascribe, Tracr, UBS, and Applicature will benefit [149]. +Data Interoperability: A cross-chain protocol is the ideal approach to guarantee interoperability across virtual worlds in +the metaverse [150], [151]. This enables the transfer of goods like avatars, NFTs, and money between virtual worlds. This +protocol will lay the foundation for broad acceptance of the metaverse. Cross-blockchain technology will make it possible for +virtual worlds to communicate with one another, doing away with the necessity for middlemen in the metaverse [152]. In the +metaverse, connecting users and apps will be made simple via blockchain. +Data Privacy Preservation: Through the use of private and public keys, blockchain technology enables users of the metaverse +to govern their data, effectively giving them ownership over it. Third-party intermediates are prohibited from misusing or +obtaining data from other parties in the blockchain-enabled metaverse. Owners of personal data stored in the blockchain- +enabled metaverse will have control over when and how third parties can access that data [153]. Blockchain ledgers come +with an audit trail as a standard, guaranteeing that the transactions in the metaverse are comprehensive and consistent. Zero- +knowledge proof has been used on the blockchain, giving people easy access to the identification of crucial data in the metaverse +while keeping their privacy and control over their belongings. Blockchain technology uses zero-knowledge proofs as a method +for users to convince apps of something without having to provide the information [154]. +IoT: Through cross-chain networks, which are created by blockchain technology, IoT devices in the metaverse may exchange +data and create tamper-proof records of shared transactions in virtual worlds. Applications and individuals will be able to +exchange and access IoT data thanks to blockchain technology without the requirement for centralized administration or +control [155]. Each transaction is documented and validated in order to reduce disputes and boost user confidence throughout +the metaverse. IoT-enabled blockchain in the metaverse makes it possible to store data in real-time. Due to the immutability of +blockchain transactions, all stakeholders can depend on the information and respond quickly and effectively [156]. By allowing +stakeholders to manage their IoT data records in shared blockchain ledgers, blockchain technology can assist in resolving +problems in the metaverse. +Digital Twins: Digital twins are attack-resistant because to blockchain’s encryption capabilities and historical data openness, +which also allow for safe data sharing [157] across many virtual worlds. With the use of an intelligent distributed ledger, data +may be exchanged between digital twins in virtual environments. Using an intelligent distributed ledger, real-world items will +be saved on the blockchain and synced to digital twins in the metaverse. The implementation of digital twins on a blockchain +will also help to resolve problems with data security and privacy [158]. Tracking sensor data and creating high-caliber digital + +twins in the metaverse will be possible by combining blockchain and AI. Every digital twin activity in the metaverse will be +documented as a transaction on the blockchain, which is unchangeable and requires consensus to modify [157]. +AI: Blockchain-based encryption gives users of the metaverse total control over their data and makes it easy to transfer +ownership of AI consent to another entity. Through the use of zero-knowledge proofs, users may convince apps and other +parties that certain information about them is true without divulging this information to the applications themselves, granting +the authority to utilize data for AI model training. Blockchain ledgers frequently offer an audit trail that may be used to +verify the legitimacy of any transactions that take place in the metaverse. People can locate crucial metaverse facts via a +zero-knowledge evidence system while still maintaining their privacy and control of their resources against deepfakes [159]. +By doing this, AI will be stopped from wasting resources in the metaverse. +Big Data: By assisting in the collecting of data from reliable data sources, blockchain technology will help to reduce the +quantity of inaccurate data received. Data modification by other parties will be prohibited, and the data owners will have +complete control over their data. This guarantees high-quality data flows across the metaverse [135]. Data scientists in the +metaverse will be able to communicate and work together on data cleaning thanks to the decentralized nature of blockchain +technology. This will greatly cut down on the time and costs involved in categorizing data and building datasets for analytics +applications, as well as the danger of data contamination. Since the data will be copied across the network and the blockchain +is immutable, it will be impossible to alter it [160]. Data accessibility for metaverse stakeholders will therefore be improved. +Interactivity: A blockchain-based distributed ledger would make it possible to validate the records of holographic telep- +resence and other XR applications in the metaverse and track the origin of inaccurate data. As a result, a more precise +recommendation system will be created. The zero-trust mechanism and cross-chain technology of the blockchain will make +it simpler for holographic telepresence and other XR applications to safely transmit data between virtual worlds [133]. Data +integrity is guaranteed for XR apps and holographic telepresence by the interplanetary file system offered by the blockchain. The +consensus technique used by these devices will make the data they gather and store on a blockchain unchangeable. Blockchain +supports confidence among AR/VR stakeholders by facilitating transparent ownership transfer and asset verification [161]. +V. FUTURE VISIONS AND DIRECTIONS +A. Content-centric Metaverse +With the ever-growing increase in using MaaS platforms, UGC is expected to be increasingly generated and transmitted +through the metaverse networks. Current IP-based host oriented content transmission protocols will face critical challenges for +securing UGC dissemination by heterogeneous end devices over the metaverse platforms. To address this challenge, content +centric networking (CCN) can be employed to rethink the current Internet architecture. According to CCN, contents will +be routed directly by their naming information instead of IP addresses [162]. For data and content sharing in CCN-based +metaverse, users request the desired UGC via sending an interest message to any CCN based node that occupies the matched +content. The main idea for securing CCN is to directly safeguard the security of every single content/data itself, rather than +securing the “pipe” or the communication channels/links [163]. In this way, flexible and content-centric secure metaverse can +be realized. Due to the inherent attributes of the CCN architectures, CCN-based metaverse can explicitly cause new security +concerns as well. For instance, content poisoning and network monitoring would be two security issues in the CCN-based +metaverse. Specifically, malicious users might inject poisoned UGCs, resulting in delayed or failed valid UGC delivery, e.g., +through flooding. A curious CCN node might observe the sensitive content disseminated by CCN users by directly monitoring +the network traffic. Therefore, further research on privacy and security protection for CCN-based metaverse is required [164]. +B. Edge computing +Hospitals, production lines, residential complexes, offshore equipment and game centers have different requirements. Hence, +solutions based on edge computing in the Metaverse should be adapted for different applications. This work requires a +methodical, repeatable and well-reasoned approach from the Metaverse operators. Otherwise, the cost of maintaining and +managing edge servers will increase and many problems will arise for users and the network. At the same time, the variety of +services and flexibility in service level agreements should still exist so that users have a better user experience. +For various applications, edge computing can be used to increase reliability, reduce latency, and offload tasks in 5G networks +[165]. More interestingly, edge computing is one of the main enablers of the 6G network because it can be used for reliable +low-latency communications, AI-empowered capabilities, and increasing energy efficiency [166]. In addition, edge computing +plays a vital role in the new generations of the Internet of Things [167]. Hence, we expect edge computing to progress along +with related technologies and add new capabilities to the metaverse. +In designing edge computing solutions, various tradeoffs must be considered. Processing power, storage volume, telecommu- +nication link bandwidth, spare parts, emergency power, security systems and many other things must be selected depending on +the needs of users. In addition, it is possible that the needs of users change over time. A scalable and flexible design approach +can help with this. In this type of design, it is possible to increase and decrease each of these facilities, depending on the +conditions. For example, it is not necessary for the Metaverse operator to have installed a lot of storage space on the edge + +sites from the beginning. However, it should have provided the possibility of increasing storage space on the edge servers. +This would enable the network to meet this critical requirement in the shortest possible time with the increase in users’ needs. +C. Blockchain’s still unsolved challenges +Despite many challenges which were discussed in this paper for MaaS developers and the solutions that blockchain technology +can provide, there are still more constraints that should be taken into consideration. There is still work to be done on creating a +robust blockchain for the metaverse to overcome unsolved problems. In the following, the authors mention some other unsolved +challenges and make some recommendations as well. +• Blockchain can be slow because of its complexity and distributed nature. +• On a blockchain, transactions can take a very long time to execute. +• Data in the metaverse will be more able to withstand copying and manipulation with the help of a consensus-based +distributed ledger, but since any new data must be duplicated throughout the entire chain, more study is needed to +overcome the latency problem. +• The number of blocks must grow along with the number of users in the metaverse, requiring the employment of enormous +computational resources [168]. As a result, users will pay a higher transaction cost for the verification of shared transactions. +For effective data sharing in the metaverse, next-generation blockchains must overcome this problem [169]. +• The existence of numerous public blockchains in various virtual reality environments that do not speak the same language +presents the biggest obstacle to cross-blockchain enabled the metaverse interoperability. It will be challenging to adjust +because different platforms will offer different degrees of smart contract capabilities. Furthermore, these virtual worlds +use a wide range of transaction architectures and consensus mechanisms, which limits interoperability [170]. +• If a small number of miners control the majority of the network’s overall mining hash rate, blockchains are susceptible. +• Due to the anonymity offered by blockchain technology, it is challenging to track down all IoT transactions involving +illicit services in the metaverse. +• To carry out the metaverse’s expansion, the blockchain needs to be regularized. +• For blockchain to be successfully used in digital twin applications in the metaverse, challenges like standardization, +privacy, and scalability must all be resolved. +• The quality of digital twins in the metaverse will increase as a result of the integration of blockchain, XAI, and federated +learning methodologies. +VI. CONCLUSIONS +This article provided an overview on privacy and security aspects of the metaverse, from different perspectives, including +the wireless access, learning algorithms, data access, and human-centric interactions. New directions towards realizing privacy- +aware and secure metaverse-as-a-service (MaaS) platforms were addressed, and less-investigated methods were reviewed to +help mobile network operators and service providers facilitate the realization of secure and private MaaS through different +layers of the metaverse, ranging from the access layer to privatizing the social interactions among clients. Additionally, edge +computing, which is one of the key enablers of the metaverse, has been discussed, along with the advantages and challenges +associated with its use in the metaverse. Later in this work, a comprehensive investigation and analyses of challenges for MaaS +developers and the blockchain’s solutions for MaaS platforms were provided. 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Sandikapura, “Improving data security, interoperability, and veracity using blockchain for one data governance, case study of local +tax big data,” in 2019 International Conference on ICT for Smart Society (ICISS), vol. 7, pp. 1–6, IEEE, 2019. + diff --git a/2dAzT4oBgHgl3EQfRvud/content/tmp_files/load_file.txt b/2dAzT4oBgHgl3EQfRvud/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2395825f84548fbff4026df9ea4f6f9483d7ab32 --- /dev/null +++ b/2dAzT4oBgHgl3EQfRvud/content/tmp_files/load_file.txt @@ -0,0 +1,2055 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf,len=2054 +page_content='Unlocking Metaverse-as-a-Service The three pillars to watch: Privacy and Security, Edge Computing, and Blockchain Vesal Ahsani Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran vesal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='ahsani@ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='edu Ali Rahimi Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='rahimi@ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='edu Mehdi Letafati Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran mletafati@ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='edu Babak Hossein Khalaj Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran khalaj@sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='edu Abstract In this article, the authors provide a comprehensive overview on three core pillars of metaverse-as-a-service (MaaS) platforms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' privacy and security, edge computing, and blockchain technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The article starts by investigating security aspects for the wireless access to the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Then it goes through the privacy and security issues inside the metaverse from data-centric, learning-centric, and human-centric points-of-view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The authors address private and secure mechanisms for privatizing sensitive data attributes and securing machine learning algorithms running in a distributed manner within the metaverse platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Novel visions and less-investigated methods are reviewed to help mobile network operators and metaverse service providers facilitate the realization of secure and private MaaS through different layers of the metaverse, ranging from the access layer to the social interactions among clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Later in the article, it has been explained how the paradigm of edge computing can strengthen different aspects of the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Along with that, the challenges of using edge computing in the metaverse have been comprehensively investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Additionally, the paper has comprehensively investigated and analyzed 10 main challenges of MaaS platforms and thoroughly discussed how blockchain technology provides solutions for these constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' At the final, future vision and directions, such as content-centric security and zero-trust metaverse, some blockchain’s unsolved challenges are also discussed to bring further insights for the network designers in the metaverse era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Index Terms Metaverse-as-a-service (MaaS), privacy and security, edge computing, blockchain I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' INTRODUCTION What is the metaverse, exactly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The metaverse is a concept in the tech world that refers to a digital living environment where conventional social structures are changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' It is a term that combines the concepts of the Greek1 prefix “meta,” which means “more complete” or “transcending,” and the acronym “Verse” for “universe,” which signifies a space-and-time container2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The idea of the metaverse was introduced in Neal Stephenson’s science fiction novel Snow Crash nearly 30 years ago [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The rapid advancements of technologies like blockchain, virtual and augmented reality, gaming, artificial intelligence, and the Internet of Things have made the metaverse one of the most buzzworthy terms in the tech world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Solutions and services are being developed for virtual worlds to allow users to have fun, intelligently engage with their surroundings, and form deeper connections with others [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Investment in the metaverse has grown significantly, with technology giants investing billions of dollars in its development and many businesses putting together their own plans for the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' A McKinsey & Company report predicts that the metaverse will be valued at over $5 trillion by 2030 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' What is Metaverse-as-a-Service (MaaS)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The phrase “as-a-Service” originally appeared in a 1985 file with the United States Patent and Trademark Office (USPTO), and it gained popularity throughout the cloud computing era [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Everything that may be considered as a service through a network can be referred to as XaaS [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Everything-as-a-service (XaaS) is a recent development in the information and communication technology sector that enables the provision of scalable computing resources on demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Accordingly, the Metaverse can profit from “as-a-service” models, in which the key elements and technologies of the Metaverse, such as platforms, infrastructures, software, and artificial intelligence (AI), could be provided as service models, 1https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='org/wiki/Meta 2https://alldimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='fandom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='com/wiki/Category:Verse arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='01221v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='CR] 1 Jan 2023 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=', Metaverse-as-a-service (MaaS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' As tech giants entering the Metaverse space, including Microsoft, Samsung, NVIDIA, and others, it won’t be long until there are many marketplaces with a Metaverse-as-a-Service (MaaS) offering allowing businesses to profit from the technology with lowered entry barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Even though MaaS is still a relatively new technology area, there are currently a number of vendors making headway there;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' such as Lovelace World, Propel MaaS, Touchcast, and MetaVerse Books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' MaaS is characterized as an on-demand subscription solution that enables companies and/or operators to create and implement different forms (such as existence, management, coordination, and implementation) in the Metaverse to support the processing of Metaverse services, collaboration, company operations, and products, among other related scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Everything in Metaverse can be thought of as a delivery model that can be simply generated and/or modified as function modules, similar to XaaS in cloud computing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' See Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The main benefits of using MaaS can be summarized as follows: Products for the Metaverse may be created by businesses without substantial digital experience or knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Without prohibitive capital requirements, even small to mid-sized firms can engage in the Metaverse economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' It promotes financial investment in a still-developing technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The majority of systems are currently only designed for consumer usage, and solutions like Microsoft Mesh and the Horizons app suite from Meta have not yet been widely released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In this setting, MaaS enables businesses to make low-risk investments and profit from technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This model also reduces the time of programming, setting up and installing systems and brings the investor a profit sooner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' MaaS may ultimately lead to industry standardization, with selected few businesses serving as Metaverse “brokers” to aid in infrastructure creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' MaaS model is a win-win situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The seller of this service does research, programming and implementation only once, but can sell this service many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' On the other hand, the buyer of the service can also use Metaverse at a competitive cost without getting into the technical, management and implementation complexities of Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Physical world Perception Layer Users IoT, sensors, … Devices Human-computer interaction AR/VR/… Holography comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Service providers Edge computing layer Edge servers Computing Storage Communication Shared, Scalable 5G/6G Network Core Cloud infrastructure Semantic comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Virtual world Metaverse service models AIaaS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' NSaaS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' BCaaS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' … SaaS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' IaaS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' PaaS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' DaaS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' … Application layer Avatars Digital twins Virtual environment Metaverse engine Security & privacy Modeling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' simulation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' … Human-centric Learning-centric Data-centric Real-time Interoperable Management Virtualization Machine learning Resource integration AI controller Metaverse service instantiation resource pool Distributed management Blockchain Consensus Smart contracts Trading Data management Policies Rules,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' SLAs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' … Regulation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Structure and important modules in the Metaverse network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Despite the outstanding advances in today’s existing MaaS platforms, the Metaverse would still need a more comprehensive and robust collection of standards and protocols to embrace interoperability, much more comprehensively than what the current internet includes, based on a set of rules and policies for communication, visuals, graphical demonstration, and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For instance, Fortnite runs on practically all popular platforms (such as iOS, Android, PlayStation, and Xbox) and supports numerous identity/account systems and payment options, which forces rivals to cooperate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=', engage in interoperability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Web2 goliaths like Apple and Google employ similar technology today, but they are not built to integrate with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Building a scalable Metaverse will depend more on interoperability than anything else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Companies might create their own Metaverse campuses using established protocols with the support of Metaverse-as-a-Service (MaaS), and then start providing immersive experiences that promote social interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The metaverse infrastructure should be supported by extensive accessibility to mediate various aspects of human-beings’ lives, to be able to provide immersive experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This opens up new venues for privacy and security risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The Internet-of-Things (IoT) plays an important role in digitalizing the physical world by utilizing pervasive sensors, cameras, wearables, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Networking connectivity is then provided via wireless networks, while the computing and storage are provisioned through cloud and edge computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The IoT networks act as “bridges” between the physical world and its digital counterpart [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The information flow between the two worlds, can help facilitate the decision making in both physical and digital worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Users, who are mainly represented as avatars, can also produce and exchange digital contents across various platforms in the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Mobile users interact with the digital world via their smartphones, wearable devices, and augmented reality/virtual reality (AR/VR) helmets, to create and share contents and gain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Information is the core resource of the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The data flow within the employed networks of metaverse has the key role in realizing the integration of physical and digital worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' To better understand the security and privacy needs in the metaverse, we indicate that there exist two main sources of information in this era: The first one is the data gathered from and exchanged by the real world that might be utilized and visualized digitally in the virtual space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The second source of information is the output of the virtual worlds, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=', the information generated by digital assets and services in a MaaS platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' At the same time, artificial intelligence and machine learning (AI/ML) algorithms, performed in the computation layer or the digital twin layer, help facilitate rendering and offering various services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Accordingly, it is crucial to safeguard the privacy and security of data flows within the MaaS platforms, as well as the learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' On the other hand, performing all metaverse computations on cloud servers is not necessarily the best option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In some situations, using edge servers close to users can facilitate low-latency services, reduce network traffic, help manage data, and improve user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In addition, the paradigm of edge computing is closely related to other technologies such as artificial intelligence and IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For example, with the help of edge servers, it is possible to train neural networks locally with the participation of end devices without having any internet connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Also, edge servers enable secure management and access control of IoT devices and sensors on-premise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Last but not least, blockchain is regarded as one of the metaverse’s core infrastructures and helps supply the metaverse with laws that are clear, open, effective, and trustworthy [7] because it can connect disparate minor sectors and create a solid economic system,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' It will be challenging to determine the worth of the commodities and resources traded in the metaverse without the assistance of blockchain technology, especially when those digital components interact economically with the real-world economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Therefore, it would be wise to investigate blockchain technology for MaaS platforms along with the other two pillars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' privacy and security and edge computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In the following, we provide a comprehensive review on guidelines to safeguard the privacy and security of MaaS platforms from different perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Additionally, in order to help metaverse operators to identify an appropriate approach for using edge computing in the metaverse, we have focused on the advantages and challenges of using the edge computing paradigm in MaaS in two separate subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Moreover, we thoroughly discuss the requirments of using blockchain technology and the actions MaaS developers should take to solve many technical challeneges by using blockcahin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Finally, we conclude the paper with future visions and directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' PRIVACY AND SECURITY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' An Overview of Privacy and Security Challenges for the metaverse Despite the ever-increasing advances in developing numerous services for the metaverse, privacy and security challenges are considered as the main concerns that need to be properly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Considering the fact that realizing the concept of MaaS requires an extremely wide variety of computation, communication, and networking, a wide range of security and privacy threats also arise in the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' See Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' A variety of recent technologies are integrated into the metaverse as its basis, hence, the intrinsic vulnerabilities of them may be inherited by the corresponding MaaS platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Recently, numerous privacy and security flaws have been identified for the emerging technologies, including the vulnerabilities of cryptography-based key management schemes against quantum computers [8], [9], the privacy leakage of distributed learning networks against honest-but-curious servers or malicious clients [10], and the misuse of massive private data by service providers [11], just to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Such threats can be intensified in the virtual world, while simultaneously other threats can also happen, which does not exist in the physical world, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=', virtual spying [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Notably, massive amount of sensitive data are utilized to create a digital replica of the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This opens up new challenges in terms of threats to privacy and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Individuals use wearables or AR/VR devices with built-in sensors collecting biometric data, brain patterns, users’ speech and facial expressions and poses, and also the surrounding environment in order to render a high-quality immersive experience for the MaaS users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' clients are under the threat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Privacy and security ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='challenges for the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Metaverse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Data fusion of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='multimodal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='private data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Distinguishing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='the realness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Highly ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='heterogeneous ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='virtual worlds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Securing the 6G- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='enabled access to the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Metaverse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Secure wireless ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='secret key ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='PHY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='authentication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='for the metaverse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='access ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Adversarial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='learning for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='wireless access ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Data-centric privacy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='and security ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Data collection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Incognito mode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Authenticating ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='synthetic data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Secure and private ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='AI/ML for the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Metaverse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Privacy for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='AI/ML ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='algorithms in the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Metaverse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Secure AI/ML ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='for the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Metaverse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Secure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='aggregation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='framework ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Human-centric privacy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='and security: privatized ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='social interactions in the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Metaverse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Social ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='applications ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Biometric data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Privacy- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='enhancing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='protocols ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Privacy and security for the Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' of being illegally tracked through their VR headsets or wearables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Such a miscellaneous set of sensitive information flow in the context of metaverse services provides adversaries with a broad attacking surface which requires proposing novel secrecy- and privacy-preserving mechanisms for mobile network operators and service providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The main privacy and security challenges in the era of metaverse can be identified as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Data fusion of multimodal private data corresponding to interactions between users, avatars, and environments imposes serious challenges for the MaaS platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Collection, communication, and processing of such sensitive information must be taken into consideration in terms of privacy and security risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The blurring boundary between the real world and its virtual counterpart causes serious confusions in terms of distin- guishing the realness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Therefore, we should focus on providing stringent authenticity guarantees from the MaaS providers’ perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Virtual worlds in the metaverse are highly heterogeneous in terms of hardware implementation, communication, and learning interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The beyond-the-fifth and the sixth generation (B5G/6G) of wireless communication networks, as the main fabric to facilitate the communications within the metaverse, pose critical challenges for conventional security solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This raises urgent needs for proposing context-aware and flexible security and privacy mechanisms [9], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' We aim to carefully review these items in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' We provide readers with useful insights through a comprehensive overview of security solutions across different layers of the MaaS platforms, ranging from the access layer to the social aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' We also review different solutions in the contexts of communication, computation, and learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Securing the 6G-enabled Access to the Metaverse For the deployment of access layer to the metaverse, the sixth generation (6G) of wireless networking will play a pivotal role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' 6G technologies, such as pencil-sharp beams, edge AI, and integrated sensing and communications (ISAC), provides users with perceived understanding of their surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' These technologies are the key enablers for implementing novel PHY layer security (PLS) schemes as promising approaches to safeguard the security of the metaverse access, while offering information-theoretic security guarantees [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Leveraging PLS, intelligent, flexible, and context-aware mechanisms are feasible for detection and mitigation of privacy and security vulnerabilities [9], [13], [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Considering the new era of truly end-to- end (E2E) quality-of-experience provisioning for the MaaS products, service level agreements (SLAs) are expected to include guarantees about the quality of security (QoSec) as well [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This will include addressing the required security level, adaptive, risk-aware, and adjustable security solutions, just to name a few [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The evolution of the metaverse systems is expected to introduce new means to harvest and interpret the “context” of the communication, where PLS techniques can be considered as a promising QoSec-assuring approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In order to secure the metaverse access, we introduce the following security mechanisms, which can provide lightweight and scalable solutions for securing the access to the MaaS platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' 1) Wireless Secret Key Generation and the Role of PHY Security: Smart wearable devices and cameras, together with head-mounted displays, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=', Oculus helmet and Vive Pro headsets, and handheld controllers are considered as the main terminals for entering the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In this regard, key generation and management is essentially required for smart devices to establish secure connections for accessing the metaverse, transmit sensory data, and receive immersive experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In [17] and [18], intrinsic features of specific wearable devices, such as gestures and motions, are taken into account to propose key agreement schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Researchers in [19] propose a heartbeat-based key generation scheme based on the measurements from electrocardiography (ECG) and photoplethysmography (PPG) sensors as the source of common randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' While the security mechanisms of the 5G standard rely on cryptography-based keys, such as the elliptic curve cryptography (ECC) to fulfill the confidentiality and authentication requirements, 6G networks with many peer-to-peer communications undermine the performance of conventional solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Besides, assumptions on the security of cryptographic methods based on the mathematical hardness of solving a certain problem will no longer hold with the advent of quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' To efficiently secure the metaverse era, data communications should to be proactively secured, where PLS has such capabilities for quantum-resilient security [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' As a promising framework to migrate from the conventional complexity-based solutions towards lightweight security techniques, PHY layer secret key generation (PHY-SKG) has been envisioned to be employed for 6G networks [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' PHY-SKG is realized through implementing lightweight mechanisms with minimum required changes at the control plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This can actually provide the access networks with different advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In particular, PHY- SKG is thoroughly decentralized without relying on any infrastructure of a particular entity, which can substantially reduce the required time for key agreement, making it suitable for extremely low latency applications in the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Moreover, continuous update of the key is realizable thanks to the dynamic variations of the PHY attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The PHY-SKG protocol under realistic assumptions of hardware impairments and adversarial attacks is developed in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Notably, the PHY-SKG has shown to be resilient against man-in-the-middle adversarial attacks in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Furthermore, for establishing secure communication between digital healthcare devices and the access point, a lightweight learning-based key agreement protocol is proposed by the authors in [20], which can guide designers to bring device-level intelligence for securing the metaverse access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' 2) Adversarial Learning for Wireless Access: Due to the proliferation of virtualized services in the metaverse era, malicious actors may have access to the network functions, such as the open-source software of virtualized access networks, and can exploit vulnerabilities of the metaverse access through adversarial machine learning (AML) [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Adversaries can poison the wireless access of the metaverse, by manipulating the inputs to the learning algorithms employed for authorizing users [22], leading to unauthorized access to the metaverse network due to the mislead access model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Malicious users can also infer sensitive information about edge devices, users, and applications which have access to the network, through membership attribute inference attacks [23] and model inversion attacks [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' As a privacy threat to the MaaS access, data characteristics such as device-level information may leak to adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Malicious users can exploit this leaked information using membership inference attacks [23] by building an inference model to determine whether a sample of interest (associated with a particular device) has been used in the training data of the MaaS provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' To mitigate such attacks, generative adversarial networks (GANs) are shown to be capable of detecting anomalies and mitigating wireless attacks for the next generation of communication and networking services [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Connection requests for the metaverse services come with quality of experience (QoE) requirements such as throughput and latency, which can be fulfilled by assigning radio resource blocks and processing power to the requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Although network slicing can manage the MaaS QoE requirements, AML techniques have the potential to attack network resource management and disrupt B5G network slicing, which can impose huge losses for the network operators, if not properly secured [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' To combat such attacks, different reactive and proactive defense mechanisms are proposed in [26], including: the induction of randomness to the decision processes to reinforce robustness against malicious nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' So far, the security challenges for accessing the metaverse were addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In the following sections, we go deeper into the privacy and security aspects inside the metaverse, in a data-centric manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' We also address synthetic and authentic realness, envisioning the privacy enhancement within the metaverse platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Data-centric Privacy and Security Inside the metaverse, the scope of data collection and processing far exceed what is traditionally performed in mobile and web applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The technologies employed for the metaverse do not just track where users click, but where they go, what they do, whom they interact with, what they look at, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Therefore, it is crucial to the users’ privacy and security to implement data-centric privacy-preserving mechanisms and push for meaningful, sensible, and aggressive regulations in terms of data privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Recent studies demonstrate how easily the metaverse users’ privacy can be compromised [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' It is shown that an adversarial program can accurately infer over 25 personal data attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' To deal with the unprecedented data-centric privacy risks in the metaverse platforms, the idea of “incognito mode” has recently been developed for the metaverse as a novel and promising solution for safeguarding VR users’ privacy [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' To realize the incognito mode in the metaverse, client-level differential privacy framework is proposed which has been shown to be capable of protecting a wide variety of sensitive data attributes in VR applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' It is show in [28] that by employing randomized mechanisms (depending on the format of the private attributes), sensitive user data attributes can be obscured effectively [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' We are facing a new trend of intelligent systems that are empowered by synthetic realness, in which AI-generated data is vastly exploited to reflect various aspects of the physical world [29], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Inside the metaverse, consumers are not simply targeted via pop-up ads, rather, they are provided with “immersive contents” in the form of virtual people and activities that look realistic [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Nevertheless, we should concern about the authenticity of synthetic data at different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Authenticating synthetic data can make AI algorithms more fair and secure through correcting data biases and safeguarding data privacy [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Synthetic data is utilized for training AI models in ways that real-world data practically cannot or should not, while protecting privacy and confidentiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' It is critically important for the network operators and MetaaS providers to leverage generative AI in an authentic way to maintain trust for their customers [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' To realize authentic realness, the provenance, the policy, and the purpose of utilizing synthetic data should be taken into account by companies and service providers [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' To this end, distributed ledger technology (DLT) can be employed to verify the provenance of digital content, hence addressing the authenticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Moreover, network operators should clarify the purpose behind the exploitation of synthetic content and the resulting advantage over non-synthetic content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Secure and Private AI/ML for the Metaverse In this section, we focus on challenges and solutions for the privacy and security of AI/ML implementations in the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Although metaverse can bring amazing AI/ML-based services to individuals and enterprises, the utilized learning algorithms are still vulnerable to privacy and security risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' There exist challenging bottlenecks for efficient deployment of AI/ML techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Among these challenges, the big concern is that the learning algorithms might leak users’ private data [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Besides, users might not be willing to share their private data in the metaverses, making it challenging for AI/ML algorithms to perform a comprehensive data analysis for enhancing the learning-based services of the metaverse platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' To address the privacy risks of AI/ML algorithms, the integration of federated learning (FL) [36] and cross-chain technologies can be considered as a promising solution to design and implement privacy-aware frameworks for the MaaS platforms [37]–[39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For instance, [39] proposes a hierarchical blockchain-based framework for decentralized FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' A main chain is employed to mimic the parameter server (aggregator server) of the FL algorithm, and multiple subchains are implemented to manage local model updates generated by smart devices (or their digital counterpart) acting as workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Despite the fact that FL algorithms leverage locally-trained models instead of the raw data of metaverse participants, sensitive information can still be inferred by analyzing the model parameters uploaded by clients [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' If the model updates are inspected by bad actors, participant users’ privacy and security would be threatened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In addition, there might exist some malicious users, who proactively upload adversarial data to the aggregator server(s) to mislead the training process, known as backdoor attacks [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' FL still has limitations in terms of security and privacy: First, malicious clients may inject mislabeled or manipulated data to mislead the global model, which is known as data poisoning attacks [21], [40];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Second, during model update of AI/ML algorithms, adversarial participants can save the model parameters and infer sensitive information, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=', private attributes of participants’ data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' They might also be able to recover the original training samples of users [24], [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In the following, we discuss the possible countermeasures to the abovementioned threats in more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' We take into account the fundamental fact that local model updates in distributed AI/ML algorithms carry extensive information about the local datasets owned by the metaverse clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Hence, secure aggregation of local models are necessary for the distributed learning entities in the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Moreover, there might exist some adversarial participants in the metaverse platform who actively upload “poisoned data” to the servers (as aggregator entities) to mislead the training processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Such attacking venues can provide the surface for the so called “backdoor attacks” on the collaborative model training procedures within the MaaS platforms [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' As an example, virtual service providers and mobile network operators employ learning algorithms as a service for their users [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Malicious users have the capability to eavesdrop on the models exchanged to infer the membership of specific clients or undermine the performance of AI/ML services [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In addition, attackers can hack edge devices in the physical world or impersonate benign avatars as a man-in-the-middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' They can then inject adversarial training sample to spoof the learning services within the MaaS products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In the following, we address secure aggregation framework to protect learning tasks from being eavesdropped by malicious participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Secure Model Aggregation: Generally speaking, state-of-the-art secure aggregation (SA) protocols rely on two main principles: i) pairwise random-seed agreement between clients to generate “masks” for hiding the metaverse users’ models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' and ii) secret sharing of the random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' A SA protocol enables secure computation of distributed learning models, while ensuring that the aggregator entities do not infer information about the local models [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Taking into account the fact that the MaaS users might be “honest-but-curious” during the learning process, a SA protocol must guarantee that nothing can be learned beyond the aggregated learning model, even if up to T users cooperate with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This is called T-privacy property of SA algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In [42], a modular system design for SA is proposed, which can help develop lightweight SA protocols for the learning infrastructure of the MaaS platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Backdoor Attacks: Malicious clients may inject mislabeled or manipulated data to mislead the learning models within the highly-distributed and heterogeneous infrastructure of the metaverse platforms, which is known as data poisoning attacks [21], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In this context, backdoor attack is a type of “data poisoning” attacks that aims to tamper a subset of training data via injecting adversarial triggers, such that AI/ML models trained on the manipulated dataset make (targeted) incorrect prediction/decisions during the inference [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Backdoor attack is aimed to mislead the trained model to infer a target label on any input data that has an adversarially- chosen embedded pattern (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='a trigger).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Distributed version of backdoor attacks, known as DBA, decomposes a global trigger pattern into separate local patterns and “embed” them into the training set of different adversarial parties respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This is in contrast to the conventional attacks on the surface of distributed AI/ML algorithms [43], where a unified global trigger pattern is embedded for all malicious parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The experiments in [44] show that the DBA method outperforms centralized attack significantly when evaluated with the global trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' To overcome backdoor attacks, adversarial training (AT) is currently considered as an empirically strong defense mechanism [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' AT corresponds to the mechanism of training a model on adversarial examples, with the aim of making the learning service more robust to attacks and reducing the inference error on benign samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' AT is comprehensively reviewed in a wide variety of researches, including [40], [45], [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Recently, a generic framework to defend against the general type of data poisoning attacks is proposed in [40] by desensitizing networks to the effects of poisoning attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' To mention an application of AT within the metaverse, an AT-based protection against facial recognition systems is proposed in [47], which can be effectively utilized for the MaaS platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Human-centric Privacy and Security: Privatized Social Interactions in the Metaverse In the metaverse, human users socially interact with others through their avatars and experience MaaS platforms by leveraging ubiquitous smart devices and network access in a real-time manner [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The metaverse as a “human-centric” virtual environment is supported by massive human-centric social applications, such as holography AR, immersive VR gaming, and the tactile Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Such immersive services can be realized through the interdisciplinary communication-computation-storage co-design techniques, integrated with the concepts of human perception, cognition, and physiology for haptic communication [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Hence, the most sensitive and personalized information will be exchanged among different parties, which significantly highlights the essence of employing novel intelligent privacy- and secrecy-preserving algorithms for the information security and social privacy of MaaS users [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The biometric data plays a significant role in terms of social interactions inside the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' To elaborate, human-driven agents and avatars, with whom the users interact, are being developed based on users’ personal data [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Such sensitive data can be fed into AI/ML algorithms to create personalized “interaction partners” and influence social interaction behaviors of the metaverse users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' By exploiting physical and mental traits inferred from biometric data, the interaction partners can maliciously lead users into undesired situations and behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Hence, the metaverse users can be manipulated not only by businesses and institutions, but by other individuals in a human-centric manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' To mitigate such human-centric threats, attention should be paid to the users themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Metaverse service providers should make users aware of the implications of biometric data extracted by AR/VR devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Developers can implement privacy- enhancing protocols such as differentially private mechanisms as proposed in [52] and [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Users can also be provided with secondary avatars, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' clone, to hide their actions in the metaverse [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The secondary avatars help users hide their real behavior within the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Furthermore, users must be able to have an adjustable level of privacy [28] to dynamically manage their personal space in the virtual world by choosing their desired privacy configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' From a governance layer, a modular bottom-up governance approach is proposed in [54], which can be adapted to different platforms and use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Decentralized autonomous organizations (DAOs) are examples of governance systems that allow users to be involved in decision-making processes for the MaaS platforms [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In this regard, organizations such as XR Safety Initiative (XRSI)3 will play an important role in encouraging MaaS providers and network operators to help facilitate enhancing trust within the MaaS implementations in terms of human-centric privacy and regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' EDGE COMPUTING To unlock the full potential of the metaverse, edge computing represents a promising paradigm for technical development of the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Edge computing is a well-known technology that allows operators to perform processing, storage, and network functions close to the edge of the network, where users are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' By using this paradigm in the Metaverse, network efficiency is improved, costs are reduced, and new features are added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' As an illustrative example, consider a smart hospital in the Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In this center, telemedicine facilities are implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Artificial intelligence is used to diagnose diseases and suggest treatment, hospitalized patients are monitored in real time with IoT sensors, and remote surgery can be done in this center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In this scenario, if all this data is to be sent directly to the 3https://xrsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='org cloud server for processing and storage in a traditional way significant network bandwidth will be occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In addition, with this method, there will be a long delay in sending data, and it is possible that the confidentiality of patients’ data will be compromised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This motivates the need for the edge computing paradigm that can process and store data near the edge of the network where data is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In short, this design can be such that several servers are installed in the hospital that can quickly collect and process large amounts of data, feed it to neural network models, then send the extracted information to the cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Edge computing can be a lucrative opportunity for network operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' One-third of network operators’ costs are spent on real estate to house infrastructure [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Converting these locations into edge computing sites is a huge revenue potential for network operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' They can also use their extra processing and storage resources to empower network edge fascilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' On the other hand, operators can have a successful entry in the development of metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' They can play key roles by using the services they provide in the 5G and 6G mobile networks [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Consequently, providing MaaS is a smart way to boost operator income and develop diverse applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In the MaaS model, organizations can flexibly create their own virtual world according to their needs and with the most recent versions of the tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This is made possible by using operator edge facilities and networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In addition to the network operator that provides a lot of hardware and software infrastructure, vendors such as application developers, edge computing equipment vendors, system integrators, and edge computing solution providers are also active in this market and new revenue lines are open to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Furthermore, various vertical markets such as automotive vehicles, virtual content productions, robotics, eHealth, entertainment, smart manufacturing, smart grids, and many others are added to this market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For a survey on the edge computing market, see [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' An edge computing system includes the internet gateway, edge servers, and perception layer [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Internet gateway is responsible for connecting the edge system with the network backbone and cloud servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Edge servers process, store, and aggregate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Perception layer includes edge devices such as IoT sensors and actuators, AR glasses, VR headsets, haptic gloves, smartphones, wearable devices, personal computers, and holographic displays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The metaverse can benefit from edge computing in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' See Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In the following, we explain some of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In addition, edge computing also brings challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' After explaining the advantages, we explain the challenges of using edge computing in the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' It should be noted that we provide a high-level review of these items and delving into their technical details is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Latency reduction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Physics emulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Graphic rendering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Real-time straming ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Performance optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Resolving computation bottlenecks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Network traffic control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Higher variety of services ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Data control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Access control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Scalability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Ubiquitous access ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Capacity enhancement: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Network traffic reduction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Free up cloud resources ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Orchestrate resource sharing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Computation offloading ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Enrichment of sensors and actuators ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Expanding into new markets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Reducing energy consumption ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Privacy preserving ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Strengthen data ownership and trust ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Leveraging encryption ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Training neural networks on local data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Monitoring & management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Edge-assisted communication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Dynamic efficient network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Remote management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Reliability & availability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Tactile Internet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Hazardous environments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Anomaly detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Edge intelligence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Edge computing for AI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Edge AI for network management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Distributed machine learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Cost reduction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Profitable edge computing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Local solutions for local problems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Resource sharing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Modern networking techniques ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Semantic communication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Virtualization of real-world edge networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Taking advantage of multi-user scenarios ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Metaverse requirements to be met by edge computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Metaverse requirements to be met by edge computing Many applications in the metaverse need seamless synchronization of the real and virtual worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Some of these applications are critical, such as remote control of a robotic assisted surgery system, and some of them, such as guiding an avatar in a virtual environment, are related to the quality of the user experience (QoE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Therefore, it is necessary to provide a metaverse infrastructure to provide low-latency services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In traditional solutions, processing and storage are outsourced to cloud servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' These servers can easily be located thousands of kilometers away from users, and this distance causes a lot of delays in telecommunications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Edge computing, as an alternative solution, brings processing and storage closer to users and the place where data is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Hence, in the literature, some metaverse processes that are very apt to run on edge servers and edge devices such as physics emulation [60], [61], graphic rendering [62]–[65], and real-time streaming protocols [66] have been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' When edge servers are set up near users, it does not mean that the operator transfers some calculations from the cloud server to the edge server forever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Instead, it flexibly uses edge facilities to optimize network performance depending on different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In other words, edge computing and edge storage capabilities add new dimensions to network performance optimization problems and enlarge their feasible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Therefore, the network has several more degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The operator can dynamically decide, depending on the situation, how the edge servers come into play and bring advantages to the metaverse network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For example, edge servers schedule computing tasks [67], provide better coverage [68], run load- balancing algorithms [69], resolve computation bottlenecks [70], and provide heterogeneous services well addressed in the MEC (Multi-access Edge Computing) standard [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' To make the metaverse network scalable, it is necessary to support issue tracking, data quality management, data protection, and risk control functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' On the one hand, we know that the metaverse uses distributed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' It also captures sensitive data such as biometric data, eye movements, and financial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In addition, the metaverse interacts with a wide range of technologies from blockchain to artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For this reason, the impact of data control functions in the metaverse is significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' On the other hand, we know that edge computing enables better and faster data control by keeping data in the local network and close to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Edge computing makes it easier for organizations to control data access to a universal metaverse [60] whilst manages data exchange in various ways such as W-LAN, cellular networks, and satellite communication systems which are supported by 5G and 6G [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' It is also possible to support distributed computing and use the processing and storage resources of end-devices, which can help network scalability [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Furthermore, small devices may not be able to run many heavy tasks in the metaverse like the inference of large neural networks and real-time renderings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In this case, they should be able to outsource these tasks to powerful servers or perform the tasks with the help of other devices in a distributed scheme [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This reduces telecommunications and thus reduces energy consumption [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Edge servers can provide more performance, flexibility, and storage than edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In addition, it is usually easier to upgrade edge servers than end devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This is because we can equip the edge server with more GPUs, RAM cards, and storage devices, or change the previous components to increase the power of the edge server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' But we may not be able to disassemble all edge devices, including IoT modules, and change their features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' So far, a lot of research has been done on the outsourcing of computing of IoT devices to edge servers, in which its advantages, challenges, architectures, and various approaches have been explained [75], [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Further, some IoT devices have limited batteries, and energy costs can be high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In any case, optimizing energy consumption in the Metaverse network is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' A practical case study is presented in [77] that explains the impact of edge computing on energy consumption in IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Even with modest Metaverse assumptions, data usage could easily expand more than 20x during this decade [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' If edge servers perform storage and processing tasks for users, there is no need to send large amounts of raw data deep into the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For this reason, the telecommunication capacity of the network is less occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Hence, edge computing can replace cloud computing at a lower cost in many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Additionally, since there are a lot of powerful edge devices in the metaverse network, it is possible to encourage resource sharing among edge computing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Thus, edge devices can serve as microcenters for storage and processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In this case, the capacity of the metaverse network will increase without occupying the resources of the cloud server [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The authors in [80] have presented an efficient method for sharing resources at the edge of the network in the framework of the resource pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For a survey of incentive mechanisms for sharing resources at the network edge, see [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Privacy protection is another challenge in the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Storing data on a local edge server (for example, at a university, hospital, or airplane) can help keep users’ data confidential [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Edge servers can monitor the passage of sensitive data and be equipped with hardware and software protection tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Many IoT sensors and devices are not smart enough to not be compromised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Sometimes their communication is not secure enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Having an edge server near these devices and possibly owned by the user can maintain confidentiality and bring more trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Users connected to an edge server can use local area network policies such as device and file sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' On the other hand, authentication mechanisms can be implemented on edge servers to secure the network against various types of hacker attacks [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Edge computing can come to the aid of end devices and participate in cryptographic algorithm calculations [84] and training neural networks on local data [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Edge servers make the network more self-aware and easy to manage and monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Spending on creating sites for edge servers can be multi-purpose and simultaneously address several different needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Edge servers provide monitoring of local network information and its devices and help with higher-level configuration and customization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Edge computing is a key technology in the achievement of 5G goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' It is also one of the enablers of 6G networks [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' It enables distributed scenarios and low-latency telecommunications which can help improve network traffic and be part of upcoming solutions in telecommunication issues such as channel selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Furthermore, to choose the most appropriate coding and communication protocol, edge servers can send network conditions to end devices and vice versa [87], [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Edge computing tools help end devices in many ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In addition to performing calculations, storage, caching, prefetching, and data traffic prediction of end devices, they can improve network efficiency and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For example, they can have different communication channels and direct the information packets of connected devices from the most efficient path [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Additionally, solutions have been offered to avoid the need for physical presence to calibrate end devices with the help of edge computing [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Environmental threats such as temperature rise, humidity, dust, and power outages are just as serious as cyber threats and can put sensitive edge devices out of service [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The presence of processing and storage facilities at edge computing sites enables edge remote management solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' With the help of these solutions, we can take care of the critical facilities of the Metaverse network at the edge and reduce the energy required for troubleshooting and repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Additionally, we want the metaverse network to be more reliable and available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Although there is no universal definition of reliability and availability, and there is no standard way to measure them, we can talk about strengthening the reliability and availability of some services compared to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In some applications that are offered under MaaS, these two features are prominent and should be considered in design, programming, and warranties given in agreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In the event of a network failure, edge computing infrastructure can still maintain the connection of the devices and perform some processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Additionally, it is possible to add redundancy to the system at edge computing sites for specific applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' By bringing the processing tasks to the edge servers, the connection of the devices to the core network is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This reduces the level of vulnerability of the system to cyber threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In [92], to increase the reliability of the Metaverse network, the authors have presented a method based on distributed coded computing algorithms that can be implemented at the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Also, a growing number of solutions are being offered for edge computing reliability in hazardous environments [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' System reliability is significantly affected by the detection and prediction of anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Artificial intelligence can train machine learning models on the large amount of local data that edge servers have access to and use them in real-time [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' 3D reconstruction, character animation, detection of harmful content, machine vision, gesture recognition, facial expressions, understanding emotions, body movement recognition, physical interactions, eye tracking, brain-computer interface, and many more are activated by artificial intelligence in the Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Some strategies for using machine learning to strengthen security in the Metaverse and to create recommendation systems are proposed in [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For various reasons, there is a rising desire to train and execute machine learning models on edge servers, which is called Edge AI or Edge Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Many applications in the metaverse such as autonomous vehicles, intelligent interaction of avatars with the environment in the virtual world, health care, smart city, and industrial complexes need machine learning algorithms [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Having computing servers on the edge equipped with artificial intelligence allows better network and telecommunication management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For example, in situations where we have many sensors in an environment and we don’t want their massive data to leave the local network raw, we can reduce the traffic volume with the help of an edge server that can process their data and reduce their size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In addition, in the edge server, algorithms can be implemented to predict traffic and cache popular data, which increases the service quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Today, machine learning has been used in different telecommunication layers, from the physical layer and coding of telecommunication signals [97] to the detection of cyberattacks in the application layer [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Designing distributed machine learning algorithms such as federated learning [99], split learning [100] and gossip learning [101] is a hot research topic because it adds new capabilities to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For example, with distributed learning, the neural network is trained without the data leaving the device and the processing load is spread across multiple devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In addition, in a local area network where Internet traffic is limited, devices can continue to train neural networks with the help of each other without having a permanent connection to the Internet [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' As previously mentioned, edge computing is not necessarily a substitute for cloud computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Rather, it is an option with its advantages and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The costs and desired quality of the various services provided in the metaverse should be examined and the appropriate way to provide them should be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In addition, attention should be paid to the interaction of cloud servers and edge servers and how to divide their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Due to the existence of various end devices and network equipment that can be used in the metaverse, the importance of this review increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Edge computing reduces network traffic load and reduces costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Additionally, it may be cheaper for some businesses to build edge servers than to outsource computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For example, in some regions, the cost of electricity makes it cost-effective to install edge servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Cost-effective solutions for edge servers are also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For example, the study [103] shows that micro data centers at the edge cost 42% less than traditional centralized data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In telecommunications, there are several techniques to reduce costs and increase capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Some of them are in the common domain of edge computing and the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Network designers can use these ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For example, semantic and goal-oriented communication is provided to increase stability and efficiency [104], which can be used in the 6G network and contribute to the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Edge servers can be equipped with various neural networks and perform semantic communication tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' On the other hand, the metaverse can help itself is by creating digital twins of edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' By monitoring the digital twins of edge devices and edge infrastructures such as edge servers, I/O ports, RIS (re-configurable intelligent surface), UAV (unmanned aerial vehicles), SAGIN (space-air-ground integrated network), and network equipment, the MaaS operator can instantly control and calibrate these physical entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Edge computing challenges An operator who wants to use the edge computing paradigm in MaaS design must consider several challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Limitations of edge servers is one of these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Due to the limitations of end devices (power, bandwidth, computing, storage, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' ), edge servers are proposed to assist end devices in tasks such as artificial intelligence processing and high-quality image rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' But it should be noted that the power of edge servers can also be limited due to hardware limitations or server room space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In this case, the edge server cannot provide all the desired functions for a large number of users at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Another challenge to be considered is the resource allocation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Users in a region will likely make many requests to the edge server at the same time and we will see network spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Managing these requests is a challenging issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Another issue is the management of faulty hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Cloud servers have more infrastructure compared to edge servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The reason why it is beneficial to have spare hardware on hand is so that in the event of a hardware failure, it is possible to replace it quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' But on edge servers, this issue is not economical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Suppose you own one device, and you buy a spare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The cost will double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' But suppose you have 100 pieces of equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' If you buy a spare, your cost will be 101 instead of 100, which is relatively not much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In addition, fairness is also essential in resource management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Because there is a lot of resource sharing in the services provided at the edge, the metaverse operator at the edge needs to be very careful about the optimal scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For more information on resource management at the edge, see [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Stragglers and malicious node are serious problems in outsourcing scenarios at the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' When a computing load is distributed among several servers or data is stored in several servers, it is possible that some of these servers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=', stragglers, will take more time than usual to respond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This issue causes our entire work to be delayed on those servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Malicious nodes in the network are entities that seek to disrupt the functioning of the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' There should be pre-prepared scenarios to deal with stragglers and malicious nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' There are various solutions to mitigate stragglers and malicious nodes in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For example, in coded computing algorithms, we can divide a task into several parts and outsource it to several servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' If some of these servers are stragglers or malignant, the computation’s final answer can still be recovered [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' There are also a couple of communication and networking challenges in using edge computing for the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Metaverse services have diverse and strict requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For example, touch Internet requires high-reliability telecommunications, AR/VR headsets require high bandwidth, and IoT devices require high coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' A large amount of data in the network and its dynamic nature should not cause friction and decrease the efficiency of the system in terms of reliability, delay, and rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Increasing the variety of services can make MaaS faster and more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For example, if different service modes and qualities are available, for example, the video size can be changed or artificial intelligence can be adjusted with different accuracy levels, some of the network challenges in overload conditions will be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The distributed and heterogeneous nature of the metaverse network at the edge complicates its management too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In addition, edge servers must be synchronized with end devices and cloud servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In addition, various functions must be executed simultaneously to maintain network performance, such as anomaly detection algorithms, resource provisioning, workload prediction, and traffic routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In places where edge servers are installed, such as a hospital, a network specialist is not always available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For this reason, another challenge of network management at the edge is the high cost of admin intervention for configuration, updates, and equipment maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Ownership considerations are another critical issue in this matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Equipment, data, and software on edge servers can have different owners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For example, in the case of equipment, some edge servers are privately owned, some are cooperatively owned, some are rented out to generate revenue, and some are created with specific access requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This issue can make it difficult for the Metaverse operator to enter into contracts and launch new services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' As the number of end devices, edge servers, and IoT modules increases, the metaverse network becomes challenged in various sectors like wireless communication, queue management, and user authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For example, in wireless communication, interference and congestion can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In queue management, there is a possibility of data being thrown away, and user authentication can take a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' We also know that edge servers hold sensitive data generated in places like hospitals, financial institutions, and homes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' But keeping data close to the user is not enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Users and businesses should be given access and data confidentiality guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For more information on edge privacy solutions, see [107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In some applications, it is necessary to know who is responsible for each operation at the edge of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' It should also be possible to receive complaints and prevent manipulation, corruption, and abuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Decision-making processes and functions should be expanded to the extent that the privacy and property rights of individuals are not compromised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The participation of different people and communities in monitoring the network not only improves the performance of edge servers in one place but also creates knowledge that can be used throughout the metaverse network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The needs of users are not the same in all regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In addition, the data formats available in the network and devices connected to the network are diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Therefore, the MaaS operator must design and install edge server equipment and software in a flexible and compatible structure and configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Another issue is changes and developments in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The network can change over time in terms of users or cloud servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Accordingly, edge equipment should be able to meet these requirements at a low cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' BLOCKCHAIN TECHNOLOGY To provide users with the best experiences possible, the metaverse gathers enormous amounts of private data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This information is required by the businesses or programs in order to construct targeted systems successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' With its authentication, access control, and consensus methods, blockchain gives consumers total control over their data, protecting their personal information [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The blockchain utilizes hash algorithms and asymmetric-key encryption to protect data in the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Additionally, the quality of the real-world data that users exchange is crucial to the construction of objects in the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This data is collected from numerous applications, including those in entertainment, healthcare, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Blockchain enables individuals and businesses to validate all transactions by providing comprehensive audit trails of all transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The metaverse’s data quality will improve as a result [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Blockchain solutions for MaaS Data acquisition Data storage Data sharing Data interoperability Data privacy preservation IoT Digital twins AI Big data Interactivity Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Blockchain solutions for MaaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' On the other hand, the successful exchange of AR and VR data is essential to Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The smooth and safe data sharing within the metaverse is made possible by the blockchain’s cutting-edge encoding information system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Moreover, stakeholders in the metaverse must have access to and control over resources in various virtual environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Due to the many settings in which these virtual worlds are created, data interchange is constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' A cross-chain protocol enables data interchange between two or more blockchains that are present in different virtual worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In terms of integrity, data in the metaverse must be constantly and accurately updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Due to the immutability offered by the blockchain, the metaverse data is kept as a copy in each block along the chain and cannot be changed or withdrawn without the agreement of a majority of the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' A number of vendors have already begun to enter the MaaS market;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' according to an announcement made by Propel, a blockchain solutions platform, it would provide MaaS solutions for smart contracts, NFT utilities, and decentralized finance (DeFi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Moreover, Lovelace, a different blockchain-based platform, offers a MaaS toolkit that gives users and developers the technology required to build and trade NFTs, run smart contracts, monetize VR gaming, interact with other metaverse platforms, and much more [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In the following, the authors concentrate on the steps that a MaaS developer should do and utilize blockchain technology to address various difficulties associated with creating and developing the Metaverse platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' See Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Open challenges for MaaS Data Acquisition: The metaverse will generate large amounts of unstructured, real-time data through decentralized appli- cations, but acquiring this data can be difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Building applications in the metaverse, such as recommender systems, will require high levels of data integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The use of virtual reality and increased streaming in the metaverse will further strain data acquisition systems [111], [112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The quality of the data may also be impacted by the acquisition of duplicative or incorrect data [113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Data Storage: Once the metaverse is fully functional, the physical world’s ability to store data may be strained to its breaking point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This could create significant challenges for the metaverse [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' If the metaverse relies on a central storage system, there is a risk of data leakage, manipulation, or loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The potential of the metaverse to offer biometric data, voice inflections, and vital signs that depend on sensitive data is also jeopardized by the likelihood of data loss and corruption in centralized applications [115], [116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Data Sharing: Data sharing on centralized platforms carries the risk of exposing private and sensitive information [117], [118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Additionally, due to data mutability, there is a risk of high latency and reduced data availability [119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This is particularly relevant in the metaverse, where many applications will generate large amounts of real-time data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' When demand for real-time data increases, data flexibility can become a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Data Interoperability: The metaverse will be created through the merger of many digital domains, but these domains are currently fragmented and disorganized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This can make it difficult for users to engage with multiple virtual worlds, as they must set up separate accounts, avatars, hardware, and payment infrastructure for each one [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' There are also few methods for users to transfer their digital assets between different digital environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In order for the metaverse to be truly interoperable, digital world apps must be able to easily exchange information with one another, regardless of their location or the technology being used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The conventional approach to interoperability is inadequate for the metaverse, so new solutions are needed [120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Data Privacy Preservation: In the early stages of the metaverse, attackers may be able to deceive users and steal important data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This could be particularly dangerous if attackers use artificial intelligence bots, as users may not be aware that they are not speaking with a real person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The metaverse also raises concerns about the confidentiality of personal data, particularly personally identifiable information (PII) [121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Finally, as the metaverse grows and more validity information is included, managing the large amounts of data will become increasingly difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' IoT: The metaverse will have a large number of interconnected IoT sensors, which raises concerns about IoT security and storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Real-time analysis of unstructured IoT data is also challenging [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' When storing data across virtual worlds, a centralized solution is not ideal, as tampering with even one piece of data could compromise the entire set of findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Additionally, data sharing across virtual worlds will depend on the cross-platform capabilities of IoT devices [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Finally, IoT data tracking is necessary for safety and legal compliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Digital Twins: The quality of the data used to build digital twin models is important for their accuracy, so the information provided by the source must be accurate and of high quality [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Digital twins from different sectors, such as healthcare and finance, must be able to communicate and connect with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' To improve the accuracy and consistency of communication, digital twins should be able to detect and correct faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' However, data security can be a challenge when using a range of devices and sensors to create digital twin models that utilize real-time data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This can be particularly vulnerable to botnets and other viruses [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' AI: The ownership of AI-powered content in the metaverse is difficult to determine, as users have no way to distinguish between communicating with a real person and an avatar created by a computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This could lead to users using AI technology to exploit other users or resources in the metaverse, such as by cheating at games or stealing from other users’ accounts [126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Additionally, AI may make mistakes, which could lead to people losing trust in the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Another challenge is the use of a similar blockchain across different AI applications in the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Big Data: One of the main challenges is the sheer volume and rate of data production in the metaverse, which can be difficult to keep up with, even with advances in data storage technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Another challenge is the variety of data produced by metaverse apps, which can make it difficult and time-consuming to collect and organize the necessary data for consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Additionally, the rapid development of big data technology [127]–[132] can make it challenging to stay up to date with technical advancements in the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Interactivity: The metaverse, which is a virtual world created through the use of technology like holographic telepresence and augmented reality, offers immersive, realistic experiences by combining audio, video, cognition, and other elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' However, the use of XR technology in the metaverse also creates challenges related to data storage, data sharing, and data interoperability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For example, the businesses can create recommendation systems using data from XR technology, but this data must be stored securely and shared transparently with stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Additionally, the metaverse must be able to handle the exchange of data between virtual worlds in an interoperable way in order to provide users with a seamless experience [133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Blockchain solutions for MaaS Data Acquisition: With the use of blockchain technology, it will be simpler to gather reliable data in the metaverse for uses like social networking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Blockchain’s distributed ledger will make it possible to trace data in the metaverse and validate transaction records [134], [135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Because the majority of nodes in the ledger must consent before any modifications to the data in the metaverse can be made, data collection is therefore resistant to attacks [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' A blockchain-specific validation process that is driven by consensus mechanisms is applied to all data collected in the metaverse [137], [138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Every action is documented as a transaction on a blockchain, and each block includes a cryptographic hash of the one before it, as well as the metadata, a date, and the activity [139].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' As a result, changing the data in one block will change the data in all the other blocks as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Any block’s data is impervious to manipulation [140].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' There will be no repetition in the data collecting process since the likelihood of producing a duplicate block is almost zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Data obtained by blockchain enabled acquisition mechanisms in the metaverse will be trustworthy since every block is approved on the blockchain [141].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Data Storage: The metaverse storage is impermeable to hacking since a new block is generated for each transaction [142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' As a result, data is stored across the chain as a copy of the original blocks, increasing data dependability and transparency in the metaverse [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' If the centralized data store is hacked, the metaverse applications, which include anything from real estate to digital things, would be very vulnerable [144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Utilizing blockchain technology will lead to a large number of blocks contributing to data distribution, enhancing data accessibility in applications like vital monitoring and life support alerts in the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Blockchain technology’s decentralized nature enables data scientists in the metaverse to work together and on data cleaning, which will greatly minimize the time and expenses involved with labeling data and getting datasets ready for analytics [145].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Data Sharing: Blockchain technology has the potential to increase the accuracy and transparency of transactions in the metaverse for applications such as education and cryptocurrency trading [118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Stakeholders would be able to access a decentralized, unchangeable record of all transactions created by applications like governance and finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Therefore, increased data openness will be advantageous to the metaverse’s stakeholders [146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Users’ confidence will increase as a result of being able to comprehend how third-party programs like Thunderbird, the Bat, and Pegasus manage data thanks to blockchain technology, which can also reduce grey market transactions [147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The owner of the data will also have total control over the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Distributed ledger technology can also be useful for data audits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Blockchain thus saves time and money by reducing the need for data validation [148].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The flexibility of data sharing will be increased through smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Usually, they are used to automate the execution of a contract so that all parties may be sure of the result right away, without the need for an intermediary or a waste of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The diverse programming of smart contracts is made possible by blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' As a result, programs like Nmusik, Ascribe, Tracr, UBS, and Applicature will benefit [149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Data Interoperability: A cross-chain protocol is the ideal approach to guarantee interoperability across virtual worlds in the metaverse [150], [151].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This enables the transfer of goods like avatars, NFTs, and money between virtual worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This protocol will lay the foundation for broad acceptance of the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Cross-blockchain technology will make it possible for virtual worlds to communicate with one another, doing away with the necessity for middlemen in the metaverse [152].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In the metaverse, connecting users and apps will be made simple via blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Data Privacy Preservation: Through the use of private and public keys, blockchain technology enables users of the metaverse to govern their data, effectively giving them ownership over it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Third-party intermediates are prohibited from misusing or obtaining data from other parties in the blockchain-enabled metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Owners of personal data stored in the blockchain- enabled metaverse will have control over when and how third parties can access that data [153].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Blockchain ledgers come with an audit trail as a standard, guaranteeing that the transactions in the metaverse are comprehensive and consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Zero- knowledge proof has been used on the blockchain, giving people easy access to the identification of crucial data in the metaverse while keeping their privacy and control over their belongings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Blockchain technology uses zero-knowledge proofs as a method for users to convince apps of something without having to provide the information [154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' IoT: Through cross-chain networks, which are created by blockchain technology, IoT devices in the metaverse may exchange data and create tamper-proof records of shared transactions in virtual worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Applications and individuals will be able to exchange and access IoT data thanks to blockchain technology without the requirement for centralized administration or control [155].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Each transaction is documented and validated in order to reduce disputes and boost user confidence throughout the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' IoT-enabled blockchain in the metaverse makes it possible to store data in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Due to the immutability of blockchain transactions, all stakeholders can depend on the information and respond quickly and effectively [156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' By allowing stakeholders to manage their IoT data records in shared blockchain ledgers, blockchain technology can assist in resolving problems in the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Digital Twins: Digital twins are attack-resistant because to blockchain’s encryption capabilities and historical data openness, which also allow for safe data sharing [157] across many virtual worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' With the use of an intelligent distributed ledger, data may be exchanged between digital twins in virtual environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Using an intelligent distributed ledger, real-world items will be saved on the blockchain and synced to digital twins in the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The implementation of digital twins on a blockchain will also help to resolve problems with data security and privacy [158].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Tracking sensor data and creating high-caliber digital twins in the metaverse will be possible by combining blockchain and AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Every digital twin activity in the metaverse will be documented as a transaction on the blockchain, which is unchangeable and requires consensus to modify [157].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' AI: Blockchain-based encryption gives users of the metaverse total control over their data and makes it easy to transfer ownership of AI consent to another entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Through the use of zero-knowledge proofs, users may convince apps and other parties that certain information about them is true without divulging this information to the applications themselves, granting the authority to utilize data for AI model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Blockchain ledgers frequently offer an audit trail that may be used to verify the legitimacy of any transactions that take place in the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' People can locate crucial metaverse facts via a zero-knowledge evidence system while still maintaining their privacy and control of their resources against deepfakes [159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' By doing this, AI will be stopped from wasting resources in the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Big Data: By assisting in the collecting of data from reliable data sources, blockchain technology will help to reduce the quantity of inaccurate data received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Data modification by other parties will be prohibited, and the data owners will have complete control over their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This guarantees high-quality data flows across the metaverse [135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Data scientists in the metaverse will be able to communicate and work together on data cleaning thanks to the decentralized nature of blockchain technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This will greatly cut down on the time and costs involved in categorizing data and building datasets for analytics applications, as well as the danger of data contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Since the data will be copied across the network and the blockchain is immutable, it will be impossible to alter it [160].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Data accessibility for metaverse stakeholders will therefore be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Interactivity: A blockchain-based distributed ledger would make it possible to validate the records of holographic telep- resence and other XR applications in the metaverse and track the origin of inaccurate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' As a result, a more precise recommendation system will be created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The zero-trust mechanism and cross-chain technology of the blockchain will make it simpler for holographic telepresence and other XR applications to safely transmit data between virtual worlds [133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Data integrity is guaranteed for XR apps and holographic telepresence by the interplanetary file system offered by the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The consensus technique used by these devices will make the data they gather and store on a blockchain unchangeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Blockchain supports confidence among AR/VR stakeholders by facilitating transparent ownership transfer and asset verification [161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' FUTURE VISIONS AND DIRECTIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Content-centric Metaverse With the ever-growing increase in using MaaS platforms, UGC is expected to be increasingly generated and transmitted through the metaverse networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Current IP-based host oriented content transmission protocols will face critical challenges for securing UGC dissemination by heterogeneous end devices over the metaverse platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' To address this challenge, content centric networking (CCN) can be employed to rethink the current Internet architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' According to CCN, contents will be routed directly by their naming information instead of IP addresses [162].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For data and content sharing in CCN-based metaverse, users request the desired UGC via sending an interest message to any CCN based node that occupies the matched content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The main idea for securing CCN is to directly safeguard the security of every single content/data itself, rather than securing the “pipe” or the communication channels/links [163].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In this way, flexible and content-centric secure metaverse can be realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Due to the inherent attributes of the CCN architectures, CCN-based metaverse can explicitly cause new security concerns as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For instance, content poisoning and network monitoring would be two security issues in the CCN-based metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Specifically, malicious users might inject poisoned UGCs, resulting in delayed or failed valid UGC delivery, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=', through flooding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' A curious CCN node might observe the sensitive content disseminated by CCN users by directly monitoring the network traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Therefore, further research on privacy and security protection for CCN-based metaverse is required [164].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Edge computing Hospitals, production lines, residential complexes, offshore equipment and game centers have different requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Hence, solutions based on edge computing in the Metaverse should be adapted for different applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This work requires a methodical, repeatable and well-reasoned approach from the Metaverse operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Otherwise, the cost of maintaining and managing edge servers will increase and many problems will arise for users and the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' At the same time, the variety of services and flexibility in service level agreements should still exist so that users have a better user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For various applications, edge computing can be used to increase reliability, reduce latency, and offload tasks in 5G networks [165].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' More interestingly, edge computing is one of the main enablers of the 6G network because it can be used for reliable low-latency communications, AI-empowered capabilities, and increasing energy efficiency [166].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In addition, edge computing plays a vital role in the new generations of the Internet of Things [167].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Hence, we expect edge computing to progress along with related technologies and add new capabilities to the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In designing edge computing solutions, various tradeoffs must be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Processing power, storage volume, telecommu- nication link bandwidth, spare parts, emergency power, security systems and many other things must be selected depending on the needs of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In addition, it is possible that the needs of users change over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' A scalable and flexible design approach can help with this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In this type of design, it is possible to increase and decrease each of these facilities, depending on the conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For example, it is not necessary for the Metaverse operator to have installed a lot of storage space on the edge sites from the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' However, it should have provided the possibility of increasing storage space on the edge servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' This would enable the network to meet this critical requirement in the shortest possible time with the increase in users’ needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Blockchain’s still unsolved challenges Despite many challenges which were discussed in this paper for MaaS developers and the solutions that blockchain technology can provide, there are still more constraints that should be taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' There is still work to be done on creating a robust blockchain for the metaverse to overcome unsolved problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' In the following, the authors mention some other unsolved challenges and make some recommendations as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Blockchain can be slow because of its complexity and distributed nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' On a blockchain, transactions can take a very long time to execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Data in the metaverse will be more able to withstand copying and manipulation with the help of a consensus-based distributed ledger, but since any new data must be duplicated throughout the entire chain, more study is needed to overcome the latency problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The number of blocks must grow along with the number of users in the metaverse, requiring the employment of enormous computational resources [168].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' As a result, users will pay a higher transaction cost for the verification of shared transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For effective data sharing in the metaverse, next-generation blockchains must overcome this problem [169].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The existence of numerous public blockchains in various virtual reality environments that do not speak the same language presents the biggest obstacle to cross-blockchain enabled the metaverse interoperability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' It will be challenging to adjust because different platforms will offer different degrees of smart contract capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Furthermore, these virtual worlds use a wide range of transaction architectures and consensus mechanisms, which limits interoperability [170].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' If a small number of miners control the majority of the network’s overall mining hash rate, blockchains are susceptible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Due to the anonymity offered by blockchain technology, it is challenging to track down all IoT transactions involving illicit services in the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' To carry out the metaverse’s expansion, the blockchain needs to be regularized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' For blockchain to be successfully used in digital twin applications in the metaverse, challenges like standardization, privacy, and scalability must all be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' The quality of digital twins in the metaverse will increase as a result of the integration of blockchain, XAI, and federated learning methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' CONCLUSIONS This article provided an overview on privacy and security aspects of the metaverse, from different perspectives, including the wireless access, learning algorithms, data access, and human-centric interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' New directions towards realizing privacy- aware and secure metaverse-as-a-service (MaaS) platforms were addressed, and less-investigated methods were reviewed to help mobile network operators and service providers facilitate the realization of secure and private MaaS through different layers of the metaverse, ranging from the access layer to privatizing the social interactions among clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Additionally, edge computing, which is one of the key enablers of the metaverse, has been discussed, along with the advantages and challenges associated with its use in the metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' Later in this work, a comprehensive investigation and analyses of challenges for MaaS developers and the blockchain’s solutions for MaaS platforms were provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf'} +page_content=' At the final, future vision, unsolved challenges, and some recommendations were also discussed to bring further insights for the network operators and engineers in the era of metaverse.' metadata={'source': 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b/4NAzT4oBgHgl3EQfuv1g/content/tmp_files/2301.01695v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fdefdfe78755c7fc63f072819bb93f91e0a469df --- /dev/null +++ b/4NAzT4oBgHgl3EQfuv1g/content/tmp_files/2301.01695v1.pdf.txt @@ -0,0 +1,4048 @@ +arXiv:2301.01695v1 [math.AG] 4 Jan 2023 +NON-FREE SECTIONS OF FANO FIBRATIONS +BRIAN LEHMANN, ERIC RIEDL, AND SHO TANIMOTO +Abstract. Let B be a smooth projective curve and let π : X → B be a smooth integral +model of a geometrically integral Fano variety over K(B). Geometric Manin’s Conjecture +predicts the structure of the irreducible components M ⊂ Sec(X/B) which parametrize non- +relatively free sections of sufficiently large anticanonical degree. Over the complex numbers, +we prove that for any such component M the sections come from morphisms f : Y → X +such that the generic fiber of Y has Fujita invariant ≥ 1. Furthermore, we prove that there +is a bounded family of morphisms f which together account for all such components M. +These results verify the first part of Batyrev’s heuristics for Geometric Manin’s Conjecture +over C. Our result has ramifications for Manin’s Conjecture over global function fields: if we +start with a Fano fibration over a number field and reduce mod p, we obtain upper bounds +of the desired form by first letting the prime go to infinity, then the height. +Contents +1. +Introduction +1 +2. +Background +9 +3. +Sections of good fibrations +16 +4. +Grauert-Mulich +18 +5. +Sections through general points +28 +6. +Twists over function fields of complex curves +31 +7. +Fujita invariant and sections +41 +8. +Boundedness statements +52 +9. +Fano fibrations +64 +10. +Examples +67 +11. +An arithmetic application +68 +References +68 +1. Introduction +Since Mori’s groundbreaking work in [Mor79] and [Mor82] the moduli space of curves +has played a central role in the analysis of Fano varieties. The irreducible components of +the moduli space that parametrize free curves – that is, curves for which the restriction of +the tangent bundle is sufficiently positive – are generically smooth and have other desirable +geometric properties. By contrast, the irreducible components that only parametrize non- +free curves frequently exhibit pathological behavior. Our goal is to classify these “exceptional +components” for Fano varieties over C. +1 + +We show that the exceptional components that parametrize curves of sufficiently large +degree must come from morphisms which increase the Fujita invariant. We call such mor- +phisms “accumulating maps”; their geometry is strongly constrained by the Minimal Model +Program. Furthermore, we show that all exceptional components can be accounted for by +a bounded family of accumulating maps. The analogous statements are still true for the +moduli space of sections of a C-Fano fibration over a curve and we will work in this more +general setting for the rest of the paper. +Our approach to this problem is motivated by arithmetic geometry. In [Bat88] Batyrev +developed a heuristic argument for Manin’s Conjecture over a global function field based on +some assumptions about the geometry of the space of curves on an Fq-Fano variety. Geo- +metric Manin’s Conjecture adapts Batyrev’s assumptions into a precise set of conjectures +about the structure of the moduli space of sections on a k-Fano fibration over a curve for +arbitrary fields k. Our work completely resolves the first prediction of Geometric Manin’s +Conjecture for a C-Fano fibration over a curve: exceptional components come from accumu- +lating maps. Our results provide evidence for Batyrev’s heuristics in characteristic p, and in +some circumstances we can deduce an arithmetic statement: if we start with a Fano fibration +over a number field and reduce mod p, we obtain upper bounds on the counting function in +Manin’s Conjecture by first letting the prime go to infinity, then the height. +1.1. Fano fibrations. Let B be a smooth irreducible projective curve over a field k and let +η denote its generic point. A Fano fibration over B is a flat k-morphism π : X → B from a +smooth projective variety X whose generic fiber Xη is a geometrically integral Fano variety +over K(B). We will denote by Sec(X /B) the moduli space of sections of π. +The following definition identifies the analogue of a free curve in the setting of fibrations. +Definition 1.1. A section C of π : X → B is relatively free if TX/B|C is globally generated +and H1(C, TX/B|C) = 0. +sect:introgmc +1.2. Geometric Manin’s Conjecture. Geometric Manin’s Conjecture is based on an influ- +ential heuristic for Manin’s Conjecture developed by Baytrev ([Bat88]). The main invariant +in Batyrev’s heuristic is the Fujita invariant. +defi:a-invariant +Definition 1.2. Let X be a smooth projective variety over a field of characteristic 0 and let +L be a big and nef Q-Cartier divisor on X. The Fujita invariant of (X, L) is +a(X, L) = min{t ∈ R | KX + tL is pseudo-effective }. +If L is nef but not big, we formally set a(X, L) = ∞. If X is singular, choose a resolution +of singularities φ : X′ → X and define a(X, L) to be a(X′, φ∗L). (The choice of resolution +does not affect the value by [HTT15, Proposition 2.7].) +Geometric Manin’s Conjecture (based on [EVW16], [LT19a], and [LST22]) predicts that +sections of Fano fibrations over any ground field are governed by two key principles: +• (Exceptional set) “Pathological” families of sections are controlled by the Fujita in- +variant. +• (Stability) “Non-pathological” families of sections exhibit homological or motivic +stability as the degree increases. +Our main theorems establish a precise version of the first principle over the complex num- +bers: the Fujita invariant controls the failure of relative freeness. In accordance with the +2 + +asymptotic nature of Manin’s Conjecture, our results apply to any family of sufficiently large +degree. +1.3. Main results. Suppose π : X → B is a Fano fibration over C. Our first main result, +Theorem 1.3, shows that every non-relatively free section of sufficiently large degree comes +from an accumulating map which does not decrease the Fujita invariant along the generic +fiber. This verifies a conjecture of [LST22] and generalizes earlier results for del Pezzo surface +fibrations ([LT22], [LT21a]) to fibrations of arbitrary dimension using completely different +techniques. +Theorem 1.3 has two cases which correspond to the two ways in which a section C could +fail to be relatively free. First, the deformations of C could fail to dominate X , in which case +the sections sweep out a subvariety Y ⊊ X . Second, the deformations of C could dominate X +but TX/B|C could have a low degree quotient. In the latter case we expect C to deform more +in some directions than in others so that there is a subvariety of X swept out by the “most +positive” deformations of C. In fact, there is an algebraic foliation on a generically finite +cover of X such that most deformations of C are tangent to the foliation; our subvariety is +swept out by the images of the leaves meeting C. In both cases Theorem 1.3 shows that the +relevant subvariety must have a large Fujita invariant along the generic fiber. +theo:maintheorem1 +Theorem 1.3. Let π : X → B be a Fano fibration. There is a constant ξ = ξ(π) with +the following properties. Let M be an irreducible component of Sec(X /B) parametrizing a +family of non-relatively free sections C which satisfy −KX/B · C ≥ ξ. Let Uν denote the +normalization of the universal family over M and let ev : Uν → X denote the evaluation +map. Then either: +(1) ev is not dominant. Then the subvariety Y swept out by the sections parametrized by +M satisfies +a(Yη, −KX/B|Yη) ≥ a(Xη, −KX/B|Xη). +(2) ev is dominant. Letting f : Y → X denote the finite part of the Stein factorization +of ev, we have +a(Yη, −f ∗KX/B|Yη) = a(Xη, −KX/B|Xη). +Furthermore, there is a dominant rational map φ : Y ��� Z over B with connected +fibers such that the dimension of Z is at least 2 and the following properties hold. Let +C′ denote a general section of Y → B parametrized by M and let W′ ⊂ Y denote the +unique irreducible component of the closure of φ−1(φ(C′)) which maps dominantly to +φ(C′). There is a resolution ψ : W → W′ such that the locus where ψ−1 is well-defined +intersects C′ and ψ has the following properties. +(a) We have a(Wη, −ψ∗f ∗KX/B|Wη) = a(Xη, −KX/B|Xη). +(b) The Iitaka dimension of KWη − a(Wη, −ψ∗f ∗KX/B|Wη)ψ∗f ∗KX/B|Wη is 0. +(c) The general deformation of the strict transform of C′ in W is relatively free in +W. +(d) There is a constant T = T(π) depending only on π, but not M, such that the +sublocus of M parametrizing deformations of the strict transform of C′ in W +has codimension at most T in M. +Remark 1.4. Suppose X is a Fano variety and we are studying irreducible components +of Mor(B, X). Any family of non-free curves on X leads to a family of non-relatively free +3 + +sections of π : X ×B → B and thus Theorem 1.3 gives a classification result for such families. +However, it is natural to ask whether non-free curves can be described using generically finite +maps f : Y → X instead of generically finite maps f : Y → X × B. The answer is “yes,” +but due to length constraints we will give the details in a supplementary paper ([LRT22]). +Theorem 1.3 has two key consequences. First, the Fujita invariant can be computed using +tools from the Minimal Model Program and thus Theorem 1.3 gives a practical way to classify +families of non-relatively free sections. +Example 1.5. In Example 10.1 we analyze the moduli spaces Sec(X /B) when π : X → B +is a cubic hypersurface fibration and B is a curve of arbitrary genus. When dim(Xη) ≥ 5, we +show that the “exceptional set” is empty so that every component of Sec(X /B) of sufficiently +high degree will generically parametrize relatively free sections. For dimension 4 a similar +analysis allows us to describe the families of non-relatively free sections of large degree. +Second, [Bir21] imposes strong finiteness constraints on Fujita invariants and Theorem 1.3 +allows us to deduce finiteness results for families of non-relatively free sections. Recall that +Theorem 1.3 shows that families of non-relatively free sections come from maps f : Y → X +such that the Fujita invariant of Yη is at least a(Xη, −KX/B|Xη) = 1. If we pass to an algebraic +closure K(B) then the set of maps fη : Yη → Xη such that a(Yη, −f ∗ +ηKXη) ≥ 1 satisfy certain +types of boundedness (see e.g. [LST22, Theorem 1.7]). However, the analogous boundedness +statements over K(B) are no longer true since a map over K(B) can correspond to infinite +families of twists over K(B). +In Section 6 we systematically study the set of twists of the map fη : Yη → Xη. Theorem +1.14 shows that amongst all the twists only a bounded subfamily carry a family of sections +which is dense in an irreducible component of Sec(X /B). In this way, we conclude that all +non-relatively free sections will come from a bounded family of maps f : Y → X . +theo:maintheorem2 +Theorem 1.6. Let π : X → B be a Fano fibration. +(1) There is a proper closed subset R ⊊ X such that if M ⊂ Sec(X /B) is an irre- +ducible component parametrizing a non-dominant family of sections then the sections +parametrized by M are contained in R. +(2) There is a constant ξ = ξ(π), a proper closed subset V ⊂ X , and a bounded family +of smooth projective B-varieties Ys, s ∈ S equipped with B-morphisms fs : Ys → X +satisfying: +(a) dim(Ys) < dim(X ) and fs is generically finite onto its image; +(b) a(Ys,η, −f ∗ +s KX/B|Ys,η) = a(Xη, −KX/B|Xη) and the Iitaka dimension of KYs,η − +f ∗ +s KX/B|Ys,η is 0; +(c) if M ⊂ Sec(X /B) is an irreducible component that generically parametrizes +non-relatively free sections C with −KX/B · C ≥ ξ then for a general section C +parametrized by M we have either +(i) C ⊂ V, or +(ii) for some fs : Ys → X in our family there is a relatively free section C′ of +Ys/B such that C = fs(C′). +Remark 1.7. Theorem 1.6 establishes geometric analogues of various conjectures about the +exceptional set in Manin’s Conjecture. For example, suppose that B is a smooth projective +Fq-curve and π : X → B is a Fano fibration equipped with an adelic metrization on the +4 + +relative canonical bundle. Weak Manin’s Conjecture predicts that there exist a constant C > +0 and a closed subset R ⊂ X such that for any ǫ > 0 the number of sections meeting X \R +of height at most d is bounded above by Cqd(1+ǫ). Using the heuristic estimate #M(Fq) ≈ +qdim M, this means that R should contain all sections parametrized by a family M such +that dim(M)/expdim(M) ≥ 1 + ǫ (with perhaps finitely many exceptions). Theorem 1.6.(1) +shows that over C there exists a closed set R with this property. +1.4. A geometric application. Suppose M is an irreducible component of Sec(X /B) and +let N ⊂ M denote the sublocus parametrizing sections C such that TX/B|C is not generically +globally generated. One would like to find a lower bound on the codimension of N. This +problem has been previously studied when X is a smooth Fano variety and M ⊂ Mor(P1, X), +in which case N ⊂ M is simply the non-free locus. For example, [BS22, Theorem 1.2] shows +that when X is a smooth hypersurface whose dimension is much larger than the degree and +M is an irreducible component of Mor(P1, X) then the codimension of N ⊂ M grows linearly +in the anticanonical degree of the curves parametrized by M. +We prove the first general statement for arbitrary Fano fibrations: the codimension of the +non-generically-globally-generated locus grows linearly in the degree unless there is a clear +geometric reason why it cannot. +theo:maintheorem3 +Theorem 1.8. Let π : X → B be a Fano fibration. There is a linear function Q(d) whose +leading coefficient is a positive number depending only on dim(X ) such that the following +property holds. +Suppose that M is an irreducible component of Sec(X /B) parametrizing a family of sec- +tions C which satisfy −KX/B · C = d. Let N ⊂ M be a subvariety parametrizing sections C +such that TX/B|C is not generically globally generated. Then either +(1) the codimension of N in M is at least sup{⌊Q(d)⌋, 0}, or +(2) the sections parametrized by N sweep out a subvariety Y ⊊ X satisfying +a(Yη, −KX/B|Yη) ≥ a(Xη, −KX/B|Xη). +In Example 10.2 we will show that the codimension of the non-generically-globally-generated +locus can be constant as the degree increases, demonstrating that case (2) of Theorem 1.8 +must be included. +Remark 1.9. Over C, Theorem 1.8 shows that families of sections such that TX/B|C is +not generically globally generated are either contained in the exceptional locus or they have +large codimension in a component of Sec(X /B). Assuming the analogous statement over a +global function field, we can expect such sections to make a negligible contribution to the +counting function for Manin’s Conjecture. Thus Theorem 1.8 supports the novel formulation +of Manin’s Conjecture due to [Pey17] which only counts rational points which are “free” in +a suitable sense. +subsec:arithmeticapp +1.5. An arithmetic application. Our results can be applied to prove an upper bound of +Manin type over a global function field. Let F be a number field and let B be a smooth +projective curve over F. Let S be a finite set of places of F including all archimedean places +and let oF,S be the ring of S-integers in F. +Let π : X → B be a Fano fibration defined over F. After perhaps enlarging S, we can find +an integral model �π : X → B of π over oF,S such that X and B are smooth over oF,S. Let +R ⊂ X be the Zariski closure of the union of the loci swept out by non-dominant families of +5 + +sections in Sec(X/B). Theorem 1.6.(1) implies that the base change of R to C is contained +in a proper closed subset, so in particular R itself is a proper closed subset. We consider the +flat closure R ⊂ X of R. +Let v be a non-archimedean place of F not contained in S and consider the reduction +πv : Xv → Bv at v which is defined over a finite field kv. Let Rv be the reduction of R at v +and let Sec(Xv/Bv, Rv)≤d be the open subset of Sec(Xv/Bv) parametrizing sections C ̸⊂ Rv +of anticanonical height ≤ d. Then we consider the following counting function: +N(Xv \ Rv, −KXv/Bv, d) = #Sec(Xv/Bv, Rv)≤d(kv). +Weak Manin’s Conjecture over K(Bv) predicts that for any ǫ > 0 we have +N(Xv \ Rv, −KXv/Bv, d) = o(qd(1+ǫ) +v +), +as d → ∞ where qv = #kv. The following approximation of this conjecture was suggested +to us by Jordan Ellenberg and Melanie Matchett Wood: +theorem:arithmeticapp +Theorem 1.10. Let F, S, �π : X → B be as above. Then assuming dǫ > dim Xη, we have +N(Xv \ Rv, −KXv/Bv, d) +qd(1+ǫ) +v +→ 0 +as v → ∞. +This result fits into the recent trend of taking hard arithmetic questions that are asymp- +totic in a different parameter and making them more accessible by first letting v go to ∞. +This technique has been explored in the contexts of Malle’s Conjecture and Cohen-Lenstra +heuristics over global function fields; see e.g. [Ach06, EVW16, FLR22, PW21, LWZB19, +ETW17]. +1.6. Strategy. The proof of our main results requires a number of statements which are +interesting in their own right. We first outline the proof of case (2) of Theorem 1.3. For +simplicity we assume that ev : Uν → X has connected fibers so that the Stein factorization +Y of ev is equal to X . We must construct a rational map φ on X that captures the geometry +of this dominant family of non-relatively free sections. Our strategy relies on the theory +of foliations and slope stability. Recall that for any curve C that deforms in a dominant +family on X [CP11] defines a notion of slope stability and Harder-Narasimhan filtrations for +torsion-free sheaves on X with respect to the numerical class [C]. +In Section 4 we prove that when E is a torsion-free sheaf on X that is semistable with +respect to the numerical class [C] of a flat family of curves then E|C is “almost” semistable. +theo:introgm +Theorem 1.11. Let π : X → B be a flat morphism with connected fibers from a smooth +projective variety X to a smooth projective curve B and let E be a torsion-free sheaf on X . +Let M be an irreducible component of Sec(X /B) and let Uν denote the normalization of +the universal family over M. Suppose that the evaluation map ev : Uν → X is dominant +with connected fibers and that for some open subset M◦ +red ⊂ Mred the restriction of ev to the +preimage of M◦ +red is flat. For a general curve C parametrized by M, write +0 = F0 ⊂ F1 ⊂ . . . ⊂ Fr = E|C +for the Harder-Narasimhan filtration of E|C. +Suppose that E is [C]-semistable. Then for every index i we have +|µ(E|C) − µ(Fi/Fi−1)| ≤ (g(B) dim(X ) − g(B) + 1)2 rk(E). +6 + +Now suppose that C is a general member of a dominant family of non-relatively free +sections of π whose evaluation map has connected fibers. +Since by hypothesis C is not +relatively free, we see that TX/B|C must have a low slope quotient. Applying Theorem 1.11 +to a birational model flattening the family, we can “lift” this quotient to all of X . The result +is a foliation F ⊂ TX of large slope and the pioneering results of [CP19] show that F is +induced by a rational map φ. We carry out this construction in Section 5. +It only remains to verify the desired properties of φ. The most difficult is the computation +of the Fujita invariant as in Theorem 1.3.(2).(b). By appealing to Birkar’s recent boundedness +results in the Minimal Model Program, we prove the following general criterion for computing +the Fujita invariant in Section 7. +theo:introainvariant +Theorem 1.12. Let π : X → B be a Fano fibration. Fix a positive rational number a and +a positive integer T. There is some constant ξ = ξ(π, a, T) with the following property. +Suppose that ψ : Y → B is a flat morphism with connected fibers from a smooth projective +variety Y and f : Y → X is a B-morphism that is generically finite onto its image. Suppose +that N is an irreducible component of Sec(Y/B) parametrizing a dominant family of sections +on Y and let M denote the irreducible component of Sec(X /B) containing f∗N. +Assume that the sections C parametrized by N satisfy −f ∗KX/B · C ≥ ξ and that +dim(N) ≥ a · dim(M) − T. +Then +a(Yη, −f ∗KX/B|Yη) ≥ a. +Remark 1.13. We prove statements analogous to Theorem 1.12 in the more general setting +of pairs (X , L) where X is a smooth projective variety admitting a flat morphism with +connected fibers π : X → B and L is a generically relatively big and semiample Cartier +divisor on X . +Returning to the setting of Theorem 1.3.(2), we prove Theorem 1.3.(2).(b) by combining +Theorem 1.12 with Theorem 1.3.(2).(d). +We next outline the proof of Theorem 1.6.(2). Suppose we have a dominant generically +finite morphism fη : Yη → Xη. As discussed earlier, the key is to understand how the set +of twists fη interacts with the behavior of sections on an integral model. We systematically +analyze this relationship in Section 6; in particular, we prove the following statement that +undergirds Theorem 1.6. +theo:introtwists +Theorem 1.14. Let π : X → B be a Fano fibration. Let Y be a normal projective vari- +ety equipped with a flat morphism with connected fibers ψ : Y → B and with a dominant +generically finite B-morphism f : Y → X . +Suppose that �Y is a B-variety which is smooth and projective and �f : �Y → X is a +dominant generically finite morphism such that �fη : �Yη → Xη is birational to a twist of +fη. Fix a positive integer T and suppose that there exists an irreducible component �N ⊂ +Sec( �Y/B) parametrizing a dominant family of sections on �Y such that the pushforward of � +N +has codimension at most T in a component of Sec(X /B). +Then there exist constants d = d(Y/X ) and n = n(Y/X , T) and a finite Galois morphism +B′ → B of degree at most d with at most n branch points such that the base changes of fη +and �fη to K(B′) are birationally equivalent. +7 + +This result implies that the set of �Y satisfying the conditions of Theorem 1.14 is birationally +bounded. We prove this boundedness by constructing a parameter space of twists which is +of finite type over the Hurwitz stack parametrizing finite covers B′ → B. +1.7. History. Ever since the seminal results due to Mori and his coauthors ([Mor79], [Mor82], +[MM86]) the moduli space of curves has played a prominent role in the study of Fano vari- +eties. The notion of free rational curves goes back to pioneering work by Koll´ar–Miyaoka– +Mori ([KMM92], [Kol96]) on rational connectedness of Fano varieties. Since then there have +been many breakthroughs in the description of the moduli spaces Mor(B, X) for Fano va- +rieties X, most notably when B = P1. One particularly influential example is the analysis +of rational curves on Fano hypersurfaces pioneered by [HRS04] and subsequently developed +by [CS09], [BK13], and [RY19]. ([BV17], [BS22] provide a different approach to this prob- +lem using an idea from analytic number theory.) Another important class of examples is the +moduli spaces of curves on various homogeneous spaces ([Tho98], [KP01], [Bou16]). However +for a long time it was unclear what structure to expect for arbitrary Fano varieties. +The situation was clarified by the introduction of ideas from arithmetic geometry. Manin’s +Conjecture is a conjectural asymptotic formula for the counting function of rational points on +Fano varieties formulated and refined in [FMT89], [BM90], [Pey95], [BT98], and [LST22]. In +[Pey17] Peyre proposed another version of Manin’s Conjecture using the notion of freeness of +rational points which is inspired by the concept of free rational curves. A motivic version of +Manin’s Conjecture has been established for equivariant compactifications of vector groups +in [CLL16] and [Bil18]. In Manin’s Conjecture, it is important to exclude the contribution to +the counting function from “exceptional sets” where rational points accumulate too quickly. +The relationship between exceptional sets and Fujita invariants was developed in [HTT15], +[LTT18], [HJ17], [LT17], [Sen21], [LST22], and [LT19b]. These developments culminated in +the main theorem of [LST22] proving that the contribution to the exceptional set coming +from maps f : Y → X such that Y has larger a and b invariants will be contained in a thin +set of rational points. This result is a source of our main theorem (Theorem 1.6) showing +that pathological components come from a bounded family. +In his influential notes ([Bat88]) Batyrev gave a heuristic for the global function field +version of Manin’s Conjecture. +Over time the principles underlying Batyrev’s heuristic +were made into precise conjectures and extended to arbitrary ground fields. First, building +upon earlier work on homological stability by [Seg79], [CJS94], and many others, [EVW16] +highlighted the connection between homological stability and rational point counts via the +Grothendieck-Lefschetz trace formula. Second, based on the analysis of the exceptional set +described earlier, [LT19a] predicted the geometry underlying “pathological” families of ra- +tional curves on Fano varieties and obtained a first prototype result of Theorem 1.3 in this +setting. Further works leveraged this intuition to study rational curves for Fano varieties of +dimension ≤ 3 and for sections of del Pezzo fibrations ([LT19a], [LT21b], [LT22], [LT21a], +[BLRT22], [ST22], and [BJ22]). +Together, the two principles in Batyrev’s heuristic (stated in Section 1.2) are known as +Geometric Manin’s Conjecture. Geometric Manin’s Conjecture unifies many disparate ex- +amples and clarifies the conjectural structure of Mor(B, X) for arbitrary Fano varieties. +Theorems 1.3 and 1.6 are the first statements in Geometric Manin’s Conjecture which have +been proved for arbitrary Fano fibrations over curves of arbitrary genus. +8 + +Acknowledgments: The authors thank Shintarou Yanagida for a helpful conversation +about stacks. The authors thank Jordan Ellenberg and Melanie Matchett Wood for suggest- +ing an arithmetic application of our work and Lars Hesselholt for recommending the reference +[BS15]. The authors also thank Izzet Coskun and Zhiyu Tian for comments on this paper. +Part of this project was conducted at the SQuaRE workshop “Geometric Manin’s Conjecture +in characteristic p” at the American Institute of Mathematics. The authors would like to +thank AIM for the generous support. +Brian Lehmann was supported by Simons Foundation grant Award Number 851129. Eric +Riedl was supported by NSF CAREER grant DMS-1945944. Sho Tanimoto was partially +supported by JST FOREST program Grant number JPMJFR212Z, by JSPS Bilateral Joint +Research Projects Grant number JPJSBP120219935, and by JSPS KAKENHI Early-Career +Scientists Grant number 19K14512. +2. Background +sect:background +Throughout all our schemes will be assumed to be separated and every connected com- +ponent will have finite type over the base ring (which is usually C or the function field of a +complex curve). Recall that in this situation the normalization of a scheme X is isomorphic +to the normalization of Xred. A variety is a separated integral scheme of finite type over the +base field. Given a coherent sheaf F on a variety V , we denote by Ftors the torsion subsheaf +and by Ftf the quotient of F by its torsion subsheaf. +Given a dominant generically finite morphism of projective varieties f : Y → X, we +denote by Aut(Y/X) the automorphism group of Y over X and by Bir(Y/X) the birational +automorphism group of Y over X. +Definition 2.1. Suppose we have a dominant morphism of varieties f : U → V such that +the general fiber of f is geometrically irreducible. +Suppose that T ⊂ V is a subvariety +that meets the open locus over which f has geometrically irreducible fibers. Then f −1(T) +has a unique irreducible component which dominates T under f. We call this the “main +component” of f −1(T). +When X is a projective variety, we will let N1(X)R denote the space of R-Cartier divisors +up to numerical equivalence. In this finite-dimensional vector space we have the pseudo- +effective cone Eff +1(X) and the nef cone Nef1(X). Dually, we will let N1(X)R denote the +space of R-1-cycles up to numerical equivalence. Inside N1(X)R we have the pseudo-effective +cone Eff1(X) and the nef cone Nef1(X). Given a curve C, we will denote its numerical class +by [C]. +We also use the standard definitions and techniques from the Minimal Model Program. +See [KM98] and [BCHM10] for more details. +Definition 2.2. Let f : X → Y be a projective morphism of varieties and let L be a +Q-Cartier divisor on X. For a property P of Q-Cartier divisors (such as ample, big, nef, +semiample, etc.), we say that L is generically relatively P if the restriction of L to the generic +fiber of f satisfies P. +2.1. Vector bundles on curves. Let B be a smooth projective curve and let E be a vector +bundle of rank r on B. Write the Harder-Narasimhan filtration of E as +0 = F0 ⊂ F1 ⊂ F2 ⊂ . . . ⊂ Fk = E. +9 + +We denote by µmax(E) the maximal slope of any torsion-free subsheaf, i.e., µmax(E) = +µ(F1). We denote by µmin(E) the minimal slope of any torsion-free quotient, i.e., µmin(E) = +µ(E/Fk−1). Note that by the mediant inequality for every index 1 < i ≤ k we have +eq:mediant +eq:mediant +(2.1) +µ(Fi) = c1(Fi−1) + c1(Fi/Fi−1) +rk(Fi−1) + rk(Fi/Fi−1) < µ(Fi−1). +lemm:minslopelower +Lemma 2.3. Let f : Y → S be a smooth projective morphism of varieties with relative +dimension 1. +Suppose that E is a locally free sheaf on Y . +Then µmin(E|Ys) is a lower- +semicontinuous function on S. +Proof. By [HL97, Theorem 2.3.2] there is a dense open set U ⊂ S and a torsion-free sheaf F +on YU such that (E/F)|Yt is the minimal slope quotient of E|Yt for the fiber Yt over any point +t ∈ U. Arguing by Noetherian induction, it suffices to show that if s denotes an arbitrary +point of S then µmin(E|Ys) ≤ µmin(E|Yt). By projectivity of the Quot scheme, for any point +s ∈ S there is a surjection E|Ys → Qs where Qs has the same degree and rank as (E/F)|Yt. +In particular µ(Qs,tf) ≤ µ((E/F)|Yt) finishing the proof. +□ +Definition 2.4. We say that a coherent sheaf E on a smooth projective curve B is generically +globally generated if the evaluation map +H0(B, E) ⊗ OB → E +is surjective at the generic point of B. +lemm:genericallygloballygenerated +Lemma 2.5. Let B be a smooth projective curve. Suppose that E is a generically globally +generated vector bundle on B. Then µmin(E) ≥ 0. +Proof. Since the evaluation map on global sections has torsion cokernel, we have +µ(E) ≥ µ(H0(B, E) ⊗ OB) = 0. +Denote the Harder-Narasimhan filtration of E by +0 = F0 ⊂ F1 ⊂ . . . ⊂ Fk = E. +Since F1 is the maximal destabilizing subsheaf, we see that +µ(F1) ≥ µ(E) ≥ 0. +Since E is generically globally generated, its quotient E/F1 is also generically globally gener- +ated, and we conclude by induction on the length k of the Harder-Narasimhan filtration. +□ +2.1.1. Cohomology bounds. We next recall some bounds on the cohomology groups of semistable +and generically globally generated vector bundles. +lemm:semistablevanishing +Lemma 2.6. Let E denote a semistable vector bundle on a smooth projective curve B. Sup- +pose that µ(E) > (2g(B) − 2). Then H1(B, E) = 0. +Proof. By Serre duality it suffices to show that H0(B, E∨ ⊗ ωB) = 0. Since E is semistable, +E∨⊗ωB is as well. Since µ(E∨⊗ωB) < 0 there are no non-zero morphisms OB → E∨⊗ωB. +□ +coro:checkinggg +Corollary 2.7. Let E denote a vector bundle on the smooth projective curve B. +Define +d = (2g(B) − 2) − µmin(E). Then: +(1) For any line bundle L of degree > d we have that H1(B, E ⊗ L) = 0. +(2) For any line bundle T of degree > d + 1 we have that E ⊗ T is globally generated. +10 + +Proof. Write the Harder-Narasimhan filtration of E as +0 = F0 ⊂ F1 ⊂ F2 ⊂ . . . ⊂ Fk = E. +(1) Since the slopes µ(Fi/Fi−1) are strictly decreasing in i we have µ(Fi/Fi−1 ⊗ L) > +2g(B) − 2 for every index i = 1, 2, . . . , k. Thus for i in this range we have H1(B, Fi/Fi−1 ⊗ +L) = 0 by Lemma 2.6. Using the exact sequences +H1(B, Fi−1 ⊗ L) → H1(B, Fi ⊗ L) → H1(B, Fi/Fi−1 ⊗ L) → 0 +and arguing by induction on i we see that H1(B, E ⊗ L) = 0. +(2) follows immediately from (1) and the LES sequence of cohomology associated to the +inclusion +E ⊗ T ⊗ OB(−p) ֒→ E ⊗ T +where p is any closed point of B. +□ +lemm:ggh1bound +Lemma 2.8. Let B be a smooth projective curve of genus g. Suppose that E is a generically +globally generated bundle on C. Then +(1) h0(C, E) ≤ deg(E) + rk(E). +(2) h1(C, E) ≤ g(B) rk(E). +Proof. Write 0 = F0 ⊂ F1 ⊂ . . . ⊂ Fk = E for the Harder-Narasimhan filtration of E. Since +E is generically globally generated, Lemma 2.5 shows that every successive quotient Fi/Fi−1 +has degree ≥ 0. +If 0 ≤ µ(Fi/Fi−1) ≤ 2g(B)−2, Clifford’s Theorem for semistable bundles as in [BPGN97, +Theorem 2.1] shows that +h0(B, Fi/Fi−1) ≤ 1 +2 deg(Fi/Fi−1) + rk(Fi/Fi−1) +and that +h1(B, Fi/Fi−1) = h0(B, Fi/Fi−1) − χ(Fi/Fi−1) ≤ −1 +2 deg(Fi/Fi−1) + g(B) rk(Fi/Fi−1). +On the other hand, if 2g(B) − 2 < µ(Fi/Fi−1) then h1(B, Fi/Fi−1) = 0 and +h0(B, Fi/Fi−1) = deg(Fi/Fi−1) + rk(Fi/Fi−1)(1 − g(B)) +by Lemma 2.6 and Riemann-Roch. Since +h0(B, E) ≤ +s +� +i=1 +h0(B, Fi/Fi−1) +and +h1(B, E) ≤ +s +� +i=1 +h1(B, Fi/Fi−1) +we obtain the desired statement using the additivity of deg and rk in exact sequences. +□ +2.2. Fujita invariant. Recall from Definition 1.2 that if X is a smooth projective variety +over a field of characteristic 0 and L is a big and nef Q-Cartier divisor on X then +a(X, L) = min{t ∈ R | KX + tL ∈ Eff +1(X)}. +By [BDPP13] the Fujita invariant will be positive if and only if X is geometrically uniruled. +We will rely on the following boundedness result. +11 + +theo:Dicerbo +Theorem 2.9 ([DC17, Theorem 1.2], [HL20, Theorem 1.3]). Fix a positive integer n and fix +ǫ > 0. As we vary X over all smooth projective varieties of dimension n defined over a field +of characteristic 0 and vary L over all big and nef Cartier divisors on X, there are only +finitely many possible values of a(X, L) in the range (ǫ, ∞). +The Fujita invariant is most useful for analyzing pairs satisfying an additional assumption. +Definition 2.10. Let X be a smooth projective variety and let L be a big and nef Q-divisor +on X. We say that (X, L) is adjoint rigid if KX + a(X, L)L has Iitaka dimension 0. If X +is singular and L is a big and nef Q-Cartier divisor, we say that (X, L) is adjoint rigid if +(X′, φ∗L) is adjoint rigid for some resolution of singularities φ : X′ → X. This definition +does not depend on the choice of resolution. +2.3. Slope stability for smooth projective varieties. The notion of slope stability with +respect to movable curve classes was developed by [CP11], [GKP14], [GKP16]. +Definition 2.11. Let X be a smooth projective variety and let α ∈ Nef1(X ). For any +torsion-free sheaf E on X, we define +µα(E) = c1(E) · α +rk(E) . +We say that E is α-semistable if for every non-zero torsion-free subsheaf F ⊂ E we have +µα(F) ≤ µα(E). +Every torsion free sheaf admits a maximal destabilizing subsheaf with respect to this slope +function. Thus we get a theory of α-Harder-Narasimhan filtrations for torsion-free sheaves +on X. The following definition captures the slopes of the pieces of the Harder-Narasimhan +filtration. +Definition 2.12. Let X be a smooth projective variety and let α ∈ Nef1(X ). Suppose that +E is a torsion-free sheaf of rank r. Write +0 = F0 ⊂ F1 ⊂ F2 ⊂ . . . ⊂ Fk = E +for the α-Harder-Narasimhan filtration of E. The slope panel SPα(E) is the r-tuple of rational +numbers obtained by combining for every index i the list of rk(Fi/Fi−1) copies of µα(Fi/Fi−1) +(arranged in non-increasing order): +SPα(E) = (µα(F1/F0), . . . +� +�� +� +rk(F1/F0) copies +, µα(F2/F1), . . . +� +�� +� +rk(F2/F1) copies +, . . . , µα(Fk/Fk−1), . . . +� +�� +� +rk(Fk/Fk−1) copies +) +We denote by µmax +α +(E) the maximal slope of any torsion-free subsheaf, i.e., µmax +α +(E) = +µα(F1). We denote by µmin +α +(E) the minimal slope of any torsion-free quotient, i.e., µmin +α +(E) = +µα(E/Fk−1). +When discussing slope panels in the case when X is a curve, we will always let α be an +ample class of degree 1 and thus we will simply write SP(E). +Variations of the next result have been proved many times in the literature. +theo:HNisfoliation +Theorem 2.13 ([Pan15, Proposition 1.3.32]). Let X be a smooth projective variety and let +α ∈ Nef1(X) be a nef curve class. Denote the α-Harder-Narasimhan filtration of TX by +0 = F0 ⊂ F1 ⊂ . . . ⊂ Fk = TX. +Then every term Fi such that µmin +α +(Fi) > 0 defines a foliation on X. +12 + +Suppose that f : X ��� Y is a rational map from a smooth projective variety X to a +normal projective variety Y . Let U be the open locus where f is defined. There is a unique +foliation on X whose restriction to U is the saturation in TU of the kernel of TU → f ∗TY . We +call this the foliation induced by f. Note that if f ′ : X ��� Y ′ is a rational map birationally +equivalent to f then f and f ′ induce the same foliation. +2.4. Numerical equivalence on Fano fibrations. Suppose π : X → B is a Fano fibration. +In this section we give some reminders about the basic properties of numerical equivalence +and the cone of nef curves on X . +lemm:relativeprops +Lemma 2.14. Let π : X → B be a Fano fibration. Then for every smooth fiber F the space +N1(F)R has the same dimension. Furthermore, if L is a Q-Cartier divisor on X then the +following are equivalent: +(1) L|F is ample for some smooth Fano fiber F. +(2) L|F is ample for all smooth Fano fibers F. +(3) L|Xη is ample. +The analogous statement is true for nefness, for bigness, and for pseudo-effectiveness. +Proof. It follows from [Kol96, IV.3.5 Corollary] that every smooth fiber is rationally con- +nected. Then our first assertion follows from standard Hodge theory. The equivalence of +the three conditions for ampleness and nefness follows from [Wi´s09, Theorem 1]. For the +equivalence of the three conditions for bigness and pseudo-effectiveness, see the paragraph +before [dFH11, Theorem 6.8]. +□ +We will also need the following version of the Cone Theorem for nef curves. This theorem +was proved by [Ara10] conditional on the Borisov-Alexeev-Borisov Conjecture which has +subsequently been proved in [Bir21]. +theo:conetheorem +Theorem 2.15 ([Ara10]). Let X be a normal Q-factorial projective variety and let ∆ be +an effective Q-Cartier divisor on X such that (X, ∆) is ǫ-lc. +There is a constant ζ = +ζ(dim(X), ǫ) such that +Eff1(X )KX +∆≥0 + Nef1(X ) = Eff1(X )KX +∆≥0 + +� +i +R≥0[Ci] +where {Ci} is a countable collection of curves which satisfy 0 < −(KX + ∆) · Ci ≤ ζ. If ∆ +is big, then the set {Ci} is finite. +For Fano fibrations, the Cone Theorem for nef curves allows us to isolate the behavior of +vertical curves: +theo:nefconetheorem +Theorem 2.16. Let π : X → B be a flat morphism from a normal Q-factorial projective +variety X to a smooth projective curve B. Suppose that ∆ is an effective Q-Cartier divisor +on X such that ∆ is π-relatively big and (X , ∆) is ǫ-lc. Let F denote a general fiber of π. +There is a positive integer m = m(dim(X ), ǫ) such that we have an equality +Eff1(X )KX +∆+mF ≥0 + Nef1(X ) = Eff1(X )KX +∆+mF ≥0 + +� +i +R≥0[Ci] +where {Ci} is a finite set of π-vertical moving curves which satisfy 0 < −(KX+∆+mF)·Ci ≤ +m. +13 + +Proof. Since ∆ is effective and π-relatively big, we see that ∆ + δF is big for any δ > 0. +By choosing δ sufficiently small we may ensure that δ < 1 and that (X , ∆ + δF) is ǫ/2-lc. +Applying Theorem 2.15 there is a constant ζ = ζ(dim(X ), ǫ) such that +Eff1(X )KX +∆+δF ≥0 + Nef1(X ) = Eff1(X )KX +∆+δF ≥0 + +� +j +[Cj] +where the Cj are a finite set of movable curves satisfying 0 ≤ −(KX ′ + ∆ + δF) · Ci ≤ ζ. +Choose a positive integer m > ζ + 1. Then (KX + ∆ + mF) · Cj > 0 for every one of our +movable curves Cj that dominates B under π. Thus we have +Eff1(X )KX +∆+δF ≥0 + Nef1(X ) = Eff1(X )KX +∆+mF ≥0 + +� +i +R≥0[Ci] +where now the Ci are π-vertical and still satisfy 0 ≤ −(KX ′ + ∆ + mF) · Ci ≤ ζ < m. +□ +2.5. Boundedness and the Fujita invariant. In this section we recall some results of +[LST22]. Our first construction shows that the family of subvarieties of X which are adjoint +rigid and have the same Fujita invariant as X is bounded. +cons:rigidsubvarieties +Construction 2.17. Let k be a field of characteristic 0. Let X be a geometrically uniruled +geometrically integral smooth projective k-variety and let L be a big and nef Q-Cartier +divisor on X. +By [LST22, Theorem 4.19] there exist a proper closed subset V , finitely +many projective varieties Wi ⊂ Hilb(X), proper families pi : Ui → Wi where Ui is a smooth +birational model of the universal family U′ +i → Wi, and dominant generically finite morphisms +si : Ui → X such that +• over k, a general fiber of pi,k : Ui,k → Wi,k is an integral uniruled projective variety +which is mapped birationally by si,k onto the subvariety of Xk parametrized by the +corresponding point of Hilb(Xk); +• a general fiber Z of pi is a smooth projective variety satisfying a(Z, s∗ +i L|Z) = a(X, L) +and is adjoint rigid with respect to s∗ +i L|Z; and +• for every subvariety Y ⊂ X not contained in B+(L) which satisfies a(Y, L|Y ) ≥ +a(X, L) and which is adjoint rigid with respect to L, either Y is contained in V or +there is some index i and a smooth fiber of pi that is mapped birationally to Y under +the map si. +In fact more is true: the next result shows that the morphisms f : Y → X such that Y +is adjoint rigid and has the same Fujita invariant as X also form a bounded family up to +twisting. +theo:ainvboundedandtwists +Theorem 2.18. Let k be a field of characteristic 0. +Let X be a geometrically uniruled +geometrically integral smooth projective k-variety and let L be a big and nef Q-Cartier divisor +on X. Denote by {pi : Ui → Wi} the finite set of families equipped with maps si : Ui → X +and by V the closed subset of Construction 2.17. There is a closed set R ⊂ X and a finite +set of smooth projective varieties Yi,j equipped with dominant morphisms ri,j : Yi,j → Ti,j +14 + +with connected fibers and dominant morphisms hi,j : Yi,j → Ui forming commuting diagrams +Yi,j +hi,j +� +ri,j +� +Ui +pi +� +Ti,j +ti,j +� Wi +that satisfy the following properties: +(1) each map hi,j is generically finite and fi,j = si ◦ hi,j is not birational; +(2) ti,j is a finite Galois cover and Ti,j is normal; +(3) Bir(Yi,j/Ui) = Aut(Yi,j/Ui); +(4) every twist Y σ +i,j of Yi,j over Ui admits a morphism rσ +i,j : Y σ +i,j → T σ +i,j which is a twist of +ri,j; +(5) we have a(Yi,j, f ∗ +i,jL) = a(X, L); +(6) suppose that Y is a geometrically integral smooth projective variety and that f : Y → +X is a morphism that is generically finite onto its image but not birational such that +a(Y, f ∗L) ≥ a(X, L). Suppose furthermore that y ∈ Y (k) satisfies f(y) ̸⊂ R. Then: +(a) there are indices i, j and a twist hσ +i,j : Y σ +i,j → Ui of hi,j such that f(y) ∈ +si(hσ +i,j(Y σ +i,j(k))), and +(b) if (Y, f ∗L) is adjoint rigid then furthermore f factors rationally through hσ +i,j and +f maps Y birationally to a fiber of rσ +i,j. +Proof. Consider the families pi : Ui → Wi. By applying [LST22, Lemma 7.3] there exists a +Zariski open subset W ◦ +i such that each map U◦ +i → W ◦ +i is a good fibration in the sense of +[LST22, Definition 8.2]. +We may then apply [LST22, Lemma 8.3] to each Ui equipped with s∗ +i L. The result is +a closed set Di ⊂ Ui and a finite set of smooth projective varieties Yi,j equipped with +morphisms ri,j : Yi,j → Ti,j, hi,j : Yi,j → Ui, and ti,j : Ti,j → Wi that have the following +properties. First, since ri,j is constructed as the Stein factorization of Yi,j → Ui → Wi we +see that every twist Y σ +i,j of Yi,j over Ui admits a morphism rσ +i,j that is a twist of ri,j. Second, +suppose that q : Y → Ui is a generically finite morphism such that a(Y, q∗s∗ +i L) = a(X, L) +and a general fiber of the Iitaka fibration for KY +a(Y, q∗s∗ +i L)q∗s∗ +i L maps generically finitely +onto a general fiber of Ui → Wi. Suppose furthermore that y ∈ Y (k) is a rational point such +that q(y) is not contained in Di. Then there is some index j and a twist hσ +i,j : Y σ +i,j → Ui +such that q(y) is in hσ +i,j(Y σ +i,j(k)). Furthermore every general fiber of the canonical fibration +for KY + a(Y, q∗s∗ +i L)q∗s∗ +i L is birational to a fiber of rσ +i,j. +By [LST22, Theorem 4.18] there is a proper closed subset V ′ ⊂ X which is the union of +all subvarieties Y satisfying a(Y, L|Y ) > a(X, L). We let R be the union of V and V ′ with +∪isi(Di). We also enlarge R by adding the images of singular fibers of ri,j. +We verify each property. (1), (2), (3) follow from [LST22, Lemma 8.3]. We have already +verified (4). (5) follows from [LST22, Lemma 8.3 (i) and (ii)]. +Now we verify (6). Suppose f : Y → X is as in the statement. By assumption f(Y ) ̸⊂ V ′. +In particular this implies that a(Y, f ∗L) = a(f(Y ), L|f(Y )) = a(X, L). Let F be the closure of +a general fiber of the canonical fibration for KY + a(Y, f ∗L)f ∗L so that (F, f ∗L|F) is adjoint +rigid. Then by [LST22, Lemma 4.9] we see that (f(F), L|f(F )) is also adjoint rigid and thus +is birational to a fiber of some map pi : Ui → Wi. This induces a rational map T ��� Hilb(X) +15 + +where T is the base of the canonical fibration of KY + a(Y, f ∗L)f ∗L. Since Ui is birational +to the universal family over Wi, we also obtain a rational map Y ��� Ui. Since the desired +statement only depends on the birational equivalence class of f : Y → X (and not the choice +of birational model of Y ), after blowing up Y we may suppose that Y admits a morphism to +Ui such that the general fiber of the canonical fibration on Y maps generically finitely onto +a fiber of the map Ui → Wi. Then the desired containment of rational points follows from +[LST22, Lemma 8.3 (vi)] as described above. When (Y, f ∗L) is adjoint rigid, the factoring +statement also follows from [LST22, Lemma 8.3 (vi)]. +□ +3. Sections of good fibrations +sect:goodfibration +Definition 3.1. We say that a morphism π : Z → B is a good fibration if: +(1) Z is a smooth projective variety, +(2) B is a smooth projective curve, and +(3) π is flat and has connected fibers. +Suppose that π : Z → B is a good fibration. We let Sec(Z/B) denote the open subset +of the Hilbert scheme parametrizing sections of π. +If M ⊂ Sec(Z/B) is an irreducible +component, the expected dimension of M is +χ(TZ/B|C) = −KZ/B · C + (dim Z − 1)(1 − g(B)) +where C is any section parametrized by M. The expected dimension is a lower bound for +the dimension of M. An upper bound is +dim H0(B, TZ/B|C) = −KZ/B · C + (dim Z − 1)(1 − g(B)) + dim H1(B, TZ/B|C). +One of the basic facts about sections of a good fibration is the Northcott property, which +in our setting should be interpreted in the following way. +coro:boundednegativity +Corollary 3.2 ([LT21a, Lemma 2.2]). Let π : Z → B be a good fibration and let L be a +generically relatively ample Q-Cartier divisor on Z. If we fix a constant Q, then there are +only finitely many components of Sec(Z/B) parametrizing sections C satisfying L · C ≤ Q. +3.1. Relatively free sections and general points. Suppose π : Z → B is a good fi- +bration. Fix points q1, . . . , qm ∈ Z which are contained in different fibers of π. +We let +Sec(Z/B, q1, . . . , qm) denote the sublocus of Sec(Z/B) parametrizing sections containing the +points q1, . . . , qm. In particular, if M ⊂ Sec(Z/B, q1, . . . , qm) is an irreducible component +then the expected dimension of M is +χ(TZ/B|C(−q1 − . . . − qm)) = −KZ/B · C + (dim Z − 1)(1 − g(B)) − m(dim(Z) − 1) +where C is any section parametrized by M. The expected dimension is a lower bound for +the dimension of M. An upper bound is +dim H0(B, TZ/B|C(−q1 − . . . − qm)) = −KZ/B · C + (dim Z − 1)(1 − g(B)) − m(dim(Z) − 1) ++ dim H1(C, TZ/B|C(−q1 − . . . − qm)). +The following result describes how the normal bundle of a section C controls the number of +general points contained in deformations of C. +16 + +prop:deffixpoints +Proposition 3.3 ([LT21a] Proposition 3.3). Let π : Z → B be a good fibration. Fix points +q1, . . . , qm of Z contained in different fibers of π. Let M denote an irreducible component of +Sec(Z/B, q1, . . . , qm) and suppose that the sections parametrized by M dominate Z. Then +for a general section C parametrized by M and for a general point p ∈ B we have that +H0(C, TZ/B|C(−q1 − . . . − qm)) → TZ/B|C|p is surjective. +Conversely, suppose we fix a section C. +Suppose that q1, . . . , qm are distinct points of +C such that H1(C, TZ/B|C(−q1 − . . . − qm)) = 0. Let M ⊂ Sec(Z/B, q1, . . . , qm) denote +the unique irreducible component containing C. If for a general point p ∈ C we have that +H0(C, TZ/B|C(−q1 − . . . − qm)) → TZ/B|C|p is surjective, then M parametrizes a dominant +family of sections on Z. +coro:domfamilyexpdim +Corollary 3.4. Let π : Z → B be a good fibration. +Suppose that M is an irreducible +component of Sec(Z/B) parametrizing a dominant family of sections. Letting C denote a +general section parametrized by M, we have +−KZ/B · C + (dim Z − 1)(1 − g(B)) ≤ dim(M) ≤ −KZ/B · C + dim Z − 1. +Proof. By Proposition 3.3 the bundle TZ/B|C is generically globally generated. Thus we have +h1(C, TZ/B|C) ≤ g(B)(dim(Z) − 1) by Lemma 2.8. The desired statement follows. +□ +Recall that a section C is relatively free if H1(C, TZ/B|C) = 0 and TZ/B|C is globally +generated. Proposition 3.3 shows that any relatively free section deforms in a dominant +family on Z. +It is easiest to work with relatively free sections when we impose further +conditions on the positivity of the terms of the Harder-Narasimhan filtration of TZ/B|C. +Definition 3.5. Let π : Z → B be a good fibration. We say that a section C is HN-free if +µmin(TZ/B|C) ≥ 2g(B). +The following result summarizes the key properties of HN-free sections. +lemma:hnfreecurves +Lemma 3.6. Let π : Z → B be a good fibration. Suppose that C is a HN-free section of π. +Then: +(1) H1(C, TZ/B|C) = 0 and for any closed point p ∈ B we have H1(C, TZ/B|C(−p)) = 0. +(2) TZ/B|C is globally generated. +(3) C is relatively free. +(4) Let b = µmin(TZ/B|C). Then deformations of C can pass through at least b−2g(B)+1 +general points of Z. +Proof. (1) and (2) follow from Corollary 2.7 and (3) follows from (1) and (2). To see (4) we +apply Corollary 2.7 to see that for any points q1, . . . , qm on C the twist TZ/B|C(−q1−. . .−qm) +is globally generated and has vanishing H1 so long as m ≤ b−2g(B). The desired statement +follows from Proposition 3.3. +□ +The next proposition shows that sections through sufficiently many general points must +be HN-free. +prop:generalimplieshnfree +Proposition 3.7. Let π : Z → B be a good fibration. Let M be an irreducible component of +Sec(Z/B). Suppose that the sections parametrized by M pass through ≥ 2g(B) + 1 general +points of Z. Then the general section parametrized by M is HN-free. +17 + +Proof. If we fix a general section C parametrized by M and a set of 2g(B) general points +{qi}2g(B) +i=1 +on C then Proposition 3.3 shows that TZ/B|C(−q1 − . . . − q2g(B)) is generically +globally generated. Lemma 2.5 shows that +µmin(TZ/B|C(−q1 − . . . − q2g(B))) ≥ 0 +and we conclude that µmin(TZ/B|C) ≥ 2g(B). +□ +We will also need to know the following avoidance property of HN-free sections. +lemm:HNavoidscodim2 +Lemma 3.8. Let π : Z → B be a good fibration. Suppose that C is a HN-free section of +π. Then for any codimension 2 closed subset W ⊂ Z there is a deformation of C which is +HN-free and avoids W. +Proof. Assume for a contradiction that every deformation of C meets with W. Then there +exists p ∈ W such that a general deformation of C containing p is HN-free and the dimension +of the family parametrizing such deformations is greater than or equal to −KZ/B · C + +(dim Z − 1)(1 − g(B)) − dim W. +Note that this is larger than the expected dimension +−KZ/B · C − (dim Z − 1)g(B) for the parameter space of sections through p. +But this +contradicts with Lemma 3.6 which shows that H1(C, TZ/B|C(−p)) = 0. +□ +4. Grauert-Mulich +sect:gm +For a good fibration π : Z → B the deformation theory of a section C is controlled by +the Harder-Narasimhan filtration of the restriction TZ/B|C. In this section, we show that +(under certain hypotheses) the Harder-Narasimhan filtration of TZ/B|C is “approximately” +the restriction of the [C]-Harder-Narasimhan filtration of TZ/B. Due to the similarity to the +Grauert-Mulich theorem ([GM75]) describing the restriction of semistable bundles to lines +in Pn, we will refer to such statements as “Grauert-Mulich” results. The material in this +section is motivated by [PRT20, Section 3] and by [OSS80, Chapter II, Section 2]. +Suppose that Z is a smooth projective variety and W is a variety parametrizing a family +of maps s : C → Z. Let E be a torsion-free sheaf on Z and let F be a term in the relative +Harder-Narasimhan filtration of E pulled back to the universal family over an open subset +of W. We would like to determine when F is the pullback of a sheaf FZ from Z. In this +case we can expect FZ to be a term in the Harder-Narasimhan filtration of E with respect +to the numerical class of the curves s∗C. +In Section 4.1 we prove a general criterion for determining when a torsion-free sheaf on +a variety is isomorphic to a pullback. We apply this result in Section 4.2 to show that our +ability to descend F to Z is controlled by the comparison between several invariants of the +normal sheaf of s and the “gaps” in slope between F and adjacent terms of the relative +Harder-Narasimhan filtration. Finally, in Section 4.3 we show that for sections of a good +fibration π : Z → B these invariants of the normal sheaf are bounded by functions of dim(Z) +and g(B). +sect:descent +4.1. Descending sheaves. The first step is to develop a criterion for identifying when a +sheaf is pulled back from the base of a morphism. +lemm:rigidity +Lemma 4.1. Suppose we have morphisms f : U → V , g : U → G satisfying the following +properties: +18 + +(1) U, V, G are smooth varieties. +(2) f is dominant. +(3) Every fiber of f is contracted to a point by g. +Then there is some open set V ◦ ⊂ V such that g|f−1(V ◦) factors through f|f−1(V ◦). +Proof. Consider the induced map (f, g) : U → V × G and let Γ denote the closure of the +image. Note that Γ is still irreducible. For a general point v ∈ V there is a unique point in +(f, g)(U)∩π−1 +1 (v). Taking closures, we see that the general fiber of Γ → V is set-theoretically +a single point. Since we are in characteristic 0, by generic smoothness we see that the general +fiber is scheme-theoretically a single point. We deduce that Γ → V is birational. If we let +V ◦ denote an open subset where Γ → V is an isomorphism, then the desired statement +follows. +□ +Recall that for a coherent sheaf F on a variety we denote by Ftors the torsion subsheaf +and by Ftf the quotient of F by its torsion subsheaf. +lemm:descent +Lemma 4.2 (Descent Lemma). Let U and Z be smooth varieties with a dominant flat mor- +phism ev : U → Z such that the general fiber of ev is connected. Let E be a locally free sheaf +on Z. Suppose that ev∗E → Q is a surjection onto a locally free sheaf and let S denote the +kernel. If +Hom(Hom(Q, S), (ΩU/Z)tf) = 0 +then there is a subsheaf SZ ⊂ E such that ev∗SZ = S as subsheaves of ev∗E. +Proof. Note that the surjection ev∗E → Q corresponds to a map φ : U → G(E, k) = G, +where k is the rank of Q and G(E, k) is the relative Grassmannian of rank k quotients. The +map φ is a map of Z-schemes. We first show that after replacing U by an open subset the +map φ factors through ev : U → Z. The map φ induces a map +(dφ)∗ : φ∗ΩG/Z → ΩU/Z. +Since φ∗ΩG/Z = Hom(Q, S), our assumption implies that (dφ)∗ is the 0 map on the com- +plement of the support of the torsion subsheaf of ΩU/Z. We denote this open subset by +�U. +Fix a general point z ∈ Z and consider the map of fibers �Uz → Gz. Using compatibility +of cotangent bundles with base change, we see that φ|∗ +�UzΩGz → Ω �Uz must be 0. Dually, the +map T �Uz → φ|∗ +�UzTGz is zero, and thus if we precompose by the the locally closed embedding +( �Uz,red)smooth → �Uz,red → �Uz the induced map on tangent sheaves still vanishes. By generic +smoothness, it follows that φ must contract ( �Uz,red)smooth to a point; taking closures, it +also contracts each fiber �Uz to a point. There is an open subset Z◦ ⊂ Z with preimage +�U◦ = ev−1(Z◦) ∩ �U such that ev| �U◦ has connected fibers. By Lemma 4.1, after possibly +shrinking Z◦ and �U◦ we can ensure that φ| �U◦ factors through ev| �U◦. Hence, S|U◦ must be +the pullback of a locally free sheaf R ⊂ E|Z◦ on Z◦. +Let SZ denote the unique torsion-free saturated subsheaf of E whose restriction to Z◦ is +R. Since ev is flat, ev∗SZ is a torsion-free saturated subsheaf of ev∗E whose restriction to +U◦ agrees with S|U◦. This implies that ev∗SZ = S. +□ +19 + +sect:slopecomputation +4.2. Slope computations. The key to using the descent lemma is to understand homo- +morphisms into ΩU/Z. When U is a family of curves mapping to Z, we will control the +existence of such homomorphisms using slope calculations. The next step is to show that in +this situation the slope of ΩU/Z is controlled by Lazarsfeld bundles. +Definition 4.3. Let Y be a variety and E be a globally generated vector bundle on Y . The +Lazarsfeld bundle ME is the kernel of the evaluation map OY ⊗ H0(Y, E) → E. +Given a morphism s : C → Z, we will denote by Ns the normal sheaf of s, i.e. the cokernel +of TC → s∗TZ. We will also let Mg,0(Z) denote the Kontsevich moduli stack of maps from +genus g curves to Z. +lem-LazMukHoms +Lemma 4.4. Let Z be a smooth projective variety, let W be a variety equipped with a gener- +ically finite morphism W → Mg,0(Z) and let p : UW → W be the universal family over W +equipped with the evaluation map evW : UW → Z. Suppose that a general fiber of p is smooth +and irreducible and that evW is dominant. +Let C denote a general fiber of UW → W equipped with the induced morphism s : C → Z. +Let t be the length of the torsion part of Ns, let G be the subsheaf of (Ns)tf generated by global +sections, and let V be the tangent space to W at s. Let q be the dimension of the cokernel +of the composition +V → TMg,0(Z),s = H0(C, Ns) → H0(C, (Ns)tf). +Then +µmax((ΩUW /Z|C)tf) ≤ (q + 1)µmax(M∨ +G ) + t. +Proof. Since the conclusion only involves a general curve, we may shrink W and thus assume +that W is smooth. After perhaps shrinking W further we may assume that p is smooth, and +thus UW is a smooth variety. We denote by h the map (p, evW) : UW → W × Z. +Fix s : C → Z as in the statement of the lemma. Let K1 denote the kernel of s∗ΩZ → ΩC +and let K2 denote the kernel of h∗ΩW ×Z → ΩUW . Note that K1 is isomorphic to the dual of +(Ns)tf. We claim that the following diagram has exact rows and columns: +0 +� +0 +� +Odim(W ) +C += +� +� +Odim(W ) +C +� +0 +� K2|C +� +� +h∗ΩW ×Z|C +� +� +Ωim +UW |C +� +� +0 +0 +� K1 +� s∗ΩZ +� +� +Ωim +C +� +� +0 +0 +0 +where Ωim +UW is the image of h∗ΩW ×Z → ΩUW and Ωim +C +is the image of s∗ΩZ → ΩC. The +bottom row is exact by definition and the middle row is the restriction of an exact sequence +20 + +to a general fiber of p and thus remains exact. The middle column is exact since C is vertical +for the map p : UW → W. By comparing the rightmost column against the middle it is clear +that Ωim +UW |C maps surjectively onto Ωim +C and that the kernel contains p∗ΩW|C ∼= Odim(W ) +C +. On +the other hand the kernel must be contained in the kernel of ΩUW |C → ΩC which is also +isomorphic to p∗ΩW|C. So the rightmost column is exact. Finally, by the nine lemma we +deduce that K2|C ∼= K1. +Let K3 denote the kernel of the map p∗ΩW → ΩUW/Z and let Ωim +UW/Z denote the image of +this map. We can make a further comparison via the following diagram. +0 +� +0 +� +s∗ΩZ += +� +� +s∗ΩZ +� +0 +� K2|C +� +� +h∗ΩW ×Z|C +� +� +Ωim +UW |C +� +� +0 +0 +� K3|C +� p∗ΩW|C +� +� +Ωim +UW/Z|C +� +� +0 +0 +0 +We claim that the rows and columns are exact. Since C is general, an exact sequence of +torsion-free sheaves on UW will remain exact upon restriction. +Thus it suffices to show +that the rightmost column is exact. Since the map evW : UW → Z is dominant, the map +ev∗ +WΩZ → ΩUW is generically injective. Since ΩZ is locally free the map must be injective. +Restricting to the curve C, we see that s∗ΩZ → ΩUW |C is injective and it is clear that its +image is contained in Ωim +UW |C. This shows the rightmost column is left-exact. Furthermore, +we see that the composed map h∗ΩW ×Z → p∗ΩW → Ωim +UW/Z is surjective, showing that the +map Ωim +UW → Ωim +UW/Z is also surjective. Finally, the exact sequence +0 → ev∗ +WΩZ → ΩUW → ΩUW /Z → 0 +implies that +0 → s∗ΩZ → ΩUW |C → ΩUW /Z|C → 0 +is exact. Thus the rightmost column must be exact at the middle. By the nine-lemma, we +conclude that K3|C ∼= K2|C ∼= K1. +Recall that V denotes the tangent space to W at s. Let ζ denote the composition +V → H0(C, Ns) → H0(C, (Ns)tf). +Then the map K1 ∼= K3|C → p∗ΩW|C is the dual of the composition +V ⊗ OC +ζ−→ H0(C, (Ns)tf) ⊗ OC → (Ns)tf. +Let G denote the subsheaf of (Ns)tf that is generated by its global sections, so we have an +exact sequence of locally free sheaves +0 → MG → H0(C, (Ns)tf) ⊗ OC → G → 0. +21 + +Since C deforms in a dominant family, Ns is generically globally generated and thus the +inclusion G → (Ns)tf is generically surjective. Taking duals, we see that G∨ is the saturation +of K1 inside of H0(C, (Ns)tf)∨ ⊗ OC. In other words, if we let S denote the cokernel of the +map K1 → H0(C, (Ns)tf)∨ ⊗ OC then the torsion free part of S is isomorphic to M∨ +G . +Consider the following diagram of short exact sequences +0 +� K1 +� += +� +H0(C, (Ns)tf)∨ ⊗ OC +� +ζ∨ +� +S +� +h +� +0 +0 +� K1 +� p∗ΩW|C +� Ωim +UW /Z|C +� 0 +By the snake lemma we obtain an exact sequence +0 → O⊕q +C → S → Ωim +UW /Z|C → O⊕e +C → 0 +where q, e are respectively the dimensions of the cokernel and kernel of ζ. +Recall that Stf ∼= M∨ +G and note that the torsion part of S injects into the torsion part +of Ωim +UW /Z|C. We will denote by R the saturation of O⊕q +C +in M∨ +G so that we obtain an exact +sequence +0 → R → M∨ +G → (Ωim +UW /Z|C)tf → O⊕e +C → 0 +Let F denote the maximal destabilizing subsheaf of (Ωim +UW /Z|C)tf, let F ′ = F ∩ im(M∨ +G ), and +let Q denote the preimage of F ′ in M∨ +G . Our next goal is to prove an upper bound on µ(F). +First suppose that µ(F) > 0. Then the image of F in O⊕e +C is 0 and so F = F ′. This means +we have an exact sequence +0 → R → Q → F → 0 +and thus +µ(F) = deg(Q) − deg(R) +rk(Q) − rk(R) +≤ +deg(Q) +rk(Q) − q +≤ (q + 1)µ(Q) +where the final line follows from the elementary inequality +1 +b−a ≤ a+1 +b +when b − 1 ≥ a ≥ 0. +Thus in this case we see that µmax(Ωim +UW /Z|C) ≤ (q + 1)µmax(M∨ +G ). If µ(F) ≤ 0, then the +same inequality still holds: the right-hand side is a non-negative number since M∨ +G is globally +generated. +22 + +Consider the following diagram +0 +0 +0 +0 +� Ωim +C +� +� +ΩC +� +� +T +� +� +0 +0 +� s∗ΩZ +� +� +ΩUW |C +� +� +ΩUW /Z|C +� +� +0 +0 +� K3|C +� +� +p∗ΩW|C +� +� +Ωim +UW /Z|C +� +� +0 +0 +� +0 +� +0 +� +where Ωim +C is the image and T is the cokernel of s∗ΩZ → ΩC. Every row and column is exact +except possibly the rightmost column, thus by nine-lemma we see the rightmost column is +also exact. We have +len(T ) = len(cok(Ωim +C → ΩC)) = len(cok(TC → T sat +C )) = t +where T sat +C +is the saturation of TC in s∗TZ. It follows that +µmax((ΩUW /Z|C)tf) ≤ µmax((Ωim +UW /Z|C)tf) + t +≤ (q + 1)µmax(M∨ +G ) + t. +□ +By combining this analysis with the descent theorem, we obtain: +theo:gmtheorem +Theorem 4.5. Let Z be a smooth projective variety. Let W be a variety equipped with a +generically finite morphism W → Mg,0(Z) and let p : UW → W denote the universal family +over W with evaluation map evW : UW → Z. Assume that a general map parametrized +by W has smooth irreducible domain, that evW is dominant, that the general fiber of the +composition of the normalization map for UW with evW is connected, and that a general fiber +of p is contained in the locus where evW is flat. +Suppose that E is a torsion-free sheaf on Z that is semistable with respect to a general +curve s : C → Z parametrized by W. Write +0 = F0 ⊂ F1 ⊂ F2 ⊂ . . . ⊂ Fk = s∗E +for the Harder-Narasimhan filtration of s∗E. Let t be the length of the torsion part of Ns, let +G be the subsheaf of (Ns)tf generated by global sections, and let V be the tangent space to W +at s. Let q be the dimension of the cokernel of the composition +V → TMg,0(Z),s = H0(C, Ns) → H0(C, (Ns)tf). +Then for every index 1 ≤ i ≤ k − 1 we have +µ(Fi/Fi−1) − µ(Fi+1/Fi) ≤ (q + 1)µmax(M∨ +G ) + t. +23 + +Note that by the flatness assumption we may ensure that the image of a general map s +parametrized by W will avoid any codimension 2 locus on Z. In particular this implies that +s∗E will be a locally free sheaf. +Proof. Since we are assuming that the general fiber of the composition of the normalization +map for UW with evW is connected, if we replace W by a smaller open subset then the general +fiber of the evaluation map will still have connected fibers. Thus to prove the statement, +we may replace W by a smaller open subset. (For ease of notation we continue to call the +smaller subset W and its p-preimage by UW.) Thus we may suppose that W and UW are +smooth and that evW|UW is flat. After possibly shrinking W further, by [HL97, Theorem +2.3.2] we may suppose that for every index 1 ≤ i ≤ k − 1 there is a torsion-free sheaf +Si ⊂ ev∗ +WE obtained from the relative Harder-Narasimhan filtration of ev∗ +WE over W such +that Si|C ∼= Fi for every fiber C over W. Since torsion-free sheaves are locally free on the +complement of a codimension 2 subset, after perhaps shrinking W again we may suppose +that each Si is locally free. +Suppose for a contradiction that there is an index i such that +µ(Fi/Fi−1) − µ(Fi+1/Fi) > (q + 1)µmax(M∨ +G ) + t. +If there were a non-zero homomorphism Hom(ev∗ +W E/Si, Si) → (ΩUW /Z)tf, then its restriction +to a general fiber C of p would yield a map that is non-zero on the generic point of C, and +thus would induce a non-zero map Hom(ev∗ +WE/Si|C, Si|C) → ((ΩUW /Z)|C)tf. But then +µmin(Hom(ev∗ +WE/Si|C, Si|C)) = µmin(Si|C) − µmax(ev∗ +WE/Si|C) +> (q + 1)µmax(M∨ +G ) + t +≥ µmax((ΩUW /Z|C)tf) +where the last inequality follows from Lemma 4.4. We conclude that there is no non-zero +homomorphism Hom(ev∗ +WE/Si, Si) → (ΩUW /Z)tf. By Lemma 4.2, we see that there is a sheaf +SZ on Z such that ev∗ +WSZ = Si. But such a sheaf would destabilize E: we have +µ[s∗C](SZ) = µ(Si|C) > µ(ev∗ +WE|C) = µ[s∗C](E) +where the equalities follow from the flatness of the evaluation map and the inequality follows +from (2.1). This gives a contradiction and we conclude the desired inequalities for every +i. +□ +It will be helpful to have a version of the previous theorem that holds for non-semistable +sheaves as well. The next theorem controls the difference between the Harder-Narasimhan +filtration of E|C and the restriction to C of the [C]-Harder-Narasimhan filtration of E. It is +a formal consequence of Theorem 4.5. +coro:hnfversion +Corollary 4.6. Let Z be a smooth projective variety and let E be a torsion free sheaf on Z +of rank r. Let W be a variety equipped with a generically finite morphism W → Mg,0(Z) and +let p : UW → W denote the universal family over W with evaluation map evW : UW → Z. +Assume that a general map parametrized by W has smooth irreducible domain, that evW is +dominant, that the general fiber of the composition of the normalization map for UW with +evW is connected, and that a general fiber of p is contained in the locus where evW is flat. +Let C denote a general fiber of UW → W equipped with the induced morphism s : C → Z. +Let t be the length of the torsion part of Ns, let G be the subsheaf of (Ns)tf generated by global +24 + +sections, and let V be the tangent space to W at s. Let q be the dimension of the cokernel +of the composition +V → TMg,0(Z),s = H0(C, Ns) → H0(C, (Ns)tf). +Let ∥ − ∥ denote the sup norm on Q⊕r. Then we have +∥ SPZ,[C](E) − SPC(s∗E)∥ ≤ 1 +2 +� +(q + 1)µmax(M∨ +G ) + t +� +rk(E). +Proof. Set δ = (q + 1)µmax(M∨ +G ) + t. Given two r-tuples of real numbers a•, b•, we write +a• ≥ b• if for every index i = 1, 2, . . . , r we have ai ≥ bi. +Write 0 = F0 ⊂ F1 ⊂ . . . ⊂ Fk = E for the [C]-Harder-Narasimhan filtration of E. We +prove this statement by induction on the length k. We start with the base case k = 1. By +Theorem 4.5 the slopes of successive quotients of the Harder-Narasimhan filtration of s∗E +differ by at most δ. We conclude the desired statement by Lemma 4.7 (where the pairs (ai, bi) +record the degrees and ranks of the various successive quotients in the Harder-Narasimhan +filtration of s∗E). +For the induction step, write 0 = G0 ⊂ G1 ⊂ . . . ⊂ Gt = s∗E for the Harder-Narasimhan +filtration of s∗E. For convenience we define three tuples: +• c• = (c1, c2, . . . , cr) to be SPC(s∗E). +• g• = (g1, g2, . . . , gr) defined as follows: we start with the tuple SPC(s∗Fk−1) (of length +rk(Fk−1)) and then replace every entry that is below µ[C](E/Fk−1) + δ +2 rk(E) by this +number. We then append entries equal to µ[C](E/Fk−1) + δ +2 rk(E) to the end so that +g• has length equal to rk(E). +• h• = (h1, h2, . . . , hr) defined as follows: we start with the tuple SPC(s∗(E/F1)) (of +length rk(E/F1)) and then replace every entry that is above µ[C](F1) − δ +2 rk(E) by +this number. We then insert entries equal to µ[C](F1) − δ +2 rk(E) at the beginning so +that h• has length equal to rk(E). +Since C is a general member of a flat family every term Fi is locally free along C. Thus we +have an exact sequence +0 → s∗Fk−1 → s∗E → s∗(E/Fk−1) → 0. +Fix an index 1 ≤ j ≤ t and suppose that µmin(Gj) > µ[C](E/Fk−1) + δ +2 rk(E). By the base +case of our result we see that µmin(Gj) > µmax(s∗(E/Fk−1)) so that Gj ⊂ s∗Fk−1. Thus we +have g• ≥ c•. +Similarly, consider the exact sequence +0 → s∗F1 → s∗E → s∗(E/F1) → 0. +Fix an index 1 ≤ j ≤ t and suppose that µmax(s∗E/Gj) < µ[C](F1) − δ +2 rk(E). By the base +case of our result we see that µmax(s∗E/Gj) < µmin(s∗F1) so that there can be no non-zero +homomorphism from s∗F1 to s∗E/Gj. We conclude that s∗F1 ⊂ Gj and thus s∗E/Gj is a +quotient of s∗(E/F1). Thus we have c• ≥ h•. +Altogether, suppose we define +• q+ +• to be SPZ,C(E) + ( δ +2 rk(E), δ +2 rk(E), . . . , δ +2 rk(E)). +• q− +• to be SPZ,C(E) − ( δ +2 rk(E), δ +2 rk(E), . . . , δ +2 rk(E)). +25 + +The induction assumption for Fk−1 shows that q+ +• ≥ g• and the induction assumption for +E/F1 shows that h• ≥ q− +• . By the argument above we conclude that +q+ +• ≥ g• ≥ c• ≥ h• ≥ q− +• +yielding the desired result. +□ +lemm:weightedAverage +Lemma 4.7. Let {(ai, bi)}k +i=1 be pairs of integers with bi > 0 such that the fractions ai +bi are +nonincreasing as i gets larger. Let m denote the mediant of fractions ai +bi and let δ denote the +maximum of the successive differences ai +bi − ai+1 +bi+1 . Then for every i we have +���� +ai +bi +− m +���� ≤ δ +2 +r +� +i=1 +bi. +Proof. It suffices to prove the inequality for the extremal fractions a1 +b1 , ak +bk . Replacing each +ai by −ai, we see that the latter case is implied by the former. So it suffices to prove the +statement for a1 +b1 . Since a1 +b1 − ai +bi ≤ (i − 1)δ, we have +k +� +i=1 +(a1bi − aib1) ≤ +k +� +i=1 +b1biδ(i − 1) +≤ b1δ +� +k +� +i=1 +bi +� i−1 +� +j=1 +bj +�� += b1δ +2 +�� +i̸=j +bibj +� +≤ b1δ +2 +� +k +� +i=1 +bi +�2 +Dividing by b1 +��k +i=1 bi +� +gives us the result. +□ +sect:genusanddimbounds +4.3. Genus and dimension bounds. Note that the Hilbert scheme of sections admits an +embedding into the stack Mg,0(Z). To apply Corollary 4.6 to moduli spaces of sections one +needs to be able to bound the quantities q, t, and µmax(M∨ +G ) appearing in the statement. We +will show how to bound these quantities for sections of a good fibration π : Z → B using the +genus of B and the dimension of Z. Note that since sections are always smooth the quantity +t = 0. +We first discuss the slope of Lazarsfeld bundles. These can be bounded using only the +genus of B using the following result of Butler. +theo:butler +Theorem 4.8 ([But94]). Let E be a globally generated locally free sheaf on a curve C of +genus g and let ME be its Lazarsfeld bundle. +(1) If µmin(E) ≥ 2g then µmin(ME) ≥ −2. +(2) If µmin(E) < 2g then µmin(ME) ≥ −2g rk(E) − 2. +Proof. Statement (1) follows from [But94, 1.3 Corollary] except when C is a rational curve. +When C is rational ME is a direct sum of O(−1)’s (see for example [PRT20, Lemma 3.10]) +and thus the statement still holds. +26 + +For statement (2), first suppose that µmax(E) ≥ 2g. Let F denote the maximal destabi- +lizing subsheaf of E. Our degree assumption implies that F is globally generated and that +h1(C, F) = 0. By the nine lemma we obtain an exact sequence +0 → MF → ME → ME/F → 0. +This implies that µmin(ME) ≥ min{µmin(MF), µmin(ME/F)}. By (1) the first quantity is at +least −2. Arguing by induction on the rank we reduce to the case when µmax(E) < 2g. +When µmax(E) < 2g, [But94, 1.5 Proposition] proves a statement stronger than (2) except +in two cases. The first is when g = 1. In this case, since ME is a subsheaf of H0(C, E) ⊗ OC +every subsheaf of ME has non-positive slope. Since deg(ME) = − deg(E), we conclude that +every quotient of ME has degree at least − deg(E), which implies that it has slope at least +− deg(E). +Since we are in the case where deg(E)/ rk(E) < 2 we conclude µmin(ME) ≥ +−2 rk(E). The second is when g ≥ 2 and E has a trivial summand. Write E = E′ ⊕ O⊕k +C . +Note that ME = ME′. Since we still have µmax(E′) < 2g we can apply [But94, 1.5 Proposition] +to E′ to obtain the desired lower bound. +□ +Next we discuss the quantity q. +lemm:sbound +Lemma 4.9. Let π : Z → B be a good fibration. +Suppose that M ⊂ Sec(Z/B) is an +irreducible component parametrizing a dominant family of sections on Z and let W = Mred. +For a general section C parametrized by M let V ⊂ H0(B, TZ/B|C) denote the tangent space +to W at C. Then the codimension of V in H0(B, TZ/B|C) is at most g(B)(dim(Z) − 1). +Proof. We have h0(B, TZ/B|C)−dim(V ) ≤ h0(B, TZ/B|C)−dim(M) and Corollary 3.4 shows +that this latter quantity is bounded above by g(B)(dim(Z) − 1). +□ +Putting these results together, we obtain a version of the Grauert-Mulich theorem for +sections. +theo:hnfforsections +Theorem 4.10. Let π : Z → B be a good fibration and let E be a torsion-free sheaf on +Z. Let M be an irreducible component of Sec(Z/B) parameterizing a dominant family of +sections of π and let p : Uν → Mred be the normalization of the universal family over Mred +with evaluation map ev : Uν → Z. Assume that ev has connected fibers. +Let C be a general section parametrized by M. Then: +(1) Suppose there is an open subset M◦ +red ⊂ Mred such that if we define Uν,◦ = p−1M◦ +red +then ev|Uν,◦ is flat. Then we have +∥ SPZ,[C](E) − SPC(E|C)∥ ≤ (g(B) dim(Z) − g(B) + 1)2 rk(E) +where ∥ − ∥ denotes the sup norm on Q⊕r. +(2) Suppose that the general curve C is HN-free. Then we have +∥ SPZ,[C](E) − SPC(E|C)∥ ≤ rk(E) +where ∥ − ∥ denotes the sup norm on Q⊕r. +Proof. (1) Theorem 4.8 shows that µmax(M∨ +G ) ≤ 2g(B)(dim(Z) − 1) + 2. By Lemma 4.9 we +have q ≤ g(B)(dim(Z) − 1). We then apply Corollary 4.6 with t = 0. +(2) When C is HN-free then M is generically smooth and TZ/B|C′ is globally generated for +a general deformation C′ of C. Furthermore, there is an open subset of M over which the +evaluation map is smooth and thus flat. Theorem 4.8 shows that µmax(M∨ +G ) ≤ 2. We then +apply Corollary 4.6 with q = t = 0. +□ +27 + +5. Sections through general points +sect:genpoints +Suppose that π : Z → B is a good fibration and M is an irreducible component of +Sec(Z/B) parametrizing a dominant family of sections. Let C be a general section parametrized +by M. +By Proposition 3.3 we can identify a lower bound on the slopes in the Harder- +Narasimhan filtration of TZ/B|C by computing how many general points of Z we can impose +on the sections parametrized by M. +Even when C has very large anticanonical degree, deformations of C do not need to +go through many general points of Z. In this section we construct a dominant family of +subvarieties Y ⊂ Z such that deformations of C go through many general points in Y. +Results of this type were used earlier in [She12], [LT22], and [LT21a]. +Here is the idea behind the construction. Suppose that M parametrizes a family of sections +C which have large degree but do not go through many points of Z. This implies that the +Harder-Narasimhan filtration of TZ/B|C has a large gap in the slopes between two consecutive +terms Gk ⊂ Gk+1 for some index k. When M satisfies the conditions of Corollary 4.6, we +can deduce that there is a foliation F on X that restricts to Gk. Appealing to the results +developed in the sequence of papers [BM16], [KSCT07], [CP19], the foliation is induced by +a rational map φ : Z ��� W. We can expect that there will be many deformations of C +in directions tangent to the fibers of φ. In particular, deformations of C should go through +many general points of the main component Y of φ−1(φ(C)). +sect:genpointfoliations +5.1. General points and foliations. We will need the following construction describ- +ing the relationship between foliations and relative tangent bundles which is adapted from +[KSCT07, Remark 19]. +cons:foliationrestriction +Construction 5.1. Let π : Z → B be a good fibration. Suppose that F is a foliation on +Z that is contained in the relative tangent bundle TZ/B. Assume that F is induced by a +rational map φ : Z ��� W that has connected fibers. Note that φ must be a rational map +over B and we may assume that W is a projective B-variety. +Suppose that C is a section of π that is contained in the regular locus of F and goes +through a general point of Z. Since C is transversal to F, the Frobenius theorem shows that +there is an irreducible analytic submanifold W ⊂ Z containing C such that the fibers of π|W +are smooth analytic manifolds which are open subsets of the leaves of φ with the property +NC/W ∼= F|C. +Let Y denote the main component of φ−1(φ(C)) (that is, the unique irreducible component +that dominates φ(C) under φ) and let Y′ denote its normalization. +Using the universal +property of normalization, we see that W admits an embedding into Y′. In particular, Y′ +admits a section C′ in its smooth locus that maps to C and has normal bundle NC′/Y′ ∼= F|C. +Thus we can choose a resolution �Y of Y that admits a section �C which maps to C and satisfies +T �Y/B| � +C ∼= F|C. +Grauert-Mulich only applies when a general curve is contained in the flat locus of the eval- +uation map. This can always be achieved after a birational modification as in the following +construction: +cons:flatteningfamilyofcurves +Construction 5.2. Let Z be a smooth projective variety and let W be a variety admitting +a morphism W → Mg,0(Z) that is generically finite onto its image. Let UW denote the +28 + +universal family over W and let Uν +W denote the normalization of UW. Then Uν +W is equipped +with a map p : Uν +W → W and an evaluation map evW : Uν +W → Z. +Assume that a general fiber of p is a smooth projective curve. We claim there is a birational +map φ : Z′ → Z from a smooth variety Z′ and an open subset W ◦ ⊂ W such that the +preimage Uν,◦ +W := p−1W ◦ admits a flat morphism ev′ : Uν,◦ +W → Z′ satisfying evW|Uν,◦ +W = φ◦ev′. +Indeed, suppose we take a flattening of ev, i.e. a diagram +V +� +ev +� +ψ +� +�Z +ψZ +� +Uν +W +evW +� Z +where V and �Z are varieties, ψ and ψZ are projective birational morphisms, and �ev is flat. +Let ρ : Z′ → �Z be a resolution of singularities. Since �ev is flat, V′ := V × �Z Z′ is also a variety +and the projection map ev′ : V′ → Z′ is still flat. The induced map ψ′ : V′ → Uν +W is still +birational. Since p defines a family of curves, there is an open subset W ◦ ⊂ W such that +p−1W ◦ is disjoint from every ψ′-exceptional center. Then W ◦ has the desired properties. +We are now ready to state the main result of this section. +theo:betterpointsandfoliations +Theorem 5.3. Let π : Z → B be a good fibration. Fix a positive integer J ≥ 2g(B) + 3. +Suppose M is an irreducible component of Sec(Z/B) and let ev : Uν → Z denote the +evaluation map for the normalization of the universal family over Mred. Assume that ev is +dominant. Let g : S → Z denote the finite part of the Stein factorization of ev and let N +denote the family of sections on S corresponding to general members of M. Let ρ : S′ → S +be a birational map from a smooth projective variety that flattens the evaluation map for the +normalization of the universal family over N as in Construction 5.2. Let C′ denote the strict +transform on S′ of a general section on S parametrized by N. +Suppose that S′ is equipped with a dominant rational map ψ : S′ ��� T over B where T is +a normal projective B-variety and ψ has connected fibers. Let G denote the foliation induced +by ψ. Furthermore assume that +µmax +[C′] (G) ≥ (J + 2g(B) + γ − 1) > µmax +[C′] (TS′/G). +where we define γ = (g(B) dim(Z) − g(B) + 1)2(dim(Z) − 1). Then either: +(1) the deformations of C′ in the main component P of ψ−1(ψ(C′)) contain at least J +general points of P, or +(2) there is a dominant rational map φ : S′ ��� W over B to a normal projective B- +variety W such that ψ factors rationally through φ and the following holds. Let C′ +be a general section in our family. Let Y denote the main component of φ−1(φ(C′)). +Then there is a resolution �Y of Y and a section �C on �Y that maps to C′ such that: +(a) The deformations of �C in �Y contain at least J general points of �Y. +(b) The space of deformations of C′ in Y has codimension at most (dim(P)−1)(J + +2g(B) + γ) in the space of deformations of C′ in P. +(c) Letting H denote the foliation induced by φ, we have µmax +[C′] (TS′/H) < J +2g(B)+ +γ − 1 ≤ µmin +[C′] (H). +29 + +Proof. Let us assume that deformations of C′ do not go through J general points of P. Since +C′ deforms in a flat family on S′, a general section C′ in the family will be contained in the +smooth locus of G. Thus, if take a resolution �P and consider the strict transform C† then +Construction 5.1 shows that the normal bundle of C† in �P is isomorphic to G|C′. By Lemma +3.6 our deformation assumption implies that +µmin(G|C′) < J + 2g(B) − 1. +Applying Theorem 4.10 we obtain +µmin +[C′] (G) ≤ µmin(G|C′) + γ +< J + 2g(B) + γ − 1 +eq:minslopegbound +eq:minslopegbound +(5.1) +Write the Harder-Narasimhan filtration of G with respect to [C′] as +0 = F0 ⊂ F1 ⊂ · · · ⊂ Fk = G. +By assumption µmax +[C′] (G) ≥ J + 2g(B) + γ − 1 so there is some index i ≥ 1 such that we have +µmin +[C′] (Fi) ≥ J + 2g(B) + γ − 1. Let i be the maximum index for which this inequality holds. +On the one hand, since Equation (5.1) shows that µmin +[C′] (G) < J + 2g(B) + γ − 1 we must +have i < k. On the other hand, since i was selected to be as large as possible we must have +µmax +[C′] (G/Fi) < J + 2g(B) + γ − 1. +We claim that Fi is a foliation on S′. By Theorem 2.13 it suffices to check that Fi is +a term in the Harder-Narasimhan filtration of TS′ with respect to [C′]. +By assumption +µmax +[C′] (TS′/G) < J + 2g(B) + γ − 1 ≤ µmin +[C′] (Fi) and thus the Harder-Narasimhan filtration of +TS′ agrees with the Harder-Narasimhan filtration of G up to the ith entry, proving the claim. +By [CP19, Theorem 1.1] the foliation Fi is induced by a rational map φ : S′ ��� W over +B that has connected fibers. Since i < k this rational map is not trivial. By our flatness +assumption a general section C′ will be contained in the regular locus of Fi. Let �Y denote +a resolution of the main component of φ−1(φ(C′)) and let �C denote the section chosen as in +Construction 5.1. In particular we have +T �Y/B| � +C ∼= Fi|C′. +Theorem 4.10 implies that +µmin(Fi|C′) ≥ µmin +[C′] (Fi) − γ +≥ J + 2g(B) − 1 +and so by Lemma 3.6 we see that �C can go through at least J general points of �Y verifying +(a). To prove (b), let NP denote the space of deformations of the strict transform of C′ in +�P and let NY denote the space of deformations of �C in �Y. Appealing to Corollary 3.4, we +see that +dim(NP) − dim(NY) ≤ (c1(G) · C′ + (dim( �P) − 1)) − (c1(Fi) · C′ + (dim( �Y) − 1)(1 − g(B))) += c1(G/Fi) · C′ + (dim(P) − dim(Y)) + g(B)(dim( �Y) − 1) +< (dim(P) − dim(Y))(J + 2g(B) + γ) + g(B)(dim( �Y) − 1) +≤ (dim(P) − 1)(J + 2g(B) + γ) +30 + +Since the dimension of the space of sections is birationally invariant, we obtain (b). Finally, +by construction µmin +[C′] (Fi) ≥ J + 2g(B) + γ − 1. On the other hand we have already seen +that both µmax +[C′] (G/Fi) and µmax +[C′] (TS′/G) are strictly less than J +2g(B)+γ −1. This implies +(c). +□ +6. Twists over function fields of complex curves +sect:twists +Let B be a smooth projective curve over an algebraically closed field k of characteristic +0. Suppose we have a dominant generically finite morphism fη : Yη → Xη between normal +projective K(B)-varieties. +In this section we study the set of twists of fη. +Recall that +a twist of fη is a generically finite K(B)-morphism f ′ +η : Y′ +η → Xη such that there is an +Xη-isomorphism between Yη and Y′ +η (where the subscript η denotes the base change to +Spec K(B)). +In Section 6.1 we discuss the Hurwitz space as described by [Wew98]. Using this con- +struction, we show in Section 6.2 that the set of twists of a dominant generically finite map +fη : Yη → Xη can be parametrized by a scheme with countably many components. We will +not construct a universal stack, since there are some steps in the construction which might +not be valid in the setting of stacks. Instead, we will construct a morphism of schemes such +that every twist of fη is the fiber over some closed point. +The remainder of the section is devoted to analyzing the canonical divisor for twists. In +Section 6.3, we prove a local-to-global principle (Corollary 6.5) for the Galois cohomology +group parametrizing twists of fη. In particular the local invariant gives us a convenient way +to identify the places of K(B) where two twists are “the same” locally. In Section 6.4 we +apply Hensel’s Lemma to give a geometric criterion that will guarantee the vanishing of the +local invariant. Finally, in Section 6.5 we analyze how the canonical divisor changes as we +choose different twists of fη. The key point is that its positivity is controlled by the places +of K(B) where the local invariant does not vanish. In particular, we show that if we bound +the positivity of the canonical divisor then the parameter space of twists has finite type +(Corollary 6.13). +sect:hurwitzspace +6.1. Hurwitz space. The starting point is the following version of the Hurwitz space: +Theorem 6.1 ([Wew98]). Let B be a smooth projective curve. Fix a positive integer r and +a finite group G. There is a smooth Deligne-Mumford stack H(G, r, B) parametrizing pairs +(q, ψ) where q : C → B is a Galois morphism from a smooth projective curve C that has r +branch points and ψ is an isomorphism ψ : Aut(C/B) → G. +Suppose we fix a finite group G and set H(G, B) = ⊔rH(G, r, B). Then we can think of +H(G, B) as a parameter space for pairs (C/B, ψ) where C/B is a finite Galois cover and +ψ : Gal(K(C)/K(B)) → G is an isomorphism. We denote the universal family over H(G, B) +by U(G, B) → H(G, B). This means that there is a morphism U(G, B) → H(G, B) × B +which over every point of the form Spec(k) → H(G, B) is the corresponding cover C → B +with an isomorphism ψ : Gal(K(C)/K(B)) → G. +Since our parameter space includes the data of an isomorphism ψ : Aut(C/B) → G, the +fiber of G := G × H(G, B) over (C/B, ψ) ∈ H(G, B) can be canonically identified with the +Galois group Gal(K(C)/K(B)). +31 + +sect:familyoftwists +6.2. The space of twists. We fix a dominant generically finite map fη : Yη → Xη of normal +geometrically integral projective K(B)-schemes. +Let G be a finite group. To avoid some stacky issues in our constructions, we will fix an +´etale cover HG → H(G, B) from a scheme HG whose irreducible components are varieties +of finite type over C. We denote the pullback of the universal family by UHG → HG. We +will use (C/B, ψ) to denote any closed point of HG such that the corresponding fiber of +UHG → HG is the map C → B equipped with the isomorphism ψ. We let GHG → HG +denote the morphism whose fiber over (C/B, ψ) ∈ HG is the corresponding Galois group, +i.e., GHG = G×HG. We let K(Yη/Xη)HG := K(Yη/Xη)×HG denote the trivial group scheme +over HG associated to +K(Yη/Xη) = Aut(Yη/Xη). +We consider the universal family UHG → HG×B and its base change U∗ +HG → HG×Spec K(B). +Since Yη ×Spec K(B) U∗ +HG is flat and projective over HG × Spec K(B) and Xη ×Spec K(B) U∗ +HG +is projective over HG × Spec K(B), by [Kol96, I.1.10 Theorem] we can define the relative +automorphism scheme +�K(Yη/Xη)HG := AutHG×Spec K(B)(Yη ×Spec K(B) U∗ +HG/Xη ×Spec K(B) U∗ +HG). +This is a quasi-finite group scheme over HG × Spec K(B). +Since �K(Yη/Xη)HG can be embedded into K(Yη/Xη)HG ×Spec K(B) as a HG×Spec K(B)- +scheme, we conclude that �K(Yη/Xη)HG is quasi-affine over HG × Spec K(B). +Using the +functoriality of the AutHG×Spec K(B)-functor we can construct descent data for the quasi-finite +group scheme �K(Yη/Xη)HG → HG ×Spec K(B) with respect to the map HG ×Spec K(B) → +HG. +Indeed, we denote by p1, p2 the two projections HG × Spec K(B) × Spec K(B) → +HG × Spec K(B). Then both p∗ +1 �K(Yη/Xη)HG and p∗ +2 �K(Yη/Xη)HG are canonically isomorphic +to +AutHG×Spec K(B)×Spec K(B)(Yη ×Spec K(B) U∗ +HG × Spec K(B)/Xη ×Spec K(B) U∗ +HG × Spec K(B)). +This defines the canonical descent data p∗ +1 �K(Yη/Xη)HG → p∗ +2 �K(Yη/Xη)HG. Then it is easy +to check that this data satisfies the gluing condition. By the fpqc descent theory for quasi +affine schemes as in [Poo17, Theorem 4.3.5(ii)] we conclude the existence of a quasi-finite +group scheme K(Yη/Xη)HG → HG whose base change to HG × Spec K(B) is isomorphic +to �K(Yη/Xη)HG. Note that this is a locally closed subgroup scheme of K(Yη/Xη)HG, so in +particular K(Yη/Xη)HG is quasi-affine over HG. For (C/B, ψ) ∈ HG consider the Galois +action by conjugation +φC/B,ψ : Gal(K(C)/K(B)) × (K(Yη/Xη)HG)(K(C)/K(B),ψ) → (K(Yη/Xη)HG)(K(C)/K(B),ψ). +This fiberwise Galois action defines a group scheme action +φ : GHG ×HG K(Yη/Xη)HG → K(Yη/Xη)HG. +Consider the morphism scheme MorHG(GHG, K(Yη/Xη)HG). We define the space +C1(GHG, K(Yη/Xη)HG) +of 1-cocycles as the closed subscheme of MorHG(GHG, K(Yη/Xη)HG) consisting of 1-cocycles +(C/B, ψ, σ : G → (K(Yη/Xη)HG)(K(C)/K(B),ψ)) which satisfy the cocycle condition σst = +σsφ(C/B,ψ)(s)(σt). +32 + +Next our goal is to construct a family of twists of (Yη/Xη) over C1(GHG, K(Yη/Xη)HG). +Define Y′ = Yη ×Spec K(B) U∗ +HG. Then the fiber of the projection Y′ → HG over (C/B, ψ) is +isomorphic to Yη ⊗ K(C). We consider +Y′ ×HG C1(GHG, K(Yη/Xη)HG) → Xη ×Spec K(B) U∗ +HG. +There is a group scheme action of GHG on this fiber product by +(s, (C/B, ψ)) · (y, σ, (C/B, ψ)) = (σs ◦ (1 ⊗ s)(y), σ, (C/B, ψ)). +We let �Y denote the quotient of Y′×HG C1(GHG, K(Yη/Xη)HG) by the finite flat group scheme +GHG. Then �Y comes equipped with a map +�Y → C1(GHG, K(Yη/Xη)HG) × Xη. +such that the fiber over (σ, (C/B, ψ)) ∈ C1(GHG, K(Yη/Xη)HG) is the map +Yσ +η → Xη, +where Yσ +η is the quotient of Yη ⊗ K(C) by the Galois action +Gal(K(C)/K(B)) ∋ s �→ σs ◦ 1 ⊗ s ∈ Aut(Yη ⊗ K(C)/Xη). +By construction every twist of Yη → Xη is parametrized by the fiber over some point +(σ, (C/B, ψ)) ∈ C1(GHG, K(Yη/Xη)HG). +Note that the scheme C1(GHG, K(Yη/Xη)HG) constructed above need not have finite type +over K(B). However, if we fix certain invariants then the corresponding subscheme will have +finite type. +lemm:twistboundedconditions +Lemma 6.2. Fix a smooth projective curve B and positive integers d, b. Suppose we have +a generically finite dominant K(B)-morphism fη : Yη → Xη where Xη and Yη are normal +projective varieties. Let S denote the set of twists f σ +η such that Yσ +η and Yη become isomorphic +after a base change by a Galois extension K(C)/K(B) whose degree is ≤ d and whose branch +locus consists of at most b points. +There is a finite type scheme R over C and morphisms ψ : UR → R, g : UR → Xη such +that every element Yσ +η ∈ S is isomorphic to the fiber of ψ over some closed point t ∈ R and +f σ +η = g|ψ−1t. +Proof. There are finitely many isomorphism classes of finite groups G of order ≤ d. As we +vary G over all such groups and vary over all r ≤ b, we obtain a finite type Deligne-Mumford +stack ⊔H(G, r, B) parametrizing extensions K(C)/K(B) and automorphisms Aut(C/B) → +G. Let HG,r be the preimage of H(G, r, B) via HG → ⊔rH(G, r, B). Then the space +⊔G,rC1(GHG,r, K(Yη/Xη)HG,r) +is a finite type scheme over C where C1(GHG,r, K(Yη/Xη)HG,r) is the base change of +C1(GHG, K(Yη/Xη)HG) +via HG,r → HG. Thus our assertion follows. +□ +33 + +6.2.1. The spaces of twists in families. Here we perform the constructions in the previous +section in families. +As before we fix a smooth projective curve B defined over C. +Let +X → S×Spec K(B), Y → S×Spec K(B) be flat families of normal projective K(B)-varieties +where S is a scheme of finite type over C. We also assume that we have a S × Spec K(B)- +morphism f : Y → X which is fiberwise dominant and generically finite. +We fix a finite group G and take an ´etale open cover HG → H(G, B) where HG is a scheme +over C. As before we denote the pullback of the universal family by UHG → HG × B and +consider its base change U∗ +HG → HG × Spec K(B). Since Y ×Spec K(B) U∗ +HG is projective and +flat over S × H × Spec K(B) and X ×Spec K(B) U∗ +HG is projective over S × HG × Spec K(B), +we can define the relative automorphism group +�K(Y/X)S×HG = AutS×HG×Spec K(B)(Y ×Spec K(B) U∗ +HG/X ×Spec K(B) U∗ +HG). +This is a quasi-finite group scheme over S × HG × Spec K(B). Since the above relative +automorphism group is also separated over S×HG×Spec K(B), �K(Y/X)S×HG is quasi-affine +over S×HG×Spec K(B). Using fpqc descent theory, �K(Y/X)S×HG → S×HG×Spec K(B) +descends to K(Y/X)S×HG → S × HG. +Let GS×HG = G × S × HG and consider the natural conjugation group action +φ : GS×HG ×S×HG K(Y/X)S×HG → K(Y/X)S×HG. +Consider the morphism scheme MorS×HG(GS×HG, K(Y/X)S×HG) and the closed subscheme +consisting of the space of 1-cycles C1(GS×HG, K(Y/X)S×HG). Define Y′ = Y ×Spec K(B) U∗ +HG +as a scheme over S × HG. Again we have a natural group action of GS×HG on Y′ ×S×HG +C1(GS×HG, K(Y/X)S×HG). We define �Y to be the quotient of this group action. It comes +equipped with a morphism �Y → C1(GS×HG, K(Y/X)S×HG) ×S X realizing +C1(GS×HG, K(Y/X)S×HG) +as the parameter space of twists of the maps fs,η : Ys,η → Xs,η for closed points s ∈ S. +Regarding this family we have the following boundedness statement: +lemm:twistboundedconditions2 +Lemma 6.3. Fix a smooth projective curve B and positive integers d, b. +Let p : X → +S×Spec K(B), q : Y → S×Spec K(B) be flat families of normal projective K(B)-varieties +where S is a scheme of finite type over C. We also assume that we have a S × Spec K(B)- +morphism f : Y → X which is fiberwise dominant and generically finite. +Let A denote the set of twists f σ +η,s : Yσ +η,s → Xη where s is a closed point of S and Yσ +η,s +and Yη,s become isomorphic after a base change by a Galois extension K(C)/K(B) whose +degree is ≤ d and whose branch locus consists of at most b points. +There is a finite type scheme R over S and morphisms ψ : UR → R, g : UR → R ×S X +such that every element f σ +η,s : Yσ +η,s → Xη,s ∈ A is isomorphic to the fiber of ψ over some +closed point t ∈ R and f σ +η,s = g|ψ−1t. +subsec:functoriality +6.2.2. Functoriality. Let X → S × Spec K(B), Y → S × Spec K(B) be flat families of +normal projective K(B)-varieties where S is a smooth scheme of finite type over C. We also +assume that we have a S × Spec K(B)-morphism f : Y → X which is fiberwise dominant +and generically finite. We further assume that we have flat families W → S × Spec K(B), +T → S × Spec K(B) of projective varieties such that Ts is normal for any s ∈ S and we +34 + +have a commutative diagram +Y +f +� +r +� +X +p +� +T +t +� W +over S × Spec K(B) where p, r are dominant with connected fibers and t is dominant, +finite, and fiberwise Galois over S, i.e., each fiber Ts → Ws is a finite Galois cover. We +also assume that the Stein factorization of Y → X → W is given by Y → T. Since a +relative automorphism induces a relative automorphism of Stein factorizations, we obtain a +homomorphism +AutS×HG(Y×Spec K(B)U∗ +HG/X×Spec K(B)U∗ +HG) → AutS×HG(T×Spec K(B)U∗ +HG/W×Spec K(B)U∗ +HG), +and this induces a morphism +C1(GS×HG, K(Y/X)S×HG) → C1(GS×HG, K(T/W)S×HG). +sect:localtoglobal +6.3. Local-to-global principle. Let K(B) be the function field of a smooth projective +curve B. +Let f : Yη → Xη be a dominant generically finite morphism between normal +projective varieties Yη and Xη defined over K(B). +We fix a place ν of K(B) over a place ν of K(B). This specifies for every finite cover +C → B a point pν,C on C such that for every factoring C +g−→ C′ → B we have g(pν,C) = pν,C′. +Consider the decomposition group +Dν = {σ ∈ Gal(K(B)) | σ(ν) = ν}. +This is isomorphic to +Gal(K(B)ν) ∼= lim +←−(Z/NZ)×. +Note that if we have two places ν, ν′ corresponding to the same place ν on K(B), then Dν +and Dν′ are conjugate to each other in Gal(K(B)). Recall that the Galois group acts on +Aut(Yη/Xη) by conjugation, and in this way one can consider Galois cohomology +H1(Gal(K(B)), Aut(Yη/Xη)) +The injection Gal(K(B)ν) ∼= Dν ⊂ Gal(K(B)) induces a map on Galois cohomology +H1(Gal(K(B)), Aut(Yη/Xη)) → H1(Gal(K(B)ν), Aut(Yη/Xη)) +which we denote by invν. (Although the choice of isomorphism Gal(K(B)ν) ∼= Dν depends +on the choice of ν, the induced map of Galois cohomology only depends on the place ν up +to an isomorphism of the pointed set, justifying our mild abuse of notation.) +For any twist [σ] of Yη/Xη the local invariant invν([σ]) vanishes for all but finitely many +places ν of B. Thus we obtain a map +H1(Gal(K(B)), Aut(Yη/Xη)) → +� +ν∈B +H1(Gal(K(B)ν), Aut(Yη/Xη)) +and we would like to use this map to establish a local-to-global principle. Note that this map +does not need to be injective; for example, there can be twists of Yη/Xη which are trivialized +by an ´etale cover of B. However, we will show that the fibers of this map are finite, which +is good enough for our purposes. +35 + +lemm:boundingdeganddisc +Lemma 6.4. Let f : Yη → Xη be a dominant generically finite morphism between normal +projective varieties over K(B). Then there exists a positive integer d = d(Yη/Xη) and a fixed +finite subset P ⊂ B (depending only on Yη/Xη) such that the following holds. Suppose that +[σ] ∈ H1(Gal(K(B)), Aut(Yη/Xη)), +is a cohomology class and let Q ⊂ B denote the finite set of places ν ∈ B with invν([σ]) ̸= 0. +Then there exists a Galois cover C → B of degree at most d and whose branch locus is +contained in P ∪ Q such that Yσ +η → Xη splits over K(C). +Proof. Let K(B′)/K(B) be a fixed Galois extension so that +Aut(Yη/Xη) = Aut(Yη ⊗ K(B′)/Xη ⊗ K(B′)). +We define P to be the set of branch points for B′ → B. +We can restrict our cocycle σ : Gal(K(B)) → Aut(Yη/Xη) to the subgroup Gal(K(B′)) +to get a cocycle σ′. Then σ′ : Gal(K(B′)) → Aut(Yη ⊗ K(B′)/Xη ⊗ K(B′)) is a honest +homomorphism because the Galois action of Gal(K(B′)) on Aut(Yη ⊗ K(B′)/Xη ⊗ K(B′)) +is trivial. The kernel is an open subgroup and thus defines a Galois cover C over B′. Then +the induced cocycle [τ] ∈ H1(Gal(K(C)), Aut(Yη ⊗ K(C)/Xη ⊗ K(C))) is trivial. +Thus +Yσ +η → Xη splits over K(C). Now note that the degree of K(C)/K(B′) is bounded by the +order of Aut(Yη/Xη) and the degree of K(B′)/K(B) only depends on Yη/Xη. Furthermore +for any place ν ∈ B with invν([σ]) = 0 and ν ̸∈ P, we have Dν ⊂ Gal(K(C)). Thus C/B′ +cannot be ramified over any place ν ∈ B\(Q ∪ P) and our assertion follows. +□ +coro:localtoglobal +Corollary 6.5. Let f : Yη → Xη be a dominant generically finite morphism between normal +projective varieties over K(B). The fibers of the local invariant map +H1(Gal(K(B)), Aut(Yη/Xη)) → +� +ν∈B +H1(Gal(K(B)ν), Aut(Yη/Xη)) +are finite. +Proof. Suppose that ξ is a element of the direct sum and let Q ⊂ B denote the finite set of +indices for which the entries of ξ are non-zero. By Lemma 6.4, there is a fixed integer d and +a fixed finite set P ⊂ B such that any twist that lies in the fiber over ξ is split by a Galois +cover C → B of degree at most d and whose branch locus is contained in P ∪ Q. There are +only finitely many such maps C → B, and for each such map there are only a finite set of +twists trivialized by the map. +□ +sect:henselslemma +6.4. Hensel’s lemma. Let B be a smooth projective curve. Let π : X → B be a good +fibration and f : Y → X be a dominant finite morphism from a normal projective variety +such that Yη is geometrically integral. In this setting we have the equality +Bir(Yη/Xη) = Aut(Yη/Xη). +For each twist [σ] ∈ H1(Gal(K(B)), Aut(Yη/Xη)) of Yη/Xη, we can construct an integral +model Yσ → B in the following way. Suppose that K(C)/K(B) is a Galois extension such +that Yη/Xη and Yσ +η /Xη are isomorphic after base change to K(C). Then the cohomology class +[σ] is represented by a cocycle σ : Gal(K(C)/K(B)) → Aut(Yη ⊗K(C)/Xη ⊗K(C)). Let �YC +be the normalization of Y ×B C. Then σ defines a homomorphism from Gal(K(C)/K(B)) +to the birational automorphism group of �YC,η over Xη, or equivalently, to the birational +36 + +automorphism group of �YC over X . By construction a birational automorphism of �YC over +X is actually an automorphism, so that Gal(K(C)/K(B)) acts on �YC via σ. We let Yσ +denote the quotient. Note that Yσ is normal and comes equipped with a finite B-morphism +f σ : Yσ → X . +Here we prove the following birational version of Hensel’s lemma. +lemma:birationalHensel +Lemma 6.6. Let π : X → B be a good fibration and let Y be a normal projective variety +equipped with a dominant morphism Y → B such that Yη is geometrically integral. Suppose +that f : Y → X is a dominant finite B-morphism. Fix a place ν ∈ B and assume that Xν +is smooth. We also assume that AutB(Y/X ) → B is flat at ν ∈ B and Yν is reduced and +normal. +Suppose that Yσ is an integral model of a twist of fη as constructed above and that there +is a birational Xν-map hν : Yν ��� Yσ +ν . Then there is an X ⊗ K(B)ν-isomorphism between +Y ⊗η K(B)ν and Yσ ⊗η K(B)ν. In particular invν(σ) = 0. +Proof. Since Yν is reduced and normal, the automorphism group Bir(Yν/Xν) = Aut(Yν/Xν) +is a reduced finite group. The flatness of AutB(Y/X ) → B at ν implies that the lengths of +Aut(Yη/Xη) and Aut(Yν/Xν) are equal. +Also note that since Yν is normal and finite over Xν and Yσ +ν is also finite over Xν, our +birational map hν extends to a birational morphism Yν → Yσ +ν . +Let us consider the relative X -birational morphism scheme over Spec( �OB,ν): +B = BirMorSpec( � +OB,ν)(Y ×B Spec( � +OB,ν), Yσ ×B Spec( � +OB,ν)) +equipped with a morphism B → Spec( �OB,ν). (This scheme can be constructed as an open +subscheme of the relative Hilbert scheme parametrizing graphs in (Y ×X Yσ)×B Spec( � +OB,ν).) +Note that B is an AutSpec( � +OB,ν)(Y ×B Spec( � +OB,ν)/X ×B Spec( �OB,ν))-torsor. Using the +fact that AutB(Y/X ) → B is flat at ν ∈ B we conclude that the above relative birational +morphism scheme is also flat at ν ∈ B. +Since we have a birational morphism of fibers Yν → Yσ +ν , Hensel’s lemma (see, e.g., [Gro67, +Th´eor`em 18.5.17]) implies that we have an X × Spec( � +OB,ν)-birational morphism from Y ×B +Spec( �OB,ν) to Yσ ×B Spec( �OB,ν). Since Y ⊗η K(B)ν and Yσ ⊗η K(B)ν are normal and +Y ⊗η K(B)ν → X ⊗K(B)ν, Yσ ⊗η K(B)ν → X ⊗K(B)ν are finite, this birational morphism +induces an isomorphism of the generic fibers. +□ +sect:splittingfields +6.5. Splitting fields and ramification. Suppose given an algebraic fiber space π : Y → B +with Y a normal projective variety and a dominant finite morphism of smooth projective +curves B′ → B. We will use the term “normalized base change” to refer to the normalization +of Y ×B B′ equipped with the structure morphism to B′. Note that the normalized base +change Y′ admits a dominant finite morphism Y′ → Y. +lemm:genreducedpreserved +Lemma 6.7. Let Y be a normal projective variety equipped with a surjective morphism π : +Y → B with connected fibers. Suppose B′ → B is a dominant finite morphism of smooth +projective curves and that π′ : Y′ → B′ is the normalized base change. Fix a closed point +t ∈ B and let t′ ∈ B′ be any point mapping to it. If Yt is generically reduced, then Y′ +t′ is also +generically reduced and the morphism Y′ +t′ → Yt is birational. +37 + +Proof. Since Y is normal, the open set U ⊂ Yt consisting of points that lie in the smooth +locus of Y and the smooth locus of Yt is dense in Yt. It is clear that the fiber of Y ×B B′ +over t′ is generically smooth, hence generically reduced. Furthermore, the preimage of U in +Y ×B B′ will be contained in the smooth locus of Y ×B B′. Thus the normalization map +restricts to a birational morphism on this fiber. +□ +corollary:fiberwisebirational +Corollary 6.8. Let π : Y → B and πσ : Yσ → B be morphisms from normal projective +varieties with connected fibers. Suppose that f : Y → X and f σ : Yσ → X are dominant +finite B-morphisms whose generic fibers are twists of each other over X . Suppose further +that t ∈ B is a closed point such that the fibers Yt, Yσ +t are generically reduced. Then the +maps ft, f σ +t are birationally equivalent. +Proof. Choose a dominant finite morphism B′ → B such that the normalized base changes +f ′ : Y′ → X ′, f ′σ : Y′σ → X ′ are birationally equivalent. Since f ′, f ′σ are finite, they are +equal to their own Stein factorizations and thus f ′ and f ′σ are isomorphic to each other. In +particular the maps f ′ +t′ : Y′ +t′ → Xt and f ′σ +t′ : Y′σ +t′ → Xt are isomorphic to each other. But by +Lemma 6.7 these are birationally equivalent to ft and f σ +t respectively. +□ +We next analyze how the canonical divisor changes upon normalized base change. This is +well-understood, e.g., in the context of semistable reduction. +Definition 6.9. Suppose h : Y′ → Y is a finite morphism of normal projective varieties. +Let U ⊂ Y and U′ ⊂ Y′ denote the smooth loci and set V = U′ ∩ h−1(U). Note that the +complement of V in Y′ has codimension ≥ 2. The Riemann-Hurwitz formula gives us a +distinguished effective representative E in the linear equivalence class of KV/U. We define +the relative canonical divisor KY′/Y to be the effective Weil divisor obtained by taking the +closure of E. +lemm:canonicalandnbc +Lemma 6.10. Let Y be a normal projective variety equipped with a surjective morphism +π : Y → B with connected fibers. Suppose g : B′ → B is a dominant finite morphism of +curves and consider the normalized base change +Y′ +φ +� +π′ +�■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +Y ×B B′ +h +� +� +Y +π +� +B′ +g +� B +We define R to be the π′-pullback of the ramification divisor of g and for any point t′ ∈ B′ +we let Rt′ denote the intersection of R with the preimage of t′. +(1) We have KY′/Y ≤ R. +(2) If t ∈ B is a closed point such that the fiber Yt is generically reduced then for any +t′ ∈ B′ mapping to t we have Rt′ ≤ KY′/Y. +Proof. (1) First note that the support of KY′/Y is contained in the support of R. Indeed, the +map h is ´etale away from φ(Supp(R)) and thus φ is an isomorphism over this locus. +Suppose t′ ∈ B′ is a ramification point for g with index e. Let t ∈ B be the image of +t′. We choose local coordinates s′ and s at t′ and t respectively so that g is defined locally +by s = s′e. Let T ′ be an irreducible component of the fiber π′−1(t′) and let q′ denote its +multiplicity in its component. Let u′ be a generic local equation of T ′ so that generically +the map Y′ → B′ is given by by s′ = u′q′. Let T ⊂ U be the image of T ′ and let u denote +38 + +a local equation of T at a generic point. We denote the multiplicity of T in π−1(t) by q so +that Y is generically defined by s = uq. The coefficient of T ′ in R is given by (e − 1)q′. Also +uq = s = s′e = u′eq′ so the coefficient of T ′ in KY′/Y is given by +eq:multiplicityincanonical +eq:multiplicityincanonical +(6.1) +eq′ +q − 1. +Since e > 1, we have (e − 1)q′ ≥ eq′ +q − 1 when q > 1. When q = 1, q′ = 1 by Lemma 6.7 and +so the inequality still holds. Thus our assertion follows. +(2) As explained above when q = 1 we have q′ = 1 as well by Lemma 6.7. Thus we have +our assertion. +□ +prop:ramificationdivisor +Proposition 6.11. Let X → B be a good fibration. Let Y, Yσ be normal projective varieties +which admit surjective morphisms π : Y → B and πσ : Yσ → B with connected fibers. +Suppose there are dominant finite B-morphisms f : Y → X and f σ : Yσ → X whose generic +fibers are twists of each other over K(B). Choose a finite Galois morphism g : B′ → B such +that the normalized base changes of Y and Yσ over B′ are isomorphic. We let �Y denote this +abstract variety; it is equipped with finite morphisms ρ1 : �Y → Y and ρ2 : �Y → Yσ. We +denote the degree of B′ → B by d. +There is a Weil divisor E on �Y , which we write as E = E+ − E− where E+, E− are +effective with no common divisor in their support, such that +• K �Y/Y − K �Y/Yσ ≥ E; +• we have E+ ≥ � +t′ �Yt′ as t′ ∈ B′ varies over closed points whose image t ∈ B satisfies +that Yt is normal but the fiber Yσ +t is not generically reduced; +• we have E− ≤ d · � +t′ �Yt′ as t′ ∈ B′ varies over closed points whose image t ∈ B +satisfies that Yt is not normal. +Proof. We compare these divisors along each fiber separately. Let t′ ∈ B′ be a closed point +and let t ∈ B be its image. If the fiber Yt is not normal, then it follows from Lemma 6.10 +that +(K �Y/Y)t′ − (K �Y/Yσ)t′ ≥ −(e − 1) �Yt′, +where e is the ramification index of t′. In particular this difference is ≥ −d �Yt′. If Yt is normal, +then in particular Yt is irreducible and reduced. Thus the fiber �Yt′ is also irreducible and +reduced, and we conclude that Yσ +t is irreducible. Let us denote the multiplicity of Yσ +t by q. +It follows from Lemma 6.10 that +(K �Y/Y)t′ − (K �Y/Yσ)t′ ≥ 0 +and equality holds when q = 1. The two inequalities above together prove the upper bound +on E−. To prove the lower bound on E+, we analyze those fibers such that Yt is normal and +which satisfy q > 1. Since �Yt′ is reduced we must have e > 1. Since the multiplicities of Yt +and �Yt′ are 1, Equation (6.1) in Lemma 6.10 shows that +(K �Y/Y)t′ = (e − 1) �Yt′, +(K �Y/Yσ)t′ = +� +e +q − 1 +� +�Yt′ +so that we conclude +(K �Y/Y)t′ − (K �Y/Yσ)t′ ≥ e +� +1 − 1 +q +� +�Yt′ ≥ �Yt′. +39 + +Thus our assertion follows. +□ +prop:curveintandrambound +Proposition 6.12. Let X → B be a good fibration. Let Y, Yσ be normal projective varieties +which admit surjective morphisms π : Y → B and πσ : Yσ → B with connected fibers. +Suppose there are dominant finite B-morphisms f : Y → X and f σ : Yσ → X whose generic +fibers are twists of each other over K(B). Choose a finite Galois morphism g : B′ → B such +that the normalized base changes of Y and Yσ over B′ are isomorphic. We let �Y denote this +abstract variety; it is equipped with finite morphisms ρ1 : �Y → Y and ρ2 : �Y → Yσ. We +denote the degree of B′ → B by d. +Let Yσ′ be a smooth birational model of Yσ equipped with a birational morphism β : Yσ′ → +Yσ. Assume that there exists a section C on Yσ′ and a constant R > 0 such that C corre- +sponds to a rational point on the smooth locus of Yσ +η and +(KYσ′/B − β∗(f σ)∗KX/B) · C ≤ R. +Let r denote the number of closed points t ∈ B such that the fiber Yt is normal but the fiber +Yσ +t is not generically reduced. Then we have r ≤ dR. +Proof. We choose smooth models Y′, �Y′ of Y, �Y respectively such that there are birational +morphisms α : Y′ → Y, γ : �Y′ → �Y and generically finite morphisms �ρ1 : �Y′ → Y′, +�ρ2 : �Y′ → Yσ′ which are birationally equivalent to ρ1, ρ2 respectively. We may ensure that +γ−1 is well-defined along the smooth locus of Yσ +η ⊗ K(B′). Our intersection bound implies +that +KYσ′/X · C ≤ R. +Since by assumption C is not contained in the ρ2-image of the γ-exceptional centers, there +is a section C′ of �Y′/B′ such that (�ρ2)∗C′ = dC. Then we have +�ρ∗ +2KYσ′/X · C′ ≤ dR. +Note that +�ρ∗ +2KYσ′/X = K �Y′/X − K �Y′/Yσ′ += �ρ∗ +1KY′/X + K �Y′/Y′ − K �Y′/Yσ′ ≥ K �Y′/Y′ − K �Y′/Yσ′ +Let E be the divisor on �Y defined by Proposition 6.11, so E+ is at least as effective as the +sum of the r fibers of �Y corresponding to the r closed points t ∈ B such that the fiber Yt +is normal but the fiber Yσ +t is not generically reduced. Taking strict transforms, we see that +K �Y′/Y′ −K �Y′/Yσ′ is at least as effective as the sum of the strict transforms of these r fibers of +�Y. Furthermore every exceptional divisor of β : Yσ′ → Yσ is contracted by f σ ◦ β : Yσ′ → X +as well, and thus appears with positive coefficient in the ramification divisor KYσ′/X. We +conclude that the support of the effective divisor �ρ∗ +2KYσ′/X contains the r reduced fibers over +these points. Since our section C′ must meet each fiber in a component of multiplicity one, +we conclude that +r ≤ �ρ∗ +2KYσ′/X · C′ ≤ dR. +□ +coro:boundedintimpliesboundedtwists +Corollary 6.13. Let X → B be a good fibration. Let Y, Yσ be normal projective varieties +which admit surjective morphisms π : Y → B and πσ : Yσ → B with connected fibers. +40 + +Suppose there are dominant finite B-morphisms f : Y → X and f σ : Yσ → X whose generic +fibers are twists of each other over K(B). +Let �Yσ be a smooth birational model of Yσ equipped with a birational morphism β : �Yσ → +Yσ. Assume that there exists a section C on �Yσ and a constant R > 0 such that C corre- +sponds to a rational point on the smooth locus of Yσ +η and +(K �Yσ/B − β∗(f σ)∗KX/B) · C ≤ R +Then there exists constants d = d(Y/X ) and n = n(Y/X , R) such that there exists a finite +Galois morphism B′ → B of degree at most d with at most n branch points such that the +normalized base changes of Y/B and Yσ/B by B′ → B become X ×B B′-isomorphic. +In particular, the set of such twists is a bounded family. +Proof. It follows from Lemma 6.4 that there exists d = d(Y/X ) and a finite Galois morphism +�B → B of degree at most d such that the normalizations of Y ×B �B and Yσ ×B �B are +isomorphic. +Let s = s(Y/X ) be the number of t ∈ B such that Yt is not normal or AutB(Y/X ) → B +is not flat at t ∈ B. Let r be the number of t ∈ B such that Yt is normal but Yσ +t is not +generically reduced. By Proposition 6.12 we have r ≤ dR. +For t ∈ B such that Yt is normal, AutB(Y/X ) → B is flat at t ∈ B and Yσ +t is generically +reduced, it follows from Corollary 6.8 and Lemma 6.6 that invt(σ) = 0. Thus the number +of t ∈ B such that invt(σ) ̸= 0 is bounded above by s + dR. Thus our first assertion follows +from Lemma 6.4. +The final statement then follows from Lemma 6.2. +□ +7. Fujita invariant and sections +sec:fujinv +Suppose that π : X → B is a good fibration and L is a generically relatively big and +semiample Cartier divisor on X . In this section the goal is to classify the generically finite +B-morphisms f : Y → X such that Y carries a family of sections N with the property that +f∗N has small codimension in an irreducible component of Sec(X /B). Assuming the sections +have large L-degree but small degree against f ∗(KX/B + a(Xη, L|Xη)L), we show that the +Fujita invariant of Yη must be at least as large as the Fujita invariant of Xη. This puts a +strong constraint on the set of morphisms f which have this property. +After addressing some preliminaries in Section 7.1 and Section 7.2, we show the funda- +mental result discussed above in Section 7.3. When working with the Fujita invariant it is +often helpful to know that the pair (Yη, f ∗L|Yη) is adjoint rigid; in Section 7.4 we show that +if we additionally assume that the sections parametrized by N go through many general +points of Y then we can also guarantee adjoint rigidity. +sect:pivertical +7.1. Modifying by π-vertical divisors. Let π : X → B be a good fibration and let L +be a generically relatively big and semiample Cartier divisor on X . We know that L|Xη +is Q-linearly equivalent to a divisor which has smooth support. The following proposition +discusses how to reframe this property as a global statement by adding π-vertical divisors. +prop:generalsurjectiontofiber +Proposition 7.1. Let π : X → B be a good fibration, let L be a generically relatively big +and semiample Cartier divisor on X , and let a be a positive rational number. Let b > a be a +positive integer such that bL|Xη defines a basepoint free linear series. There is some effective +π-vertical Q-Cartier divisor E on X such that the following property holds. +41 + +Suppose ψ : Y → B is a good fibration and f : Y → X is a B-morphism that is generically +finite onto its image. Then there is an effective Q-Cartier divisor D on Y that is Q-linearly +equivalent to f ∗(aL + E) such that D|Yη has smooth irreducible support and coefficient a +b. In +particular (Yη, D|Yη) is a terminal pair. +Proof. Let T1, . . . , Tr be a K(B)-basis for |bL|Xη|. Note that ∩iTi = ∅. +We denote by T i the closure of Ti in X . There is some effective π-vertical Q-Cartier divisor +�E such that for every i there is an effective π-vertical divisor Fi satisfying T i+Fi ∼ b(L+ �E). +Let f : Y → X be a morphism as in the statement. By construction we have ∩if ∗(T i +Fi) +does not intersect Yη. Thus f ∗(b(L+ �E)) is linearly equivalent to a divisor �D whose restriction +to Yη is smooth and irreducible. Then D = a +b �D and E = a �E have the desired properties. +□ +We will use the following definition to capture the effect of the extra divisor E. +defi: +invariant_tau +Definition 7.2. Let π : X → B be a good fibration. Suppose that E is an effective π-vertical +Q-Cartier divisor. Define +τ(π, E) = +sup +sections C +E · C. +Note that this supremum is achieved by some section C since the intersection number is +bounded above by the sum of the coefficients of E and is contained in 1 +rZ where r is the +least common multiple of the denominators of the coefficients of E. +sect:relvsabs +7.2. Relative versus absolute positivity. We will also need a couple results comparing +relative and absolute positivity for a fibration over a curve. +lemm:relvsabsmmp +Lemma 7.3. Let π : Z → B be a good fibration. Suppose that D is an effective Q-Cartier +divisor on Z such that (Z, D) is a terminal pair. +(1) Suppose that g(B) ≥ 1. Suppose that ρ : Z ��� ˜Z is a rational map obtained by +running the (KZ + D)-MMP. Then ρ is also a run of the relative (KZ + D)-MMP +over B. +(2) Suppose that g(B) = 0. There is a constant m = m(dim(Z)) such that the following +holds. +Fix a general fiber F of π and suppose that ρ : Z ��� +˜Z is a birational +morphism obtained by running the (KZ + D + mF)-MMP. Then ρ is also a run of +the relative (KZ + D + mF)-MMP over B. +In particular, in case (1) (resp. case (2)) if KZ + D (resp. KZ + D + mF) is not pseudo- +effective then its restriction to Zη is also not pseudo-effective. +Proof. (1) By [Kaw91] each step of the (KZ + D)-MMP contracts an extremal ray that is +spanned by a rational curve. This rational curve must be vertical with respect to π because +B has genus ≥ 1. +(2) Since (Z, D) is 1 +2-lc, by Theorem 2.16 there is an integer m = m(dim(Z)) such that +Nef1(Z) + Eff1(Z)KZ+D+mF ≥0 = Eff1(Z)KZ+D+mF ≥0 + +� +j +[Cj] +where the Cj are π-vertical moving curves. In particular, any contraction of a (KZ+D+mF)- +negative extremal ray must define a relative contraction over B. Furthermore the analogous +equality of cones holds for any birational model of Z obtained by running the MMP (since +42 + +the 1 +2-lc condition is preserved). Thus we see that every step of the (KZ + D + mF)-MMP +is actually a step of the relative MMP over B. +To see the final statement, suppose we are in case (1) and KZ + D is not pseudo-effective. +Then we can run the (KZ + D)-MMP with scaling of an ample divisor and the outcome will +be a Mori fibration. But then this Mori fibration must be a relative fibration over B, so +that (KZ + D) is not relatively pseudo-effective over B. The same argument applies in case +(2). +□ +Our next result shows how to turn intersection inequalities into Fujita invariant inequali- +ties. +lemm:terminalsectiontofiber +Lemma 7.4. Let π : Z → B be a good fibration. Suppose that D is an effective Q-Cartier +divisor on Z such that (Zη, D|Zη) is a terminal pair and D|Zη is big and nef. +(1) Suppose that g(B) ≥ 1. If there is a dominant family of HN-free sections C on Z +which satisfy −(KZ + D) · C > 0 then a(Zη, D|Zη) > 1. +(2) Suppose that g(B) = 0. There is a constant Ξ = Ξ(dim(Z)) such that the following +holds. If there is a dominant family of HN-free sections C on Z which satisfy −(KZ + +D) · C > Ξ then a(Zη, D|Zη) > 1. +Proof. Let φ : Z′ → Z be a log resolution of (Z, D) and let D′ be the strict transform of the +π-horizontal components of D. After perhaps taking a further blow-up, we may assume that +two irreducible components of D′ intersect if and only if their restrictions to the generic fiber +intersect. Along the central fiber we can write KZ′η + D′|Z′η = φ∗(KZη + D|Zη) + Eη where +Eη is an effective φ-exceptional divisor. We conclude that KZ′η + D′|Z′η is pseudo-effective if +and only if KZη + D|Zη is pseudo-effective. +We claim that the pair (Z′, D′) has terminal singularities. Since Supp(D′) is an SNC +divisor, by [Kol97, 3.11 Lemma] the pair (Z′, D′) will be terminal if and only if when we +write D′ = � +i diD′ +i in terms of irreducible components we have +min +i {1 − di} > 0 +and +min +i,j|Di∩Dj̸=∅{1 − di − dj} > 0. +Recall that by construction two irreducible components of D′ intersect if and only if their +restrictions to the generic fiber intersect. Thus this computation can be done on the generic +fiber, where the desired inequalities follow from the fact that (Zη, D|Zη) is terminal. +Let C′ be the strict transform of a general deformation of C. Since C is HN-free, by +Lemma 3.8 we can assume that C′ avoids any codimension 2 locus in Z and thus C′ has +vanishing intersection against every φ-exceptional divisor. We have +(KZ′ + D′) · C′ ≤ (KZ′ + φ∗D) · C′ += φ∗(KZ + D) · C′ +(1) We are in the case g(B) ≥ 1 and +(KZ′ + D′) · C′ ≤ φ∗(KZ + D) · C′ < 0 +Since C′ is a movable curve, we see that KZ′ + D′ is not pseudo-effective. By Lemma 7.3 we +see that (KZ′ + D′)|Z′η is also not pseudo-effective. As demonstrated above this means that +(KZ + D)|Zη also fails to be pseudo-effective, showing that a(Zη, D|Zη) > 1. +43 + +(2) We are in the case g(B) = 0. Let m = m(dim(Z)) be the constant from Lemma 7.3.(2) +and set Ξ = m + 1. We have +(KZ′ + D′) · C′ ≤ φ∗(KZ + D) · C′ < −Ξ +and thus (KZ′+D′+mF)·C′ < 0. Since C′ is a movable curve, we see that that KZ′+D′+mF +is not pseudo-effective. By Lemma 7.3 we see that (KZ′ +D′)|Z′η is also not pseudo-effective. +As demonstrated above this means that (KZ+D)|Zη also fails to be pseudo-effective, showing +that a(Zη, D|Zη) > 1. +□ +sect:fujinvongenfiber +7.3. Fujita invariant along the generic fiber. In this section we show that the Fujita +invariant along the generic fiber controls the expected dimension for families of sections. We +will use the following easy lemma many times. +lemm:easyintcalc +Lemma 7.5. Let π : X → B be a good fibration and let L be a generically relatively big and +semiample Cartier divisor. Assume that Xη is geometrically uniruled. Fix a positive rational +number arel and define a = arela(Xη, L|Xη). Fix a rational number β. Fix a positive integer +T. +Suppose that ψ : Y → B is a good fibration equipped with a B-morphism f : Y → X that +is generically finite onto its image. Suppose that N is an irreducible component of Sec(Y/B) +parametrizing a dominant family of sections C on Y which satisfy f ∗(KX/B +a(Xη, L|Xη)L)· +C ≤ β. Finally, suppose that +eq:dimension +eq:dimension +(7.1) +dim(N) ≥ arel(−f ∗KX/B · C + (dim(X ) − 1)(1 − g(B))) − T. +Then +(KY + af ∗L) · C ≤ arelβ + T + arel(dim(X ) − 1)(g(B) − 1) + (dim(X ) − 1) + 2g(B) − 2. +Proof. Let C denote a general section parametrized by N. By Corollary 3.4 +dim(N) ≤ −KY/B · C + (dim(Y) − 1). +Combining this equality with Equation (7.1) and rearranging we get +(KY/B − arelf ∗KX/B) · C ≤ T + arel(dim(X ) − 1)(g(B) − 1) + (dim(Y) − 1) +≤ T + arel(dim(X ) − 1)(g(B) − 1) + (dim(X ) − 1) +eq:stupidinequality +eq:stupidinequality +(7.2) +Adding in the fact that f ∗(KX/B + a(Xη, L|Xη)L) · C ≤ β, we see that +eq:alternateform +eq:alternateform +(7.3) +(KY/B + af ∗L) · C ≤ arelβ + T + arel(dim(X ) − 1)(g(B) − 1) + (dim(X ) − 1). +or equivalently +(KY + af ∗L) · C ≤ arelβ + T + arel(dim(X ) − 1)(g(B) − 1) + (dim(X ) − 1) + 2g(B) − 2. +□ +We can now prove our basic result for controlling the Fujita invariant using pathological +families of sections. +theo:generalainvsections +Theorem 7.6. Let π : X → B be a good fibration and let L be a generically relatively big and +semiample Cartier divisor on X . Assume that Xη is geometrically uniruled. Fix a positive ra- +tional number arel and set a = arela(Xη, L|Xη). Fix a rational number β. Fix a positive integer +T. Fix a positive integer b > a such that bL|Xη defines a basepoint free linear series. Use b to +construct an effective π-vertical Q-Cartier divisor E satisfying the conclusion of Proposition +44 + +7.1 with respect to aL. There is some constant ξ = ξ(dim(X ), g(B), τ(π, E), arel, a, T, β, b) +with the following property. +Suppose that ψ : Y → B is a good fibration equipped with a B-morphism f : Y → X +that is generically finite onto its image. +Suppose that N is an irreducible component of +Sec(Y/B) parametrizing a dominant family of sections C on Y which satisfy f ∗L · C ≥ ξ +and f ∗(KX/B + a(Xη, L|Xη)L) · C ≤ β. Finally, suppose that +eq:eh +eq:eh +(7.4) +dim(N) ≥ arel(−f ∗KX/B · C + (dim(X ) − 1)(1 − g(B))) − T. +Then +a(Yη, f ∗L|Yη) ≥ a. +Proof. We first prove the statement when the general section C parametrized by N is HN-free +on Y. By Theorem 2.9 there is a rational number ǫ > 0 depending only on a and dim(X ) +such that no smooth variety of dimension ≤ dim X − 1 has Fujita invariant in the range +[(1 − ǫ)a, a) with respect to any big and nef Cartier divisor. Define Ξ as: +• Ξ = 0, if g(B) ≥ 1. +• Ξ is the supremum of the constants obtained by applying Lemma 7.4 to all dimensions +≤ dim(X ), if g(B) = 0. +We define ξHN(dim(X ), g(B), τ(π, E), arel, a, T, β, b) to be +1 +aǫ ((1 − ǫ)τ(π, E) + arelβ + T + arel(dim(X ) − 1)(g(B) − 1) + (dim(X ) − 1) + 2g(B) − 2 + Ξ) + 1 +and assume that our sections C satisfy f ∗L · C ≥ ξHN. +Let C denote a general section parametrized by N. Since we are assuming C moves in a +dominant family on Y Lemma 7.5 shows that +(KY + af ∗L) · C ≤ arelβ + T + arel(dim(X ) − 1)(g(B) − 1) + (dim(X ) − 1) + 2g(B) − 2. +Adding in E, we get +(KY+af ∗L+f ∗E)·C ≤ arelβ+T+arel(dim(X )−1)(g(B)−1)+(dim(X )−1)+2g(B)−2+τ(π, E). +When the section C has degree ≥ ξHN then the inequality simplifies to +(KY + a(1 − ǫ)f ∗L + (1 − ǫ)f ∗E) · C < −Ξ. +Since E satisfies the conclusion of Proposition 7.1 with respect to aL the pullback f ∗(aL + +E) is Q-linearly equivalent to an effective Q-divisor D such that (Yη, D|Yη) has terminal +singularities. Of course (Yη, (1−ǫ)D|Yη) also has terminal singularities. Applying Lemma 7.4, +we deduce that a(Yη, f ∗L|Yη) ≥ (1−ǫ)a. By construction this implies that a(Yη, f ∗L|Yη) ≥ a. +Next we prove the statement when C is not necessarily HN-free on Y. Our strategy is to +reduce to the HN-free case. Define +equation:generalainvsections +equation:generalainvsections +(7.5) +ξ = sup +� +ξHN(dim(X ), g(B), τ(π, E), arel, a, T + (dim(X ) − 1)(4g(B) + 3 + γ), β, b), +1 +a((dim(X ) − 1)(5g(B) + 3 + γ) + arelβ + T + arel(dim(X ) − 1)(g(B) − 1)) +� +45 + +where γ = (g(B) dim(X )−g(B)+1)2(dim(X )−1). Using the second term as a lower bound +on ξ and appealing to Equation (7.3) in Lemma 7.5, we have +−KY/B · C ≥ af ∗L · C − arelβ − T − arel(dim(X ) − 1)(g(B) − 1) − (dim(X ) − 1) +≥ (dim(X ) − 1)(5g(B) + 2 + γ) +≥ (dim(Y) − 1)(5g(B) + 2 + γY) +where γY = (g(B) dim(Y) − g(B) + 1)2(dim(Y) − 1). Let S′ be a smooth birational model +of the finite part of the Stein factorization of the evaluation map for the normalization of +the universal family over N. Since the dimension of N is the same as the dimension of the +corresponding family of sections C′ on S′, Corollary 3.4 shows that +−KS′/B · C′ ≥ −KY/B · C − g(B)(dim(S′) − 1). +In particular we see that µmax +[C] (TS′/B) ≥ µ[C](TS′/B) ≥ (4g(B) + 2 + γY). On the other hand +it is clear that (4g(B) + 2 + γY) > 2 ≥ µmax +[C] (π∗TB). Thus we can apply Theorem 5.3 to Y +with J = 2g(B) + 3, T = B, and G = TS′/B. +Consider the dominant family of subvarieties W on Y obtained by taking images of the +subvarieties constructed on S′ by Theorem 5.3. These subvarieties W have the following +properties. First, if we take the strict transform of C in a resolution � +W of W then de- +formations go through ≥ 2g(B) + 3 general points of � +W. In particular by Proposition 3.7 +the strict transform of a general C in � +W is HN-free. +Second, the codimension in N of +the space of sections on � +W can only increase by at most (dim(Y) − 1)(4g(B) + 3 + γY) ≤ +(dim(X ) − 1)(4g(B) + 3 + γ). Applying the HN-free version of the desired statement to � +W +with the constant Tnew = T + (dim(X ) − 1)(4g(B) + 3 + γ), we see that +a(Wη, f ∗L|Wη) ≥ a. +Since such W move in a dominant family on Y, [LST22, Lemma 4.8] shows that the generic +Fujita invariant of Y is at least as large as that of W so that +a(Yη, f ∗L|Yη) ≥ a. +□ +Remark 7.7. Let M denote the component of Sec(X /B) containing the pushforward of the +sections parametrized by N. Then the right hand side of Equation (7.4) is arel·expdim(M)− +T. Since the expected dimension is a lower bound on dim(M), we can replace Equation (7.4) +by the stronger assumption +dim(N) ≥ arel · dim(M) − T. +In particular, when arel = 1 then T should be thought of as the codimension of N in M. +The same remark holds for later theorems as well. +sect:adjrigid +7.4. Adjoint rigidity. Our next goal is to establish a strengthening of Theorem 7.6 that +allows us to conclude adjoint rigidity at the cost of increasing the constants. +lemm:boundinggeneralpoints +Lemma 7.8. Let π : Z → B be a good fibration and let M be an irreducible component of +Sec(Z/B) parametrizing sections C. Suppose that H is a Cartier divisor on Z satisfying +H · C + 1 < h0(Z, OZ(H)). +Then the sections parametrized by M go through at most H · C + 1 general points of Z. +46 + +Proof. Set Q = H ·C +1. Since general points impose codimension 1 conditions on the linear +series |H| we see that for any set of Q points in Z there is a (possibly reducible) divisor +D ∈ |H| containing all Q points. +Suppose for a contradiction that the sections parametrized by M can go through Q + 1 +general points. This means that the space of sections through Q general points of Z forms a +dominant family. In particular, if we fix Q general points and a divisor D ∈ |H| containing +those points, then we can find a section C parametrized by M that contains all the points +but is not contained in Supp(D). Thus D · C ≥ Q > H · C, yielding a contradiction. +□ +lemm:h0growthbound +Lemma 7.9. Let Z be a smooth projective variety of dimension n and let H be a Cartier +divisor on Z such that |H| defines a birational morphism. Then for any non-negative integer +m we have +h0(Z, OZ(mH)) ≥ +�n + m +n +� +Proof. The map |H| defines a morphism g : Z → PN for some N ≥ n such that OZ(H) = +g∗O(1). By composing with a generic projection, we obtain a morphism h : Z → Pn such +that OZ(H) = h∗O(1). Thus we have +h0(Z, OZ(mH)) ≥ h0(Pn, O(m)) = +�n + m +n +� +. +□ +We can now prove the criterion for adjoint rigidity. +theo:adjointrigidcriterion +Theorem 7.10. Let π : X → B be a good fibration and let L be a generically relatively +big and semiample Cartier divisor on X . Assume that Xη is geometrically uniruled. Fix +a positive rational number arel and set a = arela(Xη, L|Xη). +Fix a rational number β. +Fix a positive integer T. +Fix a positive integer b > a such that bL|Xη defines a base- +point free linear series. +Use b to construct an effective π-vertical Q-Cartier divisor E +satisfying the conclusion of Proposition 7.1 with respect to aL. +There is some constant +Γ = Γ(dim(X ), g(B), τ(π, E), arel, a, T, β, b) with the following property. +Suppose that ψ : Y → B is a good fibration equipped with a B-morphism f : Y → X that +is generically finite onto its image. Suppose that N is an irreducible component of Sec(Y/B) +parametrizing a dominant family of sections C on Y which satisfy f ∗(KX/B +a(Xη, L|Xη)L)· +C ≤ β. Suppose that +dim(N) ≥ arel(−f ∗KX/B · C + (dim(X ) − 1)(1 − g(B))) − T. +Suppose that +a(Yη, −f ∗L|Yη) = a. +Then either: +(1) (Yη, −f ∗L|Yη) is adjoint rigid, or +(2) deformations of C go through at most Γ general points of Y. +Proof. Assume that (Yη, f ∗L|Yη) is not adjoint rigid. We may assume that the general section +C is HN-free in Y, since otherwise by Proposition 3.7 the sections parametrized by N can +go through at most 2g(B) general points of Y. Applying Lemma 7.5 we see that +(KY + af ∗L) · C ≤ arelβ + T + arel(dim(X ) − 1)(g(B) − 1) + (dim(X ) − 1) + 2g(B) − 2. +47 + +Adding in E and rearranging slightly, we obtain +(KY+af ∗L+f ∗E)·C ≤ arelβ+T+arel(dim(X )−1)(g(B)−1)+τ(π, E)+(dim(X )−1)+2g(B)−2. +We denote the right hand side of this equation by R = R(dim(X ), g(B), τ(π, E), arel, a, T, β, b). +Since E satisfies the conclusion of Proposition 7.1 with respect to aL, the pullback f ∗(aL+ +E) is Q-linearly equivalent to an effective Q-divisor D such that D|Yη has smooth irreducible +support and coefficient a +b. Let φ : Y′ → Y be a log resolution of (Y, D) and let D′ denote +the strict transform of the π-horizontal components of D. +We may ensure that φ is an +isomorphism on an open neighborhood of Yη. In particular D′ is still generically relatively +big and nef and is irreducible with coefficient a +b, so (Y′, D′) has terminal singularities. Since +we are assuming the sections are HN-free, the strict transform C′ of a general deformation +of C avoids any φ-exceptional divisor and thus satisfies +(KY′ + D′) · C′ ≤ (KY′ + φ∗D) · C′ += φ∗(KY + D) · C′ +≤ R +Let F ′ denote a general fiber of Y′ → B. By Lemma 7.3, there is some integer m only +depending on dim(X ) such that the (KY′ + D′ + mF ′)-MMP is the same as a relative MMP +over B. By assumption on the Fujita invariants KY′ + D′ + mF ′ is on the boundary of the +relative pseudo-effective cone over B. Thus the result of the MMP will be a relative Iitaka +fibration ψ : Y′ ��� Z for this divisor. Furthermore since we are assuming (Yη, f ∗L|Yη) is +not adjoint rigid we know that dim(Z) ≥ 2. +Since D′ is relatively big over B, it is also relatively big over Z in the sense of [HX15, +Definition 2.3]. By [HX15, Theorem 1.4], there is a positive integer k only depending on +dim(X ) and a +b such that |k(KY′ + D′ + mF ′)| defines a rational map birational to the Iitaka +fibration. Applying the canonical bundle formula as in [FM00, Section 4], there is a birational +model ρ : W → Y′ and a morphism ψW : W → ZW birationally equivalent to ψ such that: +• W and ZW are smooth, +• there is an effective Q-Cartier divisor BW and a nef Q-Cartier divisor MW such that +(ZW, BW) is klt and k(KZW + BW + MW) is Cartier, +• KZW + BW + MW is big, and +• for every integer p divisible by k we have that +h0(Y′, OY′(p(KY′ + D′ + mF ′))) = h0(ZW, OZW (p(KZW + BW + MW))) +and the linear series |p(KZW + BW + MW)| defines a birational map. +We claim that there is an integer Q = Q(dim(X ), g(B), τ(π, E), arel, a, T, β, b) such that +Q(KZW + BW + MW) is Cartier and +h0(ZW, OZW (Q(KZW + BW + MW))) > Q(R + m) + 1 +Indeed, consider the birational map ZW ��� V defined by |k(KZW + BW + MW)|. +Let +µ : Z′ → ZW be a smooth model resolving the map and let H denote the basepoint free part +of µ∗(k(KZW + BW + MW)). There are only finitely many possible values of dim(ZW) which +satisfy dim(X ) ≥ dim(ZW) ≥ 2, and thus Lemma 7.9 gives a quadratic lower bound (that +depends only on dim(X )) on the growth rate of sections of multiples of H. In particular +48 + +there is a constant Q′ = Q′(dim(X ), g(B), τ(π, E), arel, T, β, b) such that +h0(Z′, OZ′(Q′H)) > (Q′)k(R + m) + 1. +Then we have +h0(Z′, OZ′(Q′H)) ≤ h0(Z′, OZ′(Q′µ∗(k(KZW + BW + MW)))) += h0(ZW, OZW (Q′k(KZW + BW + MW))) +finishing the proof of the claim with Q = Q′k. Using the comparison of spaces of sections +above, we conclude that also +h0(Y′, OY′(Q(KY′ + D′ + mF ′))) > Q(R + m) + 1 +≥ Q(KY′ + D′ + mF ′) · C′ + 1 +By applying Lemma 7.8 to the divisor Q(KY′ + D′ + mF ′) we obtain an upper bound +Γ(dim(X ), g(B), τ(π, E), arel, a, T, β, b) = Q(R + m) + 1 +on the number of general points that can be contained in deformations of the sections C′ +on Y′. But this also implies an upper bound Γ on the number of general points that can be +contained in deformations of the sections C on Y. +□ +Suppose that Y carries a family of sections which have large L-degree. Although we cannot +necessarily use Theorem 7.10 to show that (Yη, −f ∗L|Yη) is adjoint rigid, by combining with +the results of Section 5 we can at least find a covering family of subvarieties of Y whose +generic fibers are adjoint rigid. +coro:adjointrigidcodim +Corollary 7.11. Let π : X → B be a good fibration and let L be a generically relatively +big and semiample Cartier divisor on X . Assume that Xη is geometrically uniruled. Fix +a positive rational number arel and set a = arela(Xη, L|Xη). +Fix a rational number β. +Fix a positive integer T. +Fix a positive integer b > a such that bL|Xη defines a base- +point free linear series. Use b to construct an effective π-vertical Q-Cartier divisor E sat- +isfying the conclusion of Proposition 7.1 with respect to aL. +There are constants ξ+ = +ξ+(dim(X ), g(B), τ(π, E), arel, a, T, β, b) and T + = T +(dim(X ), g(B), τ(π, E), arel, a, T, β, b) +with the following property. +Suppose that ψ : Y → B is a good fibration equipped with a B-morphism f : Y → X +that is generically finite onto its image. +Suppose that N is an irreducible component of +Sec(Y/B) parametrizing a dominant family of sections C on Y which satisfy f ∗L · C ≥ ξ+ +and f ∗(KX/B + a(Xη, L|Xη)L) · C ≤ β. Suppose that +dim(N) ≥ arel(−f ∗KX/B · C + (dim(X ) − 1)(1 − g(B))) − T. +Suppose that +a(Yη, −f ∗L|Yη) = a. +Let g : S → Y denote the finite part of the Stein factorization of the evaluation map for +the normalization of the universal family over N. Then there is a dominant rational B-map +φ : S ��� T to a normal projective B-variety such that the following holds. For a general +section C† on S parametrized by N let W denote the main component of the closure of +φ−1(φ(C†)). Then: +(1) We have a(Wη, g∗f ∗L|Wη) = a and the pair (Wη, g∗f ∗L|Wη) is adjoint rigid, +49 + +(2) W is swept out by the sections parametrized by a sublocus NW ⊂ N whose closure +has codimension ≤ T + in N, and +(3) there is a resolution of W such that the strict transform of a general section in NW +to the resolution goes through ≥ 2g(B) + 1 general points and is HN-free. +Proof. Define d = dim(Y) and set γ = (dg(B) − g(B) + 1)2(d − 1). We also define constants +Tk and Γk for 2 ≤ k ≤ d as follows. We first set Td = 0 and +Γd = sup{2g(B) + 3, Γ(dim(X ), g(B), τ(π, E), arel, a, T, β, b) + 1} +where Γ is the constant defined in Theorem 7.10. Then for 2 ≤ k < d we define via a +downward induction +Tk = k(Γk+1 + 2g(B) + γ) + Tk+1 +and +Γk = sup{2g(B) + 3, Γk+1, Γ(dim(X ), g(B), τ(π, E), arel, a, T + Tk, β, b) + 1}. +Finally, we set T + = T + supk=2,...,d Tk, Γ+ = supk=2,...,d Γk and ξ+ to be the maximum of the +constant ξ(dim(X ), g(B), τ(π, E), arel, a, T +, β, b) as in Theorem 7.6 and of +1 +a(dim(X )(Γ+ + 2g(B) + γ + 1) + arelβ + T + + (arel + 1)(dim(X ) − 1)(g(B) − 1)). +Recall that S denotes the finite part of the Stein factorization of the evaluation map for +the normalization of the universal family over N. We let S′ denote a smooth birational +model of S that flattens the family of sections on S as in Construction 5.2. We denote the +strict transform of a general section in our family on S′ by C′ and denote the family of +deformations of C′ by N′. Appealing to Corollary 3.4 we have +−KS′/B · C′ + (dim(S′) − 1) ≥ dim(N′) += dim(N) +≥ −KY/B · C + (dim(Y) − 1)(1 − g(B)) +≥ af ∗L · C − arelβ − T + − (dim(X ) − 1) +− (arel dim(X ) + dim(Y) − arel − 1)(g(B) − 1) +where the last line follows from Equation (7.3) of Lemma 7.5. Combining with the bound +f ∗L · C ≥ ξ+, we conclude that +eq:sprimebound +eq:sprimebound +(7.6) +− KS′/B · C′ ≥ dim(S′)(Γ+ + 2g(B) + γ − 1) + 2. +We next inductively define foliations Gd, Gd−1, . . . on S′ by repeatedly applying Theorem 5.3 +to Y using the constants Γd, Γd−1, . . .. We will also denote by ψi the rational map on S′ +induced by the foliation Gi. We will inductively verify the inequalities +µmax +[C] (Gi) ≥ Γi + 2g(B) + γ − 1 +µmax +[C] (TS′/Gi) < Γi + 2g(B) + γ − 1 +which show the requirements necessary to inductively apply Theorem 5.3. +For the base case we set Gd = TS′/B. By construction we have Γd > 2 ≥ µmax +[C] (π∗TB) and +Equation (7.6) shows that +µmax +[C] (TS′/B) ≥ µ[C](TS′/B) ≥ Γ+ + 2g(B) + γ − 1 ≥ Γd + 2g(B) + γ − 1. +50 + +Thus we have verified the two necessary inequalities in the base case. +Now suppose inductively that we apply Theorem 5.3 to Y for the foliation Gi on S′ and +the constant Γi. There are two possible outcomes. The first possibility is that if we set Pi +to be the main component of ψ−1 +i (ψi(C′)) for a general section C′ in our family then the +deformations of C′ go through at least Γi general points of Pi. In this case we stop the +inductive process. The second possibility is that we obtain a new foliation Gi−1 and a new +rational map ψi−1. Note that Theorem 5.3.(c) shows that +µmax +[C] (TS′/Gi−1) < Γi + 2g(B) + γ − 1 ≤ Γi−1 + 2g(B) + γ − 1. +On the other hand, letting r denote the rank of Gi−1 we have +µmax +[C] (Gi−1) ≥ µ[C](Gi−1) = c1(TS′/B) · C − c1(TS′/B/Gi−1) · C +r += c1(TS′/B) · C − c1(TS′/Gi−1) · C + 2g(B) − 2 +r +≥ dim(S′)(Γ+ + 2g(B) + γ − 1) − (dim(S′) − r)(Γi + 2g(B) + γ − 1) +r +≥ Γ+ + 2g(B) + γ − 1 +≥ Γi−1 + 2g(B) + γ − 1 +where the third line is a consequence of Equation (7.6). Thus we have verified the necessary +inequalities for continuing the inductive process. +Since the dimension of ψ−1 +i (ψi(C′)) is always at least 2, this process stops after at most +d − 2 steps with either +• a foliation Gk such that if we set Pk to be the main component of ψ−1 +k (ψk(C′)) for a +general section C′ in our family then the deformations of C′ go through at least Γk +general points of Pk, or +• a rank 1 foliation Gk such that if we set Pk to be the main component of ψ−1 +k (ψk(C′)) +for a general section C′ in our family then the deformations of C′ go through at least +Γk+1 general points of Pk. +Then the foliation Gk induces a rational map S′ ��� T , and hence also a rational map +S ��� T . We prove that in either case this map has the desired properties. Note that the +subvarieties Pk of S′ are birational to the subvarieties W in the statement of the theorem, +and it suffices to prove that Pk has the desired properties. +By applying Theorem 5.3.(2).(b) inductively, we see that the codimension of the space of +deformations of C′ in Pk inside of N is at most +d +� +j=k +(j − 1)(Γj + 2g(B) + γ) +verifying (2). Note that by construction the deformations of C′ in Pk go through either +Γk or Γk+1 general points of Pk. Both quantities are at least 2g(B) + 1. This property is +preserved by passing to the strict transform, and curves through this many general points +must be HN-free by Proposition 3.7, proving (3). Using the lower bound ξ ≤ ξ+, we can +apply Theorem 7.6 to Pk equipped with the family of deformations of C′ to see that +a(Pk,η, f ∗L|Pk,η) ≥ a +51 + +But since the deformations of Pk,η form a dominant family of subvarieties on Yη the equality +must be achieved. To prove (1), it only remains to verify the adjoint rigidity. If Gk has rank +1, Pk is a P1-fibration over B and thus is automatically adjoint rigid. If Gk has rank > 1, +then deformations of C′ on a resolution �Pk of Pk go through at least Γk general points. We +then apply Theorem 7.10 on �Pk to determine adjoint rigidity. This proves (3). +□ +8. Boundedness statements +sect:boundedness +We now turn to proving boundedness statements for the set of morphisms f : Y → X such +that Y carries a family of sections which is “large” on X . In Section 8.1 we prove several +technical statements which combine [Bir22] with our work on twists. In Section 8.2 we state +and prove our main boundedness result, Theorem 8.10. +sect:mainboundedresults +8.1. Boundedness. Our first statement appeals to the recent results of [Bir22] to prove +birational boundedness when the generic fiber is adjoint rigid. +theo:adjointrigidbounded +Theorem 8.1. Let π : X → B be a good fibration and let L be a generically relatively big and +semiample Cartier divisor on X . Assume that Xη is geometrically uniruled. Fix a positive ra- +tional number arel and set a = arela(Xη, L|Xη). Fix a rational number β. Fix a positive integer +T. Fix a positive integer b > a such that bL|Xη defines a basepoint free linear series. Use b to +construct an effective π-vertical Q-Cartier divisor E satisfying the conclusion of Proposition +7.1 with respect to aL. There is some constant ξ = ξ(dim(X ), g(B), τ(π, E), arel, a, T, β, b) +with the following property. +Suppose that ψ : Y → B is a good fibration equipped with a B-morphism f : Y → X that +is generically finite onto its image. Suppose that a(Yη, f ∗L|Yη) = a and that (Yη, f ∗L|Yη) +is adjoint rigid. Suppose that N is an irreducible component of Sec(Y/B) parametrizing a +dominant family of HN-free sections C on Y which satisfy f ∗L · C ≥ ξ and f ∗(KX/B + +a(Xη, L|Xη)L) · C ≤ β. Finally, suppose that +(8.1) +dim(N) ≥ arel(−f ∗KX/B · C + (dim(X ) − 1)(1 − g(B))) − T. +Then: +(1) The set of such projective varieties Y is birationally bounded. +(2) Suppose that L is big and semiample. Choose a positive integer b > a such that |bL| +is basepoint free. Then there is a constant ℸ = ℸ(dim(X ), g(B), arel, a, T, β, b) such +that vol(f ∗L) ≤ ℸ. +Proof. Lemma 7.5 shows that +(KY + af ∗L) · C ≤ arelβ + T + arel(dim(X ) − 1)(g(B) − 1) + (dim(X ) − 1) + 2g(B) − 2. +Proposition 7.1 shows that f ∗(aL + E) is Q-linearly equivalent to an effective Q-Cartier +divisor D such that (Yη, D|Yη) is a terminal pair. Let φ : Y′ → Y be a log resolution of this +pair and let D′ denote the strict transform of the π-horizontal components of D. Since D′ +is irreducible we see that (Y′, D′) is a terminal pair. +52 + +Since we are assuming the sections are HN-free, the strict transform C′ of a general +deformation of C avoids any φ-exceptional divisor and thus satisfies +(KY′ + D′) · C′ ≤ (KY′ + φ∗D) · C′ += φ∗(KY + D) · C′ +≤ arelβ + T + τ(π, E) + arel(dim(X ) − 1)(g(B) − 1) + (dim(X ) − 1) + 2g(B) − 2. +Furthermore C′ is HN-free on Y′. +Run the relative MMP for KY′ + D′ over B. Due to our adjoint rigidity assumption on +the generic fiber, the result will be a birational model ρ : Y′ ��� �Y where K �Y + ρ∗D′ is +relatively Q-linearly equivalent to 0. We denote by �ψ the structural map �ψ : �Y → B. Write +K �Y + ρ∗D′ ∼Q �ψ∗P. Since the map ρ is a (KY′ + D′)-negative birational contraction, +�ψ∗P · ρ∗C′ = ρ∗(K �Y + ρ∗D′) · C′ +≤ (KY′ + D′) · C′ +≤ arelβ + T + τ(π, E) + arel(dim(X ) − 1)(g(B) − 1) + (dim(X ) − 1) + 2g(B) − 2. +Then [Bir22, Theorem 1.3] applies with Z = B and A = ⌈arelβ +T +τ(π, E)+arel(dim(X )− +1)(g(B)−1)+(dim(X )−1)+2g(B)−1⌉p for a point p ∈ B, showing that the set of minimal +models ( �Y, ρ∗D′) is log bounded. +Now suppose that L is big and semiample. Note that the effective divisor E = 0 satisfies +the conclusion of Proposition 7.1 with respect to aL, and we make this choice for E. Recall +that the divisor D is constructed by applying Proposition 7.1. However, in our setting we +can choose D ∼Q aL where Supp(D) is smooth and irreducible and has coefficient +a +b. In +particular we can take Y′ = Y and D′ = D. +Repeating the argument above, we run a relative MMP ρ : Y ��� �Y and see that the +resulting pairs ( �Y, ρ∗D) are log bounded. By construction ρ∗D is irreducible with coefficient +a/b. This implies that there is some constant ℸ such that +vol(f ∗L) = vol(D) +adim Y ≤ vol(ρ∗D) +adim Y +≤ ℸ. +□ +Remark 8.2. In the setting of Theorem 8.1.(1) the variety Y can be replaced by a higher +birational model and N can be replaced by the strict transform family of curves without +affecting the hypotheses. Thus birational boundedness is the best one can hope for. +Construction 2.17 and Theorem 2.18 construct certain families of varieties over K(B). We +next modify these constructions to apply to integral models. +cons:allsubvarieties +Construction 8.3. Let π : X → B be a good fibration and let L be a big and semiample +Cartier divisor on X . Assume that Xη is geometrically uniruled. Set a = a(Xη, L|Xη). +By applying Construction 2.17 to Xη we obtain a proper closed subset Vη ⊂ Xη and a +finite collection of families pi,η : Ui,η → Wi,η whose smooth fibers are birational to closed +subvarieties of Xη. +Let pi : Ui → Wi denote any smooth integral model such that the +structural morphism Ui,η → Xη extends to Ui and let V denote the closure of Vη. +The +subvarieties parametrized by pi,η correspond to the K(B)-points of Wi,η, or equivalently, to +sections of Wi over B. Let Wi denote Sec(Wi/B). We let W = ⊔iWi. We first shrink W +53 + +so that the generic point of every section parametrized by W is contained in the open locus +over which ⊔ipi is smooth. We enlarge V by adding the images in X of the loci where the +maps pi fail to be smooth. Consider the universal family Wi × B → Wi with the evaluation +map Wi ×B → Wi. By taking a base change of pi : Ui → Wi over this morphism, we obtain +a morphism which we denote by Zi → Wi × B. We let Z = ⊔iZi. +Note that W is a countable union of quasiprojective schemes and Z → W × B is a finite +type morphism such that for every closed point w ∈ W the B-scheme Zw → {w} × B +has the property that Zw,η is isomorphic to a fiber of pi,η over a K(B)-point of Wi,η. By +repeatedly stratifying W into locally closed subsets and taking resolutions of components of +Z, we may also ensure that the fibers of Z → W are smooth. We denote the evaluation map +by ι : Z → X . +cons:allfamilies +Construction 8.4. Let ⊔GH(G, B) be the Hurwitz stack. Fix an ´etale covering ⊔GHG → +⊔GH(G, B) from a scheme. +Let π : X → B be a good fibration and let L be a big and semiample Cartier divisor on +X . Assume that Xη is geometrically uniruled. Set a = a(Xη, L|Xη). Let Z → W × B be the +morphism constructed in Construction 8.3. +By Theorem 2.18 there is a finite set of smooth projective K(B)-varieties Yi,j,η equipped +with dominant generically finite morphisms hi,j,η : Yi,j,η → Ui,η and a closed set Rη ⊂ Xη +that has the following property. Suppose that g : Yη → Xη is a generically finite morphism +from a geometrically integral smooth projective variety such that g(Yη) is not contained in +Rη. Suppose furthermore that a(Yη, g∗L|Yη) = a and that (Yη, g∗L|Yη) is adjoint rigid. Since +Yη is geometrically rationally connected by [LTT18, Theorem 4.5], [GHS03, Theorem 1.1] +and [HT06, Theorem 12] show that Yη carries a dense set of rational points. Thus Theorem +2.18 shows that there are indices i, j such that the map g factors rationally through a twist +hσ +i,j,η and Yη maps birationally to a fiber of a morphism Yσ +i,j,η → T σ +i,j,η. +Let Vη be the union of Rη with the generic fiber of the closed set from Construction 8.3. +Then we enlarge Vη by adding si(Bi,j,η) where Bi,j,η is the union of the irreducible components +of the branch locus of hi,j,η. We further enlarge Vη by adding the Zariski closure of the union +of the images of the fibers of ri,j,η which fail to be smooth, fail to have the same a-invariant +as Yi,j, or fail to be adjoint rigid. If Yi,j,η is a component such that some twist of Yi,j,η admits +a K(B)-rational point mapping to Xη \ Vη, then we replace Yi,j,η by this twist. If Yi,j,η is +a component such that no twist of Yi,j,η admits a K(B)-rational point mapping to Xη \ Vη, +then we remove Yi,j,η from our set. +Set Di,j = ⊔GC1(GHG, K(Yi,j,η/Ui,η)HG). By construction Di,j is a countable union of finite +type schemes over C. As described in Section 6.2, there is a morphism +hi,j,η : Yi,j,η → Di,j × Ui,η +which parametrizes twists of hi,j,η : Yi,j,η → Ui,η. After perhaps replacing Di,j by a strat- +ification into locally closed subsets, we can construct integral models in families to obtain +a map Yi,j → Di,j × Ui → Di,j × X whose composition we denote by fi,j. After perhaps +again replacing Di,j by a stratification by locally closed subsets and taking resolutions of +irreducible components of Yi,j, we may ensure that for every closed point d ∈ Di,j the fiber +Yi,j,d is a good fibration equipped with a B-morphism fi,j,d : Yi,j,d → X . By construction +every twist of hi,j,η has an integral model hσ +i,j : Yσ +i,j → Ui that is a member of our fam- +ily. After again replacing Di,j by a stratification into locally closed subsets, we may ensure +54 + +that the Stein factorization of the composition Yi,j → Di,j × Ui → Di,j × Wi induces for +every fiber over a closed point in Di,j the Stein factorization of Yσ +i,j → Wi. Denote the +Stein factorization of Yi,j → Di,j × Wi by ri,j : Yi,j → Ti,j. Then ri,j : Yi,j → Ti,j and +ti,j : Ti,j → Wi define a family of Stein factorizations rσ +i,j : Yσ +i,j → T σ +i,j, tσ +i,j : T σ +i,j → Wi. Due +to the functoriality in Section 6.2.2, we see that Ti,j,η → Di,j × Wi,η parametrizes the family +of twists of Ti,j,η → Wi,η which are induced by twists of Yi,j,η → Ui,η. +Let V be the closure of Vη. We further enlarge V by adding the Zariski closure of the union +of the images of the fibers of ri,j which fail to be smooth, fail to have the same a-invariant as +Yi,j, or fail to be adjoint rigid. We let Bi,j be the closure of Bi,j,η and define B := ∪i,jBi,j. We +also define B′ +i,j ⊂ Wi as the union of components of the branch locus of ti,j which dominate +B and the closures of the images of loci where fibers of ri,j fail to be smooth, fail to have +the same a-invariant as Yi,j, or fail to be adjoint rigid. We define B′ := ∪i,jB′ +i,j. +Recall that we assume that Yi,j,η admits a K(B)-rational point y mapping to Xη \ Vη. +Let �hi,j,η : �Yi,j,η → Ui,η be a geometric Galois closure of hi,j,η : Yi,j,η → Ui,η such that +Bir( �Yi,j,η/Ui,η) = Aut( �Yi,j,η/Ui,η) and �Yi,j,η admits a K(B)-rational point �y mapping to y. +Let Yi,j +�hi,j +−−→ Pi,j +ℓi,j +−−→ Ui be the cover corresponding to the normalizer of Aut( �Yi,j,η/Yi,j,η) in +Aut( �Yi,j,η/Ui,η) such that Pi,j is normal and ℓi,j is finite. Note that every twist Yσ +i,j → Ui +factors through ℓi,j : Pi,j → Ui. By taking the Stein factorization, we have a commutative +diagram +Yi,j +�hi,j � +ri,j +� +Pi,j +bi,j +� +ℓi,j +� Ui +pi +� +Ti,j +ci,j +� Si,j +ai,j +� Wi +where Si,j is projective and normal, bi,j has connected fibers, and ai,j is finite. +We also let Mi,j = ⊔GC1(GHG, K(Ti,j,η/Wi,η)HG) denote the parameter space for all twists +of Ti,j,η → Wi,η. We denote the universal family over Mi,j by T′ +i,j → Mi,j × Wi. Then by +the functoriality established in Section 6.2.2 we have a natural isomorphism +Ti,j → T′ +i,j ×Mi,j Di,j. +We set Y = ⊔i,jYi,j, T = ⊔i,jTi,j, T′ = ⊔i,jT′ +i,j, D = ⊔i,jDi,j, and M = ⊔i,jMi,j with +morphisms Y → D × X , T → D and T′ → M. +For each Yσ +i,j → T σ +i,j, the varieties described by Theorem 2.18.(3).(b) are parametrized by +the K(B)-points of T σ +i,j,η, or equivalently, by the closed points of Sec(T σ +i,j/B). We consider +the relative space of sections S = SecD(T/B) = SecM(T′/B) ×M D. We first shrink S +so that the generic point of every section parametrized by S is contained in the locus +in T over which Y → T is smooth. Then by taking a base change of Y → T over the +evaluation map S × B → T, we obtain a morphism F′ → S × B whose fibers are closed +subvarieties of the various Yσ +i,j. Note that we have a morphism S → Sec(⊔iWi/B), and we +replace S by the the fiber product S×Sec(⊔iWi/B) W so that we have a compatible morphism +S → W. By repeatedly stratifying S into locally closed subsets, throwing away components +whose intersection with a fiber Ys does not dominate B, taking resolutions of irreducible +55 + +components of F′, and taking Stein factorizations, we obtain a commuting diagram +F +� +� +Z +� +S × B +� W × B +where for every closed point s ∈ S the fiber Fs is a normal projective B-variety such that +Fs → B has connected fibers and Fs → Zw is a finite morphism where w denotes the image +of s in W. After taking a stratification of S, we may assume that F → S is a flat family. +Altogether, we have constructed a family F → S × B whose base is a countable union of +finite type schemes and a morphism g : F → S × X such that +(1) for every closed point s ∈ S the fiber Fs is a normal projective B-variety such that +Fs → B has connected fibers; +(2) for every closed point s ∈ S the map gs : Fs → X is a B-morphism that is generically +finite onto its image and the corresponding morphism Fs → Zw is a finite morphism; +(3) for every closed point s ∈ S we have a(Fs,η, g∗ +sL|Fs,η) = a and (Fs,η, g∗ +sL|Fs,η) is adjoint +rigid, +(4) if Y is a good fibration over B and f : Y → X is a generically finite B-morphism +such that a(Yη, f ∗L|Yη) = a and (Yη, f ∗L|Yη) is adjoint rigid, either the map f is +birationally equivalent to gs for some closed point s in our family or f(Yη) ⊂ Vη. +We also have a family Y → D × X parametrizing integral models hσ +i,j : Yσ +i,j → Ui of twists +hσ +i,j,η : Yσ +i,j,η → Ui,η. Note that all such twists hσ +i,j : Yσ +i,j → Ui factor through �hi,j : Yσ +i,j → Pi,j +and that �hi,j : Yσ +i,j → Pi,j is Galois. +We will also need two additional lemmas. +lemm:deforminghnfreetonearbyfibers +Lemma 8.5. Suppose that Y → S × B is a family of good fibrations over B with S irre- +ducible. Suppose that for some closed point s ∈ S we have an HN-free section C of Ys/B. +Then the deformations of C form a dominant family on Y. +Proof. Let M denote the space of deformations of C in Y and for a closed point s′ ∈ S let Ms′ +denote the sublocus parametrizing spaces of sections of Ys′/B. Since H1(C, TYs/B|C) = 0, +[Kol96, Theorem I.2.15.(2)] shows that +dim(M) ≥ dim(Ms) + dim(S). +Since C is an HN-free section in Ys, by replacing C by a general deformation we may ensure +that it avoids any codimension 2 locus of Ys. We conclude that TY/S×B|C is locally free, +and thus the restriction of this sheaf to a general deformation of C is also locally free. As +the universal family over Sec(Y/B) is smooth, Lemma 2.3 shows that the minimal slope of a +quotient of TY/S×B|C′ is a lower semicontinuous function as we vary C′ in Sec(Y/B). Thus +there is an open subset of M parametrizing sections which are HN-free in their fiber. For +a general section C′ parametrized by M denote by s′ the closed point of S parametrizing +the good fibration Ys′ → B containing C′. As explained above C′ is HN-free in Ys′, and in +particular for such points s′ we have dim(Ms′) = dim(Ms). Thus the dimension computation +shows that sections parametrized by M must dominate Y. +□ +Since the next lemma is well-known we will omit the proof. +56 + +lemma:uniquenessoftwists +Lemma 8.6. Let k be a field of characteristic 0 and let f : Y → X be a dominant generically +finite morphism between normal projective varieties defined over k. Assume that +Bir(Y /X) = Aut(Y /X). +Let X◦ ⊂ X be a Zariski open subset such that f|f−1(X◦) : f −1(X◦) → X◦ is ´etale. Let +f σ : Y σ → X be a twist of f over X and suppose that there are k-rational points p ∈ Y (k) +and pσ ∈ Y σ(k) which define the same geometric point on f −1(X◦)k. Then f and f σ are +isomorphic as X-schemes. +We are now ready to prove our main boundedness theorems. For these results we will be +in the situation arel = 1. +theo:positivebounded +Theorem 8.7. Let π : X → B be a good fibration and let L be a big and semiample Cartier +divisor. Assume that Xη is geometrically uniruled. Set a = a(Xη, L|Xη). Fix a rational +number β. Fix a positive integer T. Fix a positive integer b > a such that bL defines a +basepoint free linear series. There is: +• a constant ξ† = ξ†(dim(X ), g(B), a, T, β, b), +• a closed subset V ⊂ X , and +• a bounded family of smooth projective varieties q : �F → �S equipped with �S-morphisms +p : �F → �S × B and g : �F → �S × X +which have the following properties: +(1) For every closed point s ∈ �S, �Fs → B is a good fibration. +(2) For every closed point s ∈ �S the morphism gs : �Fs → X is a B-morphism that is +generically finite onto its image. +(3) For every irreducible component �Fi of �F the composition of g|�Fi : �Fi → �S × X with +the projection �S × X → X is dominant. +(4) For every closed point s ∈ �S we have a(�Fs,η, g∗ +sL|�Fs,η) = a(Xη, L|Xη) and (�Fs,η, g∗ +sL|�Fs,η) +is adjoint rigid. +(5) Suppose that ψ : Y → B is a good fibration equipped with a B-morphism f : Y → X +that is generically finite onto its image and satisfies a(Yη, f ∗L|Yη) ≥ a. +Suppose +that N is an irreducible component of Sec(Y/B) parametrizing a dominant family of +sections C on Y which satisfy f ∗L · C ≥ ξ and f ∗(KX/B + aL) · C ≤ β. Let M ⊂ +Sec(X /B) be the irreducible component containing the pushforward of the sections +parametrized by N. Finally, suppose that +dim(N) ≥ dim(M) − T. +For a general section C parametrized by N, either: +• C is contained in V, or +• there is an irreducible component �Fi of �F and an irreducible component N′ of +Sec(�Fi/B) parametrizing a dominant family of sections on �Fi such that f(C) +is the image of a section C′ parametrized by N′ and if �Fi,s denotes the fiber +containing C′ then the strict transform of C′ in a resolution of �Fi,s is HN-free. +We will divide the proof into five steps. Step 1 is devoted to some preliminary work. In +Step 2, we construct a bounded family of varieties P → Q such that every Y as in Theorem +8.7.(5) is a birational to a twist of a fiber over a closed point of Q. In Step 3, we bound the +57 + +invariants from Corollary 6.13 for the varieties in our family P → Q. Then in Step 4 we use +this corollary to construct a bounded family of twists �F → �S which carry sections of low +L-degree. Finally, in Step 5 we verify that �F → �S has the desired properties. +Proof. Step 1: Let m be the maximum of the degrees of the morphisms Yi,j → Ui from +Construction 8.4 and set d = (m+1)!. Let ⊔GH(G, B) be the Hurwitz stack and fix an ´etale +covering ⊔G,|G|≤dHG → ⊔G,|G|≤dH(G, B) by a scheme. We will work over this base for the +entire proof. +Note that the divisor E = 0 satisfies the condition of Proposition 7.1. +Define ξ = +ξ(dim(X ), g(B), 0, 1, a, T, β, b) as in Theorem 7.6. We then choose +ξ+ = ξ+(dim(X ), g(B), 0, 1, a, T, β, b) +and +T + = T +(dim(X ), g(B), 0, 1, a, T, β, b) +as in Corollary 7.11. Define ℸ = ℸ(dim(X ), g(B), 1, a, T, β, b) as in Theorem 8.1. Finally we +define ξ† = sup {ξ, ξ+}. +Since L is big and semiample, there is a closed subvariety V1 ⊂ X such that the family +of subvarieties of X that are not contained in V1 and have L-degree ≤ ℸ is bounded. By +[LST22, Theorem 4.18.(2)] there is a closed sublocus V2,η ⊂ Xη that contains all subvarieties +with larger generic Fujita invariant and we let V2 denote its closure. Let V3 be the exceptional +closed set from Construction 8.4. We start by setting V to be the union of V1, V2, and V3; +we will later enlarge it. +Let Zi → Wi×B be the families in Construction 8.3. Then there is a finite-type subscheme +Ri ⊂ Wi parametrizing those varieties whose images in X have L-degree ≤ ℸ and are not +contained in V. We denote by Zi,Ri → Ri the universal family over Ri. Set R = ⊔iRi. +Let Y → T → D, T′ → M, F → S and g : F → X × B be defined as in Construction 8.4. +For any closed point s ∈ S the map gs : Fs → X has image that is birationally equivalent +to a fiber Zs of Z → W. By [LST22, Lemma 4.7] we have +a(Fs,η, g∗ +sL|Fs,η) ≤ a(Zs,η, ι∗ +sL|Zs,η) +and if equality is achieved then [LST22, Lemma 4.9] shows that (Zs,η, ι∗ +sL|Zs,η) is adjoint +rigid. If we shrink S to remove all maps gs whose image lies in V, then we will always +have equality of Fujita invariants. We let S′ denote the sublocus of S consisting of maps gs +whose image is a member of our fixed bounded family ZR → R and denote by F′ → S′ the +corresponding family. +Step 2: We next claim that there is a morphism Q → S′ ⊂ S such that Q is of finite +type over C and for every map gs parametrized by S′ the map gs,η is a twist of the generic +fiber of a map parametrized by Q. Indeed, recall from Theorem 2.18 that we have a finite +number of smooth projective varieties Yi,j,η equipped with morphisms +Yi,j,η +hi,j,η � +ri,j,η +� +Ui,η +pi,η +� +Ti,j,η +ti,j,η +� Wi,η +where ti,j,η is Galois. +It follows from our construction that Ri is contained in finitely many irreducible compo- +nents of Sec(Wi/B). Let Sec(Wi/B, B′) denote the space of sections not contained in B′ and +58 + +define Sec(Si,j/B, a−1 +i,j (B′)) analogously. Then ai,j,∗ : Sec(Si,j/B, a−1 +i,j (B′)) → Sec(Wi/B, B′) +is of finite type, so the fiber product +�Ri,j := Sec(Si,j/B, a−1 +i,j (B′)) ×Sec(Wi/B,B′) Ri +is of finite type over C. Moreover for any C ∈ �Ri,j, the fiber b−1 +i,j (Cη) is geometrically ra- +tionally connected so �Ri,j is in the image of Sec(Pi,j/B, ℓ−1 +i,j (B)) → Sec(Si,j/B, a−1 +i,j (B′)). +Thus there is a finite disjoint union of locally closed subschemes of finite type �Ri,j ⊂ +Sec(Pi,j/B, ℓ−1 +i,j (B)) with a surjective morphism �Ri,j → �Ri,j. We denote the base change +of Zi,Ri → Ri over �Ri,j → �Ri,j → Ri by Z�Ri,j → �Ri,j. +Recall that we are working over ⊔G,|G|≤dHG. We claim that every twist of Yi,j,η/Ui,η splits +over an extension K(B′)/K(B) of degree ≤ d. Indeed, if we denote Aut(Yi,j,η/Ui,η) by G, +then Lemma 6.4 shows that one may use a Galois base change of degree ≤ |G| · #Aut(G). +In particular d = (m + 1)! gives an upper bound on this degree. +Since each rational point of Wi,η not contained in B′ will lift to a unique twist of Ti,j,η +and the number of 1-cycles representing the same Galois cohomology class is at most m, the +pushforward morphism +ti,j,∗ : SecMi,j(T′ +i,j/B, t−1 +i,j (B′)) → Sec(Wi/B, B′) × (⊔G,|G|≤dHG) +is a quasi-finite morphism onto its image of degree at most m2. +Now for each C ∈ Sec(Pi,j/B, ℓ−1 +i,j (B)), �h−1 +i,j (C) decomposes into a union of curves which are +Galois conjugate to each other over B where �hi,j : Yi,j → Pi,j is the morphism constructed +in Construction 8.4. +Then after taking a stratification of �Ri,j by locally closed subsets +and replacing �Ri,j by an ´etale cover, the universal property of the Hurwitz stack yields a +morphism +ψi,j : �Ri,j → ⊔G,|G|≤dH(G, B) +that sends a section C of Pi,j/B to the cover C′ → B obtained by normalizing an irreducible +component of �h−1 +i,j (C). We denote by Ri,j the fiber product +�Ri,j ×⊔G,|G|≤dH(G,B) (⊔G,|G|≤dHG) +which is of finite type over C. Thus using the morphism Ri,j → Sec(Wi/B, Bi,j)×(⊔G,|G|≤dHG) +we define the scheme +Q′ +i,j = SecMi,j(T′ +i,j/B, t−1 +i,j (Bi,j)) ×Sec(Wi/B,Bi,j)×(⊔G,|G|≤dHG) Ri,j +which is a finite type scheme over C. Note that Q′ +i,j parametrizes the sections of T σ +i,j/B which +map to Ri such that the twist T σ +i,j is trivialized by a base change C′ → B coming from �h−1 +i,j (C) +as constructed above. Let Q′ = ⊔i,jQ′ +i,j. Then we have a morphism Q′ → SecM(T′/B) and +S′ → SecD(T/B) → SecM(T′/B). +We set Q = Q′ ×SecM(T′/B) S′. Since S′ → SecM(T′/B) is of finite type over each HG, Q +is a scheme of finite type over C. Then we denote the base change of Y → D over Q → D +by �Y → Q and we denote the base change of F → S over Q → S′ ֒→ S by P → Q. +We still must verify that P → Q satisfies the claimed property. Let s ∈ S′ and consider +the corresponding gs : F′ +s → Zr → X with r ∈ R. By Theorem 2.18, gs,η : F′ +s,η → Xη +is birationally equivalent to the map to Xη from a fiber of a twist Yσ +i,j,η → T σ +i,j,η for some +59 + +i, j. +Then since Yσ +i,j factors through Pi,j, by the construction we find a point �r ∈ Ri,j +mapping to r. This point �r specifies a point (B′/B) ∈ ⊔G,|G|≤dHG. On the other hand +one can find a twist Yτ +i,j,η → T σ +i,j,η such that the preimage (�hτ +i,j,η)−1(Cη) of the section C ∈ +Sec(Pi,j/B) corresponding to �r consists entirely of K(B)-rational points. Such a twist will +be trivialized by the base change B′ → B. This means that r is in the image of the map +Q = Q′ ×SecM(T′/B) S′ → Sec(⊔iWi/B, B′). Thus there is a point q ∈ Q mapping to r such +that F′ +s,η → Xη is a twist of Pq,η → Xη for the fiber Pq over q. This finishes the verification +of the desired property. +Step 3: Note that the degree of hσ +q,η : Pσ +q,η → Zq,η is bounded by the maximum of the +degrees of hi,j : Yi,j → Ui. In particular the size of Aut(Pσ +q,η/Zq,η) is uniformly bounded +by the integer m. It follows from Lemma 6.4 that for every closed point q ∈ Q the map +hσ +q,η : Pσ +q,η → Zq,η becomes isomorphic to hq,η : Pq,η → Zq,η after a Galois base change +�B → B of degree ≤ d. +Next we define an integer t by using the family P → Q. Since normality is a constructible +property in proper families ([Gro66, Th´eor`eme 12.2.4]), by Noetherian induction there is +a positive integer t1 that bounds the number of non-normal fibers of Pq → B as we vary +over all q ∈ Q. Since the relative automorphism scheme AutB(Pq/Zq) is quasifinite over +Q × B and since flatness is a constructible property, as we vary over all closed points q ∈ Q +there is a positive integer t2 that bounds the number of places in B where the restriction of +AutB(Pq/Zq) to {q} × B is not flat. We set t = t1 + t2. +Step 4: Lemma 6.3 and Corollary 6.13 show that as we vary the closed point q ∈ Q +the set of twists of hq : Pq → Zq which are trivialized by a base change B′ → B of degree +at most d and with at most t + d(T + T +) branch points is parametrized by a bounded +family. +We denote by �F → �S the bounded subfamily of F′ → S′ parametrizing maps +gs : Fs → Zs satisfying these properties. After taking smooth resolutions and stratifying the +base, we obtain �F → �S such that each fiber is a good fibration over B. We then shrink �S +by removing all irreducible components Sj such that the corresponding family �Fj fails to +dominate X and we enlarge V by taking the union with the closures of the images of these +families. +Step 5: We are now ready to verify the desired properties of �F → �S. Properties (1)-(4) +follow from the construction, and we only need to check (5). Suppose f : Y → X is as in +(5). Applying Theorem 7.6 with arel = 1 we see that a(Yη, f ∗L|Yη) ≥ a. If we have a strict +inequality then f(Y) ⊂ V and so the sections on Y are accounted for by V. From now on +we assume that f(Y) ̸⊂ V which implies that a(Yη, f ∗L|Yη) = a. +We apply Corollary 7.11 to construct subvarieties on the Stein factorization of the eval- +uation map over Y and then take images in Y to obtain a dominant family of subvarieties +F ⊂ Y satisfying: +• the codimension in N of the space of deformations of C in F is at most T +, +• the strict transform of C to a resolution of F is HN-free, +• (Fη, f ∗L|Fη) is adjoint rigid. +We will show that the conclusion of (5) holds for the sections on the general subvariety F +in our family. +Consider a general subvariety F in our family and set Z = f(F). Since it is not possible +for Zη to have larger a-invariant (as it is not contained in Vη), we have a(Zη, L|Zη) = a and +60 + +thus by [LST22, Lemma 4.9] (Zη, L|Zη) is adjoint rigid. Theorem 8.1.(2) shows that Z is +birationally equivalent to a smooth Z′ that is parametrized by the bounded family ZR → R. +In particular the map µ : Fη → Zη is birationally equivalent to a twist of hq : Pq,η → Zq,η +for some closed point q ∈ Q. +Choose a morphism µ′ : F ′ → Z′ birationally equivalent to µ where F ′ is smooth. Let NF +denote the moduli space of deformations of the strict transforms C′ of C in F ′ and let MZ +denote the moduli space of deformations of the image in Z′. Then +dim(MZ) − dim(NF) ≤ dim(M) − (dim(N) − T +) ≤ T + T + +so that NF has codimension at most T + + T in MZ. Then we have +−KF′/B · C′ + (dim(F ′) − 1)(1 − g(B)) = dim(NF) +≥ dim(MZ) − T + − T +≥ −KZ′/B · µ′ +∗C′ + (dim(Z′) − 1)(1 − g(B)) − T + − T +which rearranges to +(KF′/B − µ∗KZ′/B) · C′ ≤ T + + T. +This intersection bound and Corollary 6.13 imply that µ′ : F ′ → Z′ is birationally equivalent +to a twist of hq that is trivialized by a base change B′ → B that has degree at most d and +has at most t + d(T + T +) branch points. Thus µ′ is birationally equivalent to one of the +maps hs : �Fs → Zs parametrized by our bounded family �F → �S. +Consider the strict transform of our family of sections in the fiber �Fs. Since these sections +go through at least 2g(B)+1 general points of �Fs, they are HN-free in this fiber. Lemma 8.5 +shows that the sections deform to give a dominant family on the entire irreducible component +�Fi containing �Fs. Since by construction every irreducible component of �F dominates X , we +deduce that the family of sections gives a dominant family on X . Furthermore, we see that +the general section parametrized by NF is in the image of the map Sec(�Fi/B) → Sec(X /B). +Thus the same property is true for N, proving (5). +□ +Our next boundedness statement is closer in spirit to the results of [LST22]: instead of +using a bounded family �F → �S such that the fibers �Fs,η are adjoint rigid, one can instead +use a bounded family �Y → �S such that the fibers �Ys,η are twists of the finite set of universal +families constructed in Theorem 2.18. +theo:bounded_bigandsemiample +Theorem 8.8. Let π : X → B be a good fibration and let L be a big and semiample Cartier +divisor. Assume that Xη is geometrically uniruled. Set a = a(Xη, L|Xη). Fix a constant β. +Fix a positive integer b > a such that bL defines a basepoint free linear series. +There is a constant ξ† = ξ†(dim(X ), g(B), a, β, b), a proper closed subset V ⊂ X , and a +bounded family �Y → �S×B of good fibrations equipped with a �S×B-morphism �f : �Y → �S×X +such that: +(1) for every closed point s ∈ �S the map �fs is dominant and generically finite but not +birational; +(2) for every closed point s ∈ �S we have a(�Ys,η, − �f ∗ +s KX/B|�Ys,η) = a(Xη, −KX/B|Xη); +(3) as we vary over all closed points s ∈ �S the set of birational equivalence classes of the +maps { �fs,η : �Ys,η → Xη} obtained by base changing to Spec(K(B)) is finite; +61 + +(4) if M ⊂ Sec(X /B) is an irreducible component that generically parametrizes non-HN- +free sections C with L · C ≥ ξ† and (KX/B + a(Xη, L|Xη)L) · C ≤ β then a general +section C parametrized by M satisfies either C ⊂ V or C ∈ �f∗(Sec(�Ys/B)) for some +closed point s ∈ �S. +Proof. We begin by making exactly the same constructions and definitions as in the proof +of Theorem 8.7; we continue from the end of this proof. Additionally we set T = 0. +Let �f : �Y → �S × X be the family of twists of hi,j : Yi,j → Ui which becomes isomorphic +to a member of �Y → Q by a finite base change B′ → B of degree ≤ d and with at most +t + dT + branch points. By Lemma 6.3 �S has finite type over C. +Properties (1), (2), (3) follow from the construction and we only need to verify (4). Suppose +M ⊂ Sec(X /B) is an irreducible component that generically parametrizes non-relatively free +sections C with −KX/B · C ≥ ξ and (KX/B + a(Xη, L|Xη)L) · C ≤ β. We may assume that +M generically parametrizes sections which are not contained in V. Then M parametrizes a +dominant family of sections due to Theorem 8.7.(5). Let Y → X be the finite part of the +Stein factorization for the evaluation map for M. +Applying Corollary 7.11, we find a dominant family of subvarieties F ⊂ Y satisfying: +• the codimension in N of the space of deformations of C in F is at most T +, +• the strict transform of C is HN-free in a resolution of F, +• (Fη, f ∗L|Fη) satisfies a(Xη, L) = a(Fη, f ∗L|Fη) and is adjoint rigid. +Using the universal property described in Theorem 2.18, there exists a twist Yσ +i,j → T σ +i,j over +Ui such that F is birational to the main component F ′ +C of the preimage of a section C under +the map Yσ +i,j → T σ +i,j. Then note that Yσ +i,j → Ui factors through Pi,j → Ui. +We claim that there is some closed point q ∈ Q such that there is an Xη-isomorphism +between Yσ +i,j,η and �Yq,η which maps FC,η to the image P′ +q,η of Pq,η under the map Pq,η → �Yq,η. +Indeed, by the defining property of P → Q we know that FC,η is a twist of Pq′,η for some +q′ ∈ Q. The point q′ specifies a twist Yτ +i,j → T τ +i,j and a point q′′ on Sec(T τ +i,j/B, t−1 +i,j (B′)). We +let p denote the rational point on Sec(Wi/B, B′) obtained by taking the image of q′′ under +Sec(T τ +i,j/B, t−1 +i,j (B)) → Sec(Wi/B, B′). +Let Cp denote the section of Wi → B corresponding to p. Since tτ +i,j : T τ +i,j → Wi is Galois, +every geometric point in (tτ +i,j)−1(Cp,η) is a K(B)-rational point on T τ +i,j,η. +Note that the +geometric fiber corresponding to FC,η will lie over one of these points; we replace q′′ by this +point. Moreover by construction the image Zr of FC has L-degree ≤ ℸ. In particular this +point will lift to q′′′ ∈ Q′. Thus we can define a point q = (q′′′, s′) ∈ Q′ ×SecM(T′/B) S′ = Q +where s′ ∈ S′ is a point corresponding to (q′′, d′) with d′ is the image of q′ via Q → D. This +point q maps to p and Pq,η is birational to the same geometric fiber of �Yq,η as FC,η in Yσ +i,j,η. +By the construction of the families �F → �S, FC,η and P′ +q,η are trivialized by a base change +B′ → B of degree ≤ d with the number of branch points ≤ t + dT +. By Lemma 8.6 Yσ +i,j,η +and �Yq,η are trivialized by the same base change. Thus our assertion follows. +□ +sect:boundedconsequences +8.2. General statements. The following proposition will allow us to remove the global +positivity assumption in Theorem 8.8. +62 + +prop:birmodelposL +Proposition 8.9. Let π : X → B be a good fibration and let L be a generically relatively +ample Q-Cartier divisor. +There is a birational model φ : X + → X that restricts to an +isomorphism of generic fibers over B such that X + is smooth and there is a π ◦ φ-vertical +effective Q-Cartier divisor G such that φ∗L + G is a big and semiample Q-Cartier divisor. +Proof. Choose a positive integer p such that A = pL − KX is generically relatively ample. +Choose E as in Proposition 7.1 applied to A. Thus there is an effective Q-divisor D ∼Q A+E +such that D|Xη has SNC support and has positive coefficients < 1. Let ψ : X ′ → X denote +a log resolution and let D′ denote the strict transform of the π-horizontal components of +D. Note that ψ is an isomorphism over Xη and so D′ and ψ∗D only differ by a π-vertical +divisor. Thus we can choose some positive integer m such that D′ + mF −ψ∗D is Q-linearly +equivalent to an effective Q-Cartier divisor, where F denotes a general fiber of X ′ → B. +By passing to a relative canonical model, we obtain a birational map ρ : X ′ ��� +� +X +such that ρ∗(KX ′ + D′ + mF) is relatively ample. Note that ρ is an isomorphism along +X ′ +η since (KX ′ + D′ + mF)|X ′η was already ample. By increasing m, we can ensure that +ρ∗(KX ′ + D′ + mF) is ample. Let X + denote a birational model admitting morphisms to +X and to � +X . Then the difference between the pullback of 1 +pρ∗(KX ′ + D′ + mF) to X + and +the pullback of L to X + is Q-linearly equivalent to a π-vertical Q-Cartier divisor G′. Since +we may add any fiber of X + → B to the pullback of 1 +pρ∗(KX ′ + D′ + mF) without affecting +semiampleness, we may eliminate the negative part of G′ to obtain the desired effective +π-vertical Q-Cartier divisor G. +□ +Putting everything together, we obtain the following variant of Theorem 8.7. (One can +easily develop an analogous variant of Theorem 8.8 using a similar argument.) +theo:combinedbounded +Theorem 8.10. Let π : X → B be a good fibration and let L be a generically relatively +ample Q-Cartier divisor. Fix a constant β. +(1) There is a proper closed subset R ⊊ X such that if M ⊂ Sec(X /B) is an irre- +ducible component parametrizing a non-dominant family of sections with (KX/B + +a(Xη, L|Xη)L) · C ≤ β then the sections parametrized by M are contained in R. +(2) There is a constant ξ, a proper closed subset V ⊂ X , and a bounded family of smooth +projective B-varieties Y equipped with B-morphisms f : Y → X satisfying: +(a) dim(Y) < dim(X ) and f is generically finite onto its image; +(b) a(Yη, −f ∗L|Yη) = a(Xη, L|Xη) and the Iitaka dimension of KYη + f ∗L|Yη is 0; +(c) if M ⊂ Sec(X /B) is an irreducible component that generically parametrizes non- +HN-free sections C with L · C ≥ ξ and (KX/B + a(Xη, L|Xη)L) · C ≤ β then for +a general section C parametrized by M we have either +(i) C ⊂ V, or +(ii) for some f : Y → X in our family there is a HN-free section C′ of Y/B +such that C = f(C′). +Proof. Let φ : X + → X be a birational morphism and G be a π-vertical effective Q-Cartier +divisor as in Proposition 8.9. Choose a positive integer k such that kφ∗(L + G) is Cartier. +We define L+ = k(φ∗L + G) and β+ = β + τ(π ◦ φ, KX +/X + ka(X + +η , L+|X + +η )G). +Choose a b > a(X + +η , L+|X + +η ) such that bL+|X + +η defines a basepoint free linear series. We +then apply the constructions of the proof of Theorem 8.7 to (X +, L+) with our chosen +constants and with T = 0 to obtain a constant ξ†, a closed subset V+ ⊂ X +, and a bounded +63 + +family of normal projective varieties q : �F → �S equipped with �S-morphisms p : �F → �S × B +and g : �F → �S × X +. We define V to be the union of φ(V+) with the locus where φ−1 is not +defined. We define ξ = 1 +kξ†. +Suppose M ⊂ Sec(X /B) parametrizes a family of sections satisfying L · C ≥ ξ and +(KX/B+a(Xη, L|Xη)L)·C ≤ β. If the locus swept out by the curves parametrized by M meets +the locus where φ−1 is defined, by taking strict transforms we obtain a family of sections C+ +on X +. These sections satisfy L+·C+ ≥ ξ†. Furthermore since a(X + +η , L+|X + +η ) = 1 +ka(Xη, L|Xη) +we have +(KX +/B + a(X + +η , L+|X + +η )L+) · C+ ≤ β+. +First we prove the statement (1). We define R to be the union of V with the images +of the (finitely many) non-dominant families of sections satisfying L · C < ξ and (KX/B + +a(Xη, L|Xη)L) · C ≤ β. If we have a non-dominant family of sections C such that L · C ≥ ξ +then it follows from Theorem 8.7.(5) applied to (X +, L+) that the sections will be contained +in V and thus in R. Altogether we see that R has the desired property. +Next we prove (2). +By Theorem 8.7 the bounded family �F → �S equipped with the +composition �F → �S × X + → �S × X satisfies all the properties except possibly Theorem +8.10.(2).(c). If M parametrizes a non-dominant family of sections, then as explained above +the sections are contained in V. On the other hand, if M parametrizes a dominant family +of non-HN-free sections, then the inclusion TX + → φ∗TX is still injective upon restriction +to a general section C+. Thus the family of strict transforms C+ is a dominant family of +non-HN-free curves on X +. Theorem 8.7 shows that the general section parametrized by M +will be the pushforward of an HN-free section on some fiber �Fs. +□ +9. Fano fibrations +sect:fanofib +In this section we apply previous results to Fano fibrations. +9.1. The Υ-invariant. +defi: +invariant_upsilon +Definition 9.1. Let π : X → B be a Fano fibration. Fix a positive rational number a. By +[KMM92, Theorem 0.2] there is a positive integer b = b(dim(X ), a) such that | − bKXη| is +very ample and b > a. Define Υa(π) to be the minimal value of τ(π, E) as we vary over all +effective π-vertical Q-Cartier divisors E constructed as in Proposition 7.1 with respect to +our choice of b and a. (Note that there is a divisor E achieving this infimum since if a = p +q +then each τ(π, E) lies in +1 +bqZ.) +We also define Υ(π) = Υ1(π). +Remark 9.2. The invariant Υ(π) measures the “failure” of π to be a trivial fibration. For +example, if X = X × B for some Fano variety X then we have Υ(π) = 0. +Applying the results of Section 7 we obtain: +theo:ainvariantandsections +Theorem 9.3. Let π : X → B be a Fano fibration. Fix a positive rational number arel. Fix +a positive integer T. There is some constant ξ = ξ(dim(X ), g(B), Υarel(π), arel, T) with the +following property. +Suppose that ψ : Y → B is a good fibration equipped with a B-morphism f : Y → X that +is generically finite onto its image. Suppose that N is an irreducible component of Sec(Y/B) +64 + +parametrizing a dominant family of sections C on Y which satisfy −f ∗KX/B ·C ≥ ξ. Finally, +suppose that +dim(N) ≥ arel(−KX/B · C + (dim(X ) − 1)(1 − g(B))) − T. +Then +a(Yη, −f ∗KX/B|Yη) ≥ arel. +Proof. Set L = −KX/B. By [KMM92, Theorem 0.2] there is a positive integer b > arel +depending only on dim(X ) and arel such that | − bKXη| is very ample. We apply Theorem +7.6 with β = 0, with a = arel, with our choice of b, and with an effective π-vertical Cartier +divisor E as in Definition 9.1 that achieves the bound τ(π, E) = Υarel(π). The explicit bound +(7.5) for ξ is a max of two terms, which simplifies to +ξ = +1 +arelǫ((1 − ǫ)Υarel(π)+T + (dim(X ) − 1)(5g(B) + 3 + γ) +eq:fanofibrationbound +eq:fanofibrationbound +(9.1) ++ arel(dim(X ) − 1)(g(B) − 1) + 2g(B) − 2 + Ξ) + 1 +where ǫ is a rational number chosen so that no smooth projective variety of dimension +≤ dim(X ) − 1 has a Fujita invariant in the range [1 − ǫ, 1) with respect to any big and nef +Cartier divisor, γ = (g(B) dim(X ) − g(B) + 1)2(dim(X ) − 1), and +• Ξ = 0, if g(B) ≥ 1. +• Ξ is the supremum of the constants obtained by applying Lemma 7.4 to all dimensions +≤ dim(X ), if g(B) = 0. +Theorem 7.6 immediately implies the desired conclusion. +□ +rema:exceptionalrelativelyfree +Remark 9.4. The exceptional set in Geometric Manin’s Conjecture as described in [LST22] +can include families of relatively free sections as well as families of non-relatively free sec- +tions. +For example, sometimes we must discount the contributions of irreducible com- +ponents M ⊂ Sec(X /B) which parametrize relatively free sections when the evaluation +map for the universal family over M has disconnected fibers. +Let f : Y → X denote +the finite part of the Stein factorization of the evaluation map. Theorem 9.3 shows that +a(Yη, −f ∗KX/B|Yη) = a(Xη, −KX/B|Xη) so that such sections can be accounted for by the +exceptional set of [LST22]. +9.2. Proofs of main results. We now prove the theorems stated in the introduction (except +for Theorem 1.10 which is postponed to Section 11). +Proof of Theorem 1.3: (1) Let Y′ be a resolution of Y and let N parametrize the strict +transforms on Y′ of the general sections on Y parametrized by M. Since the Fujita invariant +is birationally invariant, the desired statement follows from Theorem 9.3 applied to Y′ and +N with arel = 1 and T = 0. +(2) To see the equality of Fujita invariants for Y, we let Y′ denote a resolution of singu- +larities and let N denote the family of sections on Y′ such that f∗N is dense in M. The +desired equality follows from Theorem 9.3 applied to Y′, N, and L = −KX/B with arel = 1 +and T = 0. +We next construct the rational map φ. By [KMM92, Theorem 0.2] there is a positive +integer b > arel depending only on dim(X ) and arel such that |−bKXη| is very ample. Define +ξ+ as in Corollary 7.11 applied to Y′, N, L = −KX/B, arel = 1, β = 0, T = 0, with our choice +of b, and with an effective π-vertical Cartier divisor E as in Definition 9.1 that achieves the +65 + +bound τ(π, E) = Υ(π). Then Corollary 7.11 constructs a rational map φ : Y′ ��� Z over +B that has all the desired properties. The only thing left to check is that dim(Z) ≥ 2, +or in other words, that the rational map φ is not trivial. Note that we have an inclusion +TY′/B → f ∗TX/B. Since C′ deforms in a dominant family on Y′, this map remains injective +upon restriction to a general C′ and we conclude that C′ is not an HN-free section on Y′. +But then the map Y′ → B does not satisfy Corollary 7.11.(3), showing that φ must be +non-trivial. +□ +Proof of Theorem 1.6: This follows from Theorem 8.10 applied with L = −KX/B and β = +0. +□ +Proof of Theorem 1.8. Fix a positive integer T. Define the constant ξ(T) (depending also +on dim(X ), g(B), and Υ(π)) as in Theorem 9.3 using the constants arel = 1, an effective +π-exceptional divisor E as in Definition 9.1 such that τ(π, E) = Υ(π), and the chosen value +of T. +Suppose that M is an irreducible component of Sec(X /B) parametrizing sections satis- +fying −KX/B · C ≥ ξ(T). Suppose that N ⊂ M is a subvariety of codimension ≤ T that +parametrizes sections C such that TX/B|C is not generically globally generated. In particular +this implies that the sections parametrized by N do not dominate X ; let Y ⊊ X denote the +locus swept out by these sections. Then Theorem 9.3 shows that +a(Yη, −KX/B|Yη) ≥ a(Xη, −KX/B|Xη). +Furthermore, the explicit description of ξ(T) in Equation (9.1) shows that ξ(T) is linear in +T with leading coefficient 1/ǫ where ǫ = ǫ(dim(X )) is a positive rational number such that +no smooth projective variety of dimension ≤ dim(X ) − 1 has Fujita invariant in [1 − ǫ, 1) +with respect to a big and nef Cartier divisor. By inverting this linear function we obtain the +desired function Q. +□ +Proof of Theorem 1.11: This is the special case of Theorem 4.10.(1) when E is [C]-semistable. +□ +Proof of Theorem 1.12: This follows from Theorem 9.3 applied to an irreducible component +M ⊂ Sec(X /B) using arel = a and the inequality +dim(M) ≥ −KX/B · C + (dim(X ) − 1)(1 − g(B)). +□ +Proof of Theorem 1.14: Let C be a general section parametrized by � +N. By Corollary 3.4 we +have +−K �Y/B · C + (dim( �Y) − 1) ≥ dim( �N) +≥ dim(M) − T +≥ − �f ∗KX/B · C + (dim(X ) − 1)(1 − g(B)) − T +Rearranging we see that +(K �Y/B − �f ∗KX/B) · C ≤ T + g(B)(dim(X ) − 1). +We conclude the desired statement by Corollary 6.13. +□ +66 + +10. Examples +sect:examples +Our first example illustrates how Theorem 1.3 can be used in practice to understand +sections. +exam:cubichyp +Example 10.1 (Cubic hypersurface fibrations). Suppose that π : X → B is a Fano fibra- +tion whose general fiber is a smooth cubic hypersurface of dimension n ≥ 4. We will analyze +the irreducible components of Sec(X /B) parametrizing non-relatively free sections of large +degree. (In the special case when X is a smooth cubic hypersurface and B = P1, [CS09] +proves a stronger statement for X × P1 by classifying all the irreducible components of +Mor(P1, X).) +A straightforward argument combining [H¨or10, 1.3 Proposition] with the techniques of +[LT19b, Theorem 11.1] shows that: +• If n ≥ 5 then there are no non-birational generically finite morphisms f : Yη → Xη +with a(Yη, −f ∗KXη) ≥ 1. +• If n = 4 then there are no non-birational generically finite morphisms f : Yη → Xη +with a(Yη, −f ∗KXη) ≥ 1 unless Xη contains a plane. When Xη contains a plane, the +only possibility is that f is the composition of a birational map φ : Yη → P2 +η and the +inclusion of a plane P2 +η ⊂ Xη. +Let M be a component of Sec(X /B) of sufficiently large anticanonical degree. +Then +Theorem 1.3 shows: +(1) If n ≥ 5 then M will generically parametrize relatively free sections and the evaluation +map for its universal family will have connected fibers as in Remark 9.4. +(2) If n = 4, then M can only fail to generically parametrize relatively free sections if it +parametrizes a family of sections whose intersection with Xη is contained in a plane. +This finishes the classification of irreducible components parametrizing non-free curves of +large degree. +Our second example addresses the non-generically-globally-generated locus. Let π : X → +B be a Fano fibration and let M be an irreducible component of Sec(X /B) of large degree. +Theorem 1.8 shows that the codimension in M of the non-generically-globally-generated +locus will grow linearly in degree except possibly when the sections sweep out a subvariety +with large Fujita invariant. The following example demonstrates that it is possible for the +non-generically-globally-generated locus to have constant codimension. +exam:largenonfreelocus +Example 10.2. Let X be a smooth cubic threefold. Suppose M is a component of Mor(P1, X) +parametrizing maps of anticanonical degree ≥ 3. We will see that M admits a (possibly +reducible) codimension 1 sublocus parametrizing multiple covers of non-free lines. In partic- +ular, the non-free locus in M will always have codimension 1. +[CS09] shows that for any degree d ≥ 2 the moduli stack M0,0(X, d) has two irreducible +components: a component Md that generically parametrizes irreducible free curves and a +component Rd that parametrizes degree d covers of lines. +We will be interested in the +intersection Td of these two components. +Inside of the parameter space of lines on X the sublocus parametrizing non-free lines has +codimension 1. Thus the locus Qd ⊂ Rd parametrizing multiple covers of non-free lines +also has codimension 1. Note that Td will be contained in Qd. On the other hand, since +all components of the moduli stack M0,0(X, d) have the expected dimension M0,0(X, d) has +67 + +only LCI singularities. Thus Td must have codimension 1. Altogether, we see that Td consists +of a (non-empty) union of irreducible components of Qd. +Since the general stable map parametrized by Td has irreducible domain, we see that +every irreducible component of Mor(P1, X) of degree ≥ 3 will have a codimension 1 sublocus +parametrizing non-free morphisms consisting of multiple covers of non-free lines. +This result illustrates Theorem 1.8 applied to the projection π : X ×P1 → P1. Let Y ⊂ X +denote a subvariety swept out by the curves parametrized by an irreducible component of Td. +Then Y is also swept out by a one-parameter family of non-free lines; in particular we have +a(Y, −KX|Y ) = 1. Passing to the relative situation, we see that any codimension 1 locus +of Md parametrizing sections whose normal bundle is not generically globally generated will +sweep out a subvariety Y × P1 which has generic Fujita invariant ≥ 1. +11. An arithmetic application +sect:application +In this section we prove Theorem 1.10. We freely use the notations set up in Section 1.5. +First of all, consider the open subscheme of the relative Hilbert scheme that parametrizes +sections of anticanonical height ≤ d which are not contained in R. There exists a finite set of +places Sd ⊃ S such that this open subset is flat over Spec oF,Sd. Let ψ : Hd → Spec oF,Sd be +the closure of this set inside the relative Hilbert scheme equipped with the reduced structure. +Then ψ : Hd → Spec oF,Sd is projective and flat. +We denote the generic fiber of ψ by Hd. By Theorem 1.6.(1) every irreducible component +of Hd parametrizes a dominant family of sections. Corollary 3.4 shows that the dimension +of such a component is bounded by d + dim Xη. Define +Cd := +2(d+dim Xη) +� +i=0 +hi +sing(Han +d , Q). +Since the ℓ-adic sheaf Riψ∗Qℓ is constructible in the pro-´etale topology by [BS15, Lemma +6.7.2], there exists a pro-´etale open i : U → Spec oF,Sd such that i−1Riψ∗Qℓ is a constant +sheaf in the pro-´etale topology. In particular there exists a finite set of places S′ +d ⊃ Sd such +that we have +hi +´et(Hd,v, Qℓ) = hi +sing(Han +d , Q) +for all i and v ̸∈ S′ +d where Hd,v is the base change of Hd,v to the algebraic closure. By +applying the Grothendieck-Lefschetz trace formula and a version of the Weil conjectures for +singular projective varieties ([Del80]), we conclude that +N(Xv \ Rv, −KXv/Bv, d) ≤ #Hd,v(kv) ≤ Cdqd+dim Xη +v +Thus assuming dǫ > dim Xη, we can conclude that +N(Xv \ Rv, −KXv/Bv, d) +qd(1+ǫ) +v +→ 0 +as v → ∞. +References +[Ach06] +J. D. Achter. The distribution of class groups of function fields. J. Pure Appl. Algebra, 204(2):316– +333, 2006. +68 + +[Ara10] +C. Araujo. The cone of pseudo-effective divisors of log varieties after Batyrev. Math. Z., +264(1):179–193, 2010. +[Bat88] +V. V. Batyrev. Distribution of rational points of bounded height. a lecture at Math. Inst. Berlin, +Thu 21th Jul, 1988. +[BCHM10] C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan. 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Soc., 41(5):779– +781, 2009. +71 + +Department of Mathematics, Boston College, Chestnut Hill, MA +02467 +Email address: lehmannb@bc.edu +Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, +IN 46556 +Email address: eriedl@nd.edu +Graduate School of Mathematics, Nagoya University, Furocho Chikusa-ku, Nagoya, 464- +8602, Japan +Institute for Advanced Research, Nagoya University +Email address: sho.tanimoto@math.nagoya-u.ac.jp +72 + diff --git a/4NAzT4oBgHgl3EQfuv1g/content/tmp_files/load_file.txt b/4NAzT4oBgHgl3EQfuv1g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dc8fb372ffeb65a4d9853354e48ad2587f88d6af --- /dev/null +++ b/4NAzT4oBgHgl3EQfuv1g/content/tmp_files/load_file.txt @@ -0,0 +1,3174 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf,len=3173 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='01695v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='AG] 4 Jan 2023 NON-FREE SECTIONS OF FANO FIBRATIONS BRIAN LEHMANN, ERIC RIEDL, AND SHO TANIMOTO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let B be a smooth projective curve and let π : X → B be a smooth integral model of a geometrically integral Fano variety over K(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Geometric Manin’s Conjecture predicts the structure of the irreducible components M ⊂ Sec(X/B) which parametrize non- relatively free sections of sufficiently large anticanonical degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Over the complex numbers, we prove that for any such component M the sections come from morphisms f : Y → X such that the generic fiber of Y has Fujita invariant ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Furthermore, we prove that there is a bounded family of morphisms f which together account for all such components M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' These results verify the first part of Batyrev’s heuristics for Geometric Manin’s Conjecture over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Our result has ramifications for Manin’s Conjecture over global function fields: if we start with a Fano fibration over a number field and reduce mod p, we obtain upper bounds of the desired form by first letting the prime go to infinity, then the height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Background 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Sections of good fibrations 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Grauert-Mulich 18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Sections through general points 28 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Twists over function fields of complex curves 31 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fujita invariant and sections 41 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Boundedness statements 52 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fano fibrations 64 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Examples 67 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' An arithmetic application 68 References 68 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Introduction Since Mori’s groundbreaking work in [Mor79] and [Mor82] the moduli space of curves has played a central role in the analysis of Fano varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The irreducible components of the moduli space that parametrize free curves – that is, curves for which the restriction of the tangent bundle is sufficiently positive – are generically smooth and have other desirable geometric properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By contrast, the irreducible components that only parametrize non- free curves frequently exhibit pathological behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Our goal is to classify these “exceptional components” for Fano varieties over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 1 We show that the exceptional components that parametrize curves of sufficiently large degree must come from morphisms which increase the Fujita invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We call such mor- phisms “accumulating maps”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' their geometry is strongly constrained by the Minimal Model Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Furthermore, we show that all exceptional components can be accounted for by a bounded family of accumulating maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The analogous statements are still true for the moduli space of sections of a C-Fano fibration over a curve and we will work in this more general setting for the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Our approach to this problem is motivated by arithmetic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In [Bat88] Batyrev developed a heuristic argument for Manin’s Conjecture over a global function field based on some assumptions about the geometry of the space of curves on an Fq-Fano variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Geo- metric Manin’s Conjecture adapts Batyrev’s assumptions into a precise set of conjectures about the structure of the moduli space of sections on a k-Fano fibration over a curve for arbitrary fields k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Our work completely resolves the first prediction of Geometric Manin’s Conjecture for a C-Fano fibration over a curve: exceptional components come from accumu- lating maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Our results provide evidence for Batyrev’s heuristics in characteristic p, and in some circumstances we can deduce an arithmetic statement: if we start with a Fano fibration over a number field and reduce mod p, we obtain upper bounds on the counting function in Manin’s Conjecture by first letting the prime go to infinity, then the height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fano fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let B be a smooth irreducible projective curve over a field k and let η denote its generic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' A Fano fibration over B is a flat k-morphism π : X → B from a smooth projective variety X whose generic fiber Xη is a geometrically integral Fano variety over K(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We will denote by Sec(X /B) the moduli space of sections of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The following definition identifies the analogue of a free curve in the setting of fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' A section C of π : X → B is relatively free if TX/B|C is globally generated and H1(C, TX/B|C) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' sect:introgmc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Geometric Manin’s Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Geometric Manin’s Conjecture is based on an influ- ential heuristic for Manin’s Conjecture developed by Baytrev ([Bat88]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The main invariant in Batyrev’s heuristic is the Fujita invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' defi:a-invariant Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let X be a smooth projective variety over a field of characteristic 0 and let L be a big and nef Q-Cartier divisor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The Fujita invariant of (X, L) is a(X, L) = min{t ∈ R | KX + tL is pseudo-effective }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If L is nef but not big, we formally set a(X, L) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If X is singular, choose a resolution of singularities φ : X′ → X and define a(X, L) to be a(X′, φ∗L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (The choice of resolution does not affect the value by [HTT15, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=') Geometric Manin’s Conjecture (based on [EVW16], [LT19a], and [LST22]) predicts that sections of Fano fibrations over any ground field are governed by two key principles: (Exceptional set) “Pathological” families of sections are controlled by the Fujita in- variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (Stability) “Non-pathological” families of sections exhibit homological or motivic stability as the degree increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Our main theorems establish a precise version of the first principle over the complex num- bers: the Fujita invariant controls the failure of relative freeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In accordance with the 2 asymptotic nature of Manin’s Conjecture, our results apply to any family of sufficiently large degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose π : X → B is a Fano fibration over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Our first main result, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3, shows that every non-relatively free section of sufficiently large degree comes from an accumulating map which does not decrease the Fujita invariant along the generic fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This verifies a conjecture of [LST22] and generalizes earlier results for del Pezzo surface fibrations ([LT22], [LT21a]) to fibrations of arbitrary dimension using completely different techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 has two cases which correspond to the two ways in which a section C could fail to be relatively free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' First, the deformations of C could fail to dominate X , in which case the sections sweep out a subvariety Y ⊊ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Second, the deformations of C could dominate X but TX/B|C could have a low degree quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In the latter case we expect C to deform more in some directions than in others so that there is a subvariety of X swept out by the “most positive” deformations of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In fact, there is an algebraic foliation on a generically finite cover of X such that most deformations of C are tangent to the foliation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' our subvariety is swept out by the images of the leaves meeting C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In both cases Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 shows that the relevant subvariety must have a large Fujita invariant along the generic fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' theo:maintheorem1 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a Fano fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is a constant ξ = ξ(π) with the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let M be an irreducible component of Sec(X /B) parametrizing a family of non-relatively free sections C which satisfy −KX/B · C ≥ ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Uν denote the normalization of the universal family over M and let ev : Uν → X denote the evaluation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then either: (1) ev is not dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then the subvariety Y swept out by the sections parametrized by M satisfies a(Yη, −KX/B|Yη) ≥ a(Xη, −KX/B|Xη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) ev is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Letting f : Y → X denote the finite part of the Stein factorization of ev, we have a(Yη, −f ∗KX/B|Yη) = a(Xη, −KX/B|Xη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Furthermore, there is a dominant rational map φ : Y ��� Z over B with connected fibers such that the dimension of Z is at least 2 and the following properties hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let C′ denote a general section of Y → B parametrized by M and let W′ ⊂ Y denote the unique irreducible component of the closure of φ−1(φ(C′)) which maps dominantly to φ(C′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is a resolution ψ : W → W′ such that the locus where ψ−1 is well-defined intersects C′ and ψ has the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (a) We have a(Wη, −ψ∗f ∗KX/B|Wη) = a(Xη, −KX/B|Xη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (b) The Iitaka dimension of KWη − a(Wη, −ψ∗f ∗KX/B|Wη)ψ∗f ∗KX/B|Wη is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (c) The general deformation of the strict transform of C′ in W is relatively free in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (d) There is a constant T = T(π) depending only on π, but not M, such that the sublocus of M parametrizing deformations of the strict transform of C′ in W has codimension at most T in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose X is a Fano variety and we are studying irreducible components of Mor(B, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Any family of non-free curves on X leads to a family of non-relatively free 3 sections of π : X ×B → B and thus Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 gives a classification result for such families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' However, it is natural to ask whether non-free curves can be described using generically finite maps f : Y → X instead of generically finite maps f : Y → X × B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The answer is “yes,” but due to length constraints we will give the details in a supplementary paper ([LRT22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 has two key consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' First, the Fujita invariant can be computed using tools from the Minimal Model Program and thus Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 gives a practical way to classify families of non-relatively free sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 we analyze the moduli spaces Sec(X /B) when π : X → B is a cubic hypersurface fibration and B is a curve of arbitrary genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' When dim(Xη) ≥ 5, we show that the “exceptional set” is empty so that every component of Sec(X /B) of sufficiently high degree will generically parametrize relatively free sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For dimension 4 a similar analysis allows us to describe the families of non-relatively free sections of large degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Second, [Bir21] imposes strong finiteness constraints on Fujita invariants and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 allows us to deduce finiteness results for families of non-relatively free sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Recall that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 shows that families of non-relatively free sections come from maps f : Y → X such that the Fujita invariant of Yη is at least a(Xη, −KX/B|Xη) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If we pass to an algebraic closure K(B) then the set of maps fη : Yη → Xη such that a(Yη, −f ∗ ηKXη) ≥ 1 satisfy certain types of boundedness (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' [LST22, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' However, the analogous boundedness statements over K(B) are no longer true since a map over K(B) can correspond to infinite families of twists over K(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In Section 6 we systematically study the set of twists of the map fη : Yη → Xη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='14 shows that amongst all the twists only a bounded subfamily carry a family of sections which is dense in an irreducible component of Sec(X /B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In this way, we conclude that all non-relatively free sections will come from a bounded family of maps f : Y → X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' theo:maintheorem2 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a Fano fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (1) There is a proper closed subset R ⊊ X such that if M ⊂ Sec(X /B) is an irre- ducible component parametrizing a non-dominant family of sections then the sections parametrized by M are contained in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) There is a constant ξ = ξ(π), a proper closed subset V ⊂ X , and a bounded family of smooth projective B-varieties Ys, s ∈ S equipped with B-morphisms fs : Ys → X satisfying: (a) dim(Ys) < dim(X ) and fs is generically finite onto its image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (b) a(Ys,η, −f ∗ s KX/B|Ys,η) = a(Xη, −KX/B|Xη) and the Iitaka dimension of KYs,η − f ∗ s KX/B|Ys,η is 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (c) if M ⊂ Sec(X /B) is an irreducible component that generically parametrizes non-relatively free sections C with −KX/B · C ≥ ξ then for a general section C parametrized by M we have either (i) C ⊂ V, or (ii) for some fs : Ys → X in our family there is a relatively free section C′ of Ys/B such that C = fs(C′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6 establishes geometric analogues of various conjectures about the exceptional set in Manin’s Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For example, suppose that B is a smooth projective Fq-curve and π : X → B is a Fano fibration equipped with an adelic metrization on the 4 relative canonical bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Weak Manin’s Conjecture predicts that there exist a constant C > 0 and a closed subset R ⊂ X such that for any ǫ > 0 the number of sections meeting X \\R of height at most d is bounded above by Cqd(1+ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Using the heuristic estimate #M(Fq) ≈ qdim M, this means that R should contain all sections parametrized by a family M such that dim(M)/expdim(M) ≥ 1 + ǫ (with perhaps finitely many exceptions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (1) shows that over C there exists a closed set R with this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' A geometric application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose M is an irreducible component of Sec(X /B) and let N ⊂ M denote the sublocus parametrizing sections C such that TX/B|C is not generically globally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' One would like to find a lower bound on the codimension of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This problem has been previously studied when X is a smooth Fano variety and M ⊂ Mor(P1, X), in which case N ⊂ M is simply the non-free locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For example, [BS22, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2] shows that when X is a smooth hypersurface whose dimension is much larger than the degree and M is an irreducible component of Mor(P1, X) then the codimension of N ⊂ M grows linearly in the anticanonical degree of the curves parametrized by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We prove the first general statement for arbitrary Fano fibrations: the codimension of the non-generically-globally-generated locus grows linearly in the degree unless there is a clear geometric reason why it cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' theo:maintheorem3 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a Fano fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is a linear function Q(d) whose leading coefficient is a positive number depending only on dim(X ) such that the following property holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that M is an irreducible component of Sec(X /B) parametrizing a family of sec- tions C which satisfy −KX/B · C = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let N ⊂ M be a subvariety parametrizing sections C such that TX/B|C is not generically globally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then either (1) the codimension of N in M is at least sup{⌊Q(d)⌋, 0}, or (2) the sections parametrized by N sweep out a subvariety Y ⊊ X satisfying a(Yη, −KX/B|Yη) ≥ a(Xη, −KX/B|Xη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2 we will show that the codimension of the non-generically-globally-generated locus can be constant as the degree increases, demonstrating that case (2) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8 must be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Over C, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8 shows that families of sections such that TX/B|C is not generically globally generated are either contained in the exceptional locus or they have large codimension in a component of Sec(X /B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assuming the analogous statement over a global function field, we can expect such sections to make a negligible contribution to the counting function for Manin’s Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8 supports the novel formulation of Manin’s Conjecture due to [Pey17] which only counts rational points which are “free” in a suitable sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' subsec:arithmeticapp 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' An arithmetic application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Our results can be applied to prove an upper bound of Manin type over a global function field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let F be a number field and let B be a smooth projective curve over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let S be a finite set of places of F including all archimedean places and let oF,S be the ring of S-integers in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a Fano fibration defined over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' After perhaps enlarging S, we can find an integral model �π : X → B of π over oF,S such that X and B are smooth over oF,S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let R ⊂ X be the Zariski closure of the union of the loci swept out by non-dominant families of 5 sections in Sec(X/B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (1) implies that the base change of R to C is contained in a proper closed subset, so in particular R itself is a proper closed subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We consider the flat closure R ⊂ X of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let v be a non-archimedean place of F not contained in S and consider the reduction πv : Xv → Bv at v which is defined over a finite field kv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Rv be the reduction of R at v and let Sec(Xv/Bv, Rv)≤d be the open subset of Sec(Xv/Bv) parametrizing sections C ̸⊂ Rv of anticanonical height ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then we consider the following counting function: N(Xv \\ Rv, −KXv/Bv, d) = #Sec(Xv/Bv, Rv)≤d(kv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Weak Manin’s Conjecture over K(Bv) predicts that for any ǫ > 0 we have N(Xv \\ Rv, −KXv/Bv, d) = o(qd(1+ǫ) v ), as d → ∞ where qv = #kv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The following approximation of this conjecture was suggested to us by Jordan Ellenberg and Melanie Matchett Wood: theorem:arithmeticapp Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let F, S, �π : X → B be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then assuming dǫ > dim Xη, we have N(Xv \\ Rv, −KXv/Bv, d) qd(1+ǫ) v → 0 as v → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This result fits into the recent trend of taking hard arithmetic questions that are asymp- totic in a different parameter and making them more accessible by first letting v go to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This technique has been explored in the contexts of Malle’s Conjecture and Cohen-Lenstra heuristics over global function fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' [Ach06, EVW16, FLR22, PW21, LWZB19, ETW17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The proof of our main results requires a number of statements which are interesting in their own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We first outline the proof of case (2) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For simplicity we assume that ev : Uν → X has connected fibers so that the Stein factorization Y of ev is equal to X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We must construct a rational map φ on X that captures the geometry of this dominant family of non-relatively free sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Our strategy relies on the theory of foliations and slope stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Recall that for any curve C that deforms in a dominant family on X [CP11] defines a notion of slope stability and Harder-Narasimhan filtrations for torsion-free sheaves on X with respect to the numerical class [C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In Section 4 we prove that when E is a torsion-free sheaf on X that is semistable with respect to the numerical class [C] of a flat family of curves then E|C is “almost” semistable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' theo:introgm Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a flat morphism with connected fibers from a smooth projective variety X to a smooth projective curve B and let E be a torsion-free sheaf on X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let M be an irreducible component of Sec(X /B) and let Uν denote the normalization of the universal family over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that the evaluation map ev : Uν → X is dominant with connected fibers and that for some open subset M◦ red ⊂ Mred the restriction of ev to the preimage of M◦ red is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For a general curve C parametrized by M, write 0 = F0 ⊂ F1 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' ⊂ Fr = E|C for the Harder-Narasimhan filtration of E|C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that E is [C]-semistable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then for every index i we have |µ(E|C) − µ(Fi/Fi−1)| ≤ (g(B) dim(X ) − g(B) + 1)2 rk(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 6 Now suppose that C is a general member of a dominant family of non-relatively free sections of π whose evaluation map has connected fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since by hypothesis C is not relatively free, we see that TX/B|C must have a low slope quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Applying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='11 to a birational model flattening the family, we can “lift” this quotient to all of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The result is a foliation F ⊂ TX of large slope and the pioneering results of [CP19] show that F is induced by a rational map φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We carry out this construction in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' It only remains to verify the desired properties of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The most difficult is the computation of the Fujita invariant as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By appealing to Birkar’s recent boundedness results in the Minimal Model Program, we prove the following general criterion for computing the Fujita invariant in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' theo:introainvariant Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a Fano fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive rational number a and a positive integer T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is some constant ξ = ξ(π, a, T) with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that ψ : Y → B is a flat morphism with connected fibers from a smooth projective variety Y and f : Y → X is a B-morphism that is generically finite onto its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that N is an irreducible component of Sec(Y/B) parametrizing a dominant family of sections on Y and let M denote the irreducible component of Sec(X /B) containing f∗N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume that the sections C parametrized by N satisfy −f ∗KX/B · C ≥ ξ and that dim(N) ≥ a · dim(M) − T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then a(Yη, −f ∗KX/B|Yη) ≥ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We prove statements analogous to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='12 in the more general setting of pairs (X , L) where X is a smooth projective variety admitting a flat morphism with connected fibers π : X → B and L is a generically relatively big and semiample Cartier divisor on X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Returning to the setting of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2), we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (b) by combining Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='12 with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We next outline the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose we have a dominant generically finite morphism fη : Yη → Xη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' As discussed earlier, the key is to understand how the set of twists fη interacts with the behavior of sections on an integral model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We systematically analyze this relationship in Section 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' in particular, we prove the following statement that undergirds Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' theo:introtwists Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a Fano fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Y be a normal projective vari- ety equipped with a flat morphism with connected fibers ψ : Y → B and with a dominant generically finite B-morphism f : Y → X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that �Y is a B-variety which is smooth and projective and �f : �Y → X is a dominant generically finite morphism such that �fη : �Yη → Xη is birational to a twist of fη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive integer T and suppose that there exists an irreducible component �N ⊂ Sec( �Y/B) parametrizing a dominant family of sections on �Y such that the pushforward of � N has codimension at most T in a component of Sec(X /B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then there exist constants d = d(Y/X ) and n = n(Y/X , T) and a finite Galois morphism B′ → B of degree at most d with at most n branch points such that the base changes of fη and �fη to K(B′) are birationally equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 7 This result implies that the set of �Y satisfying the conditions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='14 is birationally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We prove this boundedness by constructing a parameter space of twists which is of finite type over the Hurwitz stack parametrizing finite covers B′ → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' History.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Ever since the seminal results due to Mori and his coauthors ([Mor79], [Mor82], [MM86]) the moduli space of curves has played a prominent role in the study of Fano vari- eties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The notion of free rational curves goes back to pioneering work by Koll´ar–Miyaoka– Mori ([KMM92], [Kol96]) on rational connectedness of Fano varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since then there have been many breakthroughs in the description of the moduli spaces Mor(B, X) for Fano va- rieties X, most notably when B = P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' One particularly influential example is the analysis of rational curves on Fano hypersurfaces pioneered by [HRS04] and subsequently developed by [CS09], [BK13], and [RY19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' ([BV17], [BS22] provide a different approach to this prob- lem using an idea from analytic number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=') Another important class of examples is the moduli spaces of curves on various homogeneous spaces ([Tho98], [KP01], [Bou16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' However for a long time it was unclear what structure to expect for arbitrary Fano varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The situation was clarified by the introduction of ideas from arithmetic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Manin’s Conjecture is a conjectural asymptotic formula for the counting function of rational points on Fano varieties formulated and refined in [FMT89], [BM90], [Pey95], [BT98], and [LST22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In [Pey17] Peyre proposed another version of Manin’s Conjecture using the notion of freeness of rational points which is inspired by the concept of free rational curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' A motivic version of Manin’s Conjecture has been established for equivariant compactifications of vector groups in [CLL16] and [Bil18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In Manin’s Conjecture, it is important to exclude the contribution to the counting function from “exceptional sets” where rational points accumulate too quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The relationship between exceptional sets and Fujita invariants was developed in [HTT15], [LTT18], [HJ17], [LT17], [Sen21], [LST22], and [LT19b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' These developments culminated in the main theorem of [LST22] proving that the contribution to the exceptional set coming from maps f : Y → X such that Y has larger a and b invariants will be contained in a thin set of rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This result is a source of our main theorem (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6) showing that pathological components come from a bounded family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In his influential notes ([Bat88]) Batyrev gave a heuristic for the global function field version of Manin’s Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Over time the principles underlying Batyrev’s heuristic were made into precise conjectures and extended to arbitrary ground fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' First, building upon earlier work on homological stability by [Seg79], [CJS94], and many others, [EVW16] highlighted the connection between homological stability and rational point counts via the Grothendieck-Lefschetz trace formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Second, based on the analysis of the exceptional set described earlier, [LT19a] predicted the geometry underlying “pathological” families of ra- tional curves on Fano varieties and obtained a first prototype result of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Further works leveraged this intuition to study rational curves for Fano varieties of dimension ≤ 3 and for sections of del Pezzo fibrations ([LT19a], [LT21b], [LT22], [LT21a], [BLRT22], [ST22], and [BJ22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Together, the two principles in Batyrev’s heuristic (stated in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2) are known as Geometric Manin’s Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Geometric Manin’s Conjecture unifies many disparate ex- amples and clarifies the conjectural structure of Mor(B, X) for arbitrary Fano varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6 are the first statements in Geometric Manin’s Conjecture which have been proved for arbitrary Fano fibrations over curves of arbitrary genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 8 Acknowledgments: The authors thank Shintarou Yanagida for a helpful conversation about stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The authors thank Jordan Ellenberg and Melanie Matchett Wood for suggest- ing an arithmetic application of our work and Lars Hesselholt for recommending the reference [BS15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The authors also thank Izzet Coskun and Zhiyu Tian for comments on this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Part of this project was conducted at the SQuaRE workshop “Geometric Manin’s Conjecture in characteristic p” at the American Institute of Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The authors would like to thank AIM for the generous support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Brian Lehmann was supported by Simons Foundation grant Award Number 851129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Eric Riedl was supported by NSF CAREER grant DMS-1945944.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Sho Tanimoto was partially supported by JST FOREST program Grant number JPMJFR212Z, by JSPS Bilateral Joint Research Projects Grant number JPJSBP120219935, and by JSPS KAKENHI Early-Career Scientists Grant number 19K14512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Background sect:background Throughout all our schemes will be assumed to be separated and every connected com- ponent will have finite type over the base ring (which is usually C or the function field of a complex curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Recall that in this situation the normalization of a scheme X is isomorphic to the normalization of Xred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' A variety is a separated integral scheme of finite type over the base field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Given a coherent sheaf F on a variety V , we denote by Ftors the torsion subsheaf and by Ftf the quotient of F by its torsion subsheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Given a dominant generically finite morphism of projective varieties f : Y → X, we denote by Aut(Y/X) the automorphism group of Y over X and by Bir(Y/X) the birational automorphism group of Y over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose we have a dominant morphism of varieties f : U → V such that the general fiber of f is geometrically irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that T ⊂ V is a subvariety that meets the open locus over which f has geometrically irreducible fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then f −1(T) has a unique irreducible component which dominates T under f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We call this the “main component” of f −1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' When X is a projective variety, we will let N1(X)R denote the space of R-Cartier divisors up to numerical equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In this finite-dimensional vector space we have the pseudo- effective cone Eff 1(X) and the nef cone Nef1(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Dually, we will let N1(X)R denote the space of R-1-cycles up to numerical equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Inside N1(X)R we have the pseudo-effective cone Eff1(X) and the nef cone Nef1(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Given a curve C, we will denote its numerical class by [C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We also use the standard definitions and techniques from the Minimal Model Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' See [KM98] and [BCHM10] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let f : X → Y be a projective morphism of varieties and let L be a Q-Cartier divisor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For a property P of Q-Cartier divisors (such as ample, big, nef, semiample, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' ), we say that L is generically relatively P if the restriction of L to the generic fiber of f satisfies P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Vector bundles on curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let B be a smooth projective curve and let E be a vector bundle of rank r on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Write the Harder-Narasimhan filtration of E as 0 = F0 ⊂ F1 ⊂ F2 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' ⊂ Fk = E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 9 We denote by µmax(E) the maximal slope of any torsion-free subsheaf, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=', µmax(E) = µ(F1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We denote by µmin(E) the minimal slope of any torsion-free quotient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=', µmin(E) = µ(E/Fk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that by the mediant inequality for every index 1 < i ≤ k we have eq:mediant eq:mediant (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1) µ(Fi) = c1(Fi−1) + c1(Fi/Fi−1) rk(Fi−1) + rk(Fi/Fi−1) < µ(Fi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' lemm:minslopelower Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let f : Y → S be a smooth projective morphism of varieties with relative dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that E is a locally free sheaf on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then µmin(E|Ys) is a lower- semicontinuous function on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By [HL97, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2] there is a dense open set U ⊂ S and a torsion-free sheaf F on YU such that (E/F)|Yt is the minimal slope quotient of E|Yt for the fiber Yt over any point t ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Arguing by Noetherian induction, it suffices to show that if s denotes an arbitrary point of S then µmin(E|Ys) ≤ µmin(E|Yt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By projectivity of the Quot scheme, for any point s ∈ S there is a surjection E|Ys → Qs where Qs has the same degree and rank as (E/F)|Yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular µ(Qs,tf) ≤ µ((E/F)|Yt) finishing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We say that a coherent sheaf E on a smooth projective curve B is generically globally generated if the evaluation map H0(B, E) ⊗ OB → E is surjective at the generic point of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' lemm:genericallygloballygenerated Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let B be a smooth projective curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that E is a generically globally generated vector bundle on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then µmin(E) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since the evaluation map on global sections has torsion cokernel, we have µ(E) ≥ µ(H0(B, E) ⊗ OB) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Denote the Harder-Narasimhan filtration of E by 0 = F0 ⊂ F1 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' ⊂ Fk = E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since F1 is the maximal destabilizing subsheaf, we see that µ(F1) ≥ µ(E) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since E is generically globally generated, its quotient E/F1 is also generically globally gener- ated, and we conclude by induction on the length k of the Harder-Narasimhan filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Cohomology bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We next recall some bounds on the cohomology groups of semistable and generically globally generated vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' lemm:semistablevanishing Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let E denote a semistable vector bundle on a smooth projective curve B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Sup- pose that µ(E) > (2g(B) − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then H1(B, E) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Serre duality it suffices to show that H0(B, E∨ ⊗ ωB) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since E is semistable, E∨⊗ωB is as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since µ(E∨⊗ωB) < 0 there are no non-zero morphisms OB → E∨⊗ωB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ coro:checkinggg Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let E denote a vector bundle on the smooth projective curve B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Define d = (2g(B) − 2) − µmin(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then: (1) For any line bundle L of degree > d we have that H1(B, E ⊗ L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) For any line bundle T of degree > d + 1 we have that E ⊗ T is globally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Write the Harder-Narasimhan filtration of E as 0 = F0 ⊂ F1 ⊂ F2 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' ⊂ Fk = E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (1) Since the slopes µ(Fi/Fi−1) are strictly decreasing in i we have µ(Fi/Fi−1 ⊗ L) > 2g(B) − 2 for every index i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus for i in this range we have H1(B, Fi/Fi−1 ⊗ L) = 0 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Using the exact sequences H1(B, Fi−1 ⊗ L) → H1(B, Fi ⊗ L) → H1(B, Fi/Fi−1 ⊗ L) → 0 and arguing by induction on i we see that H1(B, E ⊗ L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) follows immediately from (1) and the LES sequence of cohomology associated to the inclusion E ⊗ T ⊗ OB(−p) ֒→ E ⊗ T where p is any closed point of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ lemm:ggh1bound Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let B be a smooth projective curve of genus g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that E is a generically globally generated bundle on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then (1) h0(C, E) ≤ deg(E) + rk(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) h1(C, E) ≤ g(B) rk(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Write 0 = F0 ⊂ F1 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' ⊂ Fk = E for the Harder-Narasimhan filtration of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since E is generically globally generated, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5 shows that every successive quotient Fi/Fi−1 has degree ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If 0 ≤ µ(Fi/Fi−1) ≤ 2g(B)−2, Clifford’s Theorem for semistable bundles as in [BPGN97, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1] shows that h0(B, Fi/Fi−1) ≤ 1 2 deg(Fi/Fi−1) + rk(Fi/Fi−1) and that h1(B, Fi/Fi−1) = h0(B, Fi/Fi−1) − χ(Fi/Fi−1) ≤ −1 2 deg(Fi/Fi−1) + g(B) rk(Fi/Fi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' On the other hand, if 2g(B) − 2 < µ(Fi/Fi−1) then h1(B, Fi/Fi−1) = 0 and h0(B, Fi/Fi−1) = deg(Fi/Fi−1) + rk(Fi/Fi−1)(1 − g(B)) by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6 and Riemann-Roch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since h0(B, E) ≤ s � i=1 h0(B, Fi/Fi−1) and h1(B, E) ≤ s � i=1 h1(B, Fi/Fi−1) we obtain the desired statement using the additivity of deg and rk in exact sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fujita invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Recall from Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2 that if X is a smooth projective variety over a field of characteristic 0 and L is a big and nef Q-Cartier divisor on X then a(X, L) = min{t ∈ R | KX + tL ∈ Eff 1(X)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By [BDPP13] the Fujita invariant will be positive if and only if X is geometrically uniruled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We will rely on the following boundedness result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 11 theo:Dicerbo Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='9 ([DC17, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2], [HL20, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive integer n and fix ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' As we vary X over all smooth projective varieties of dimension n defined over a field of characteristic 0 and vary L over all big and nef Cartier divisors on X, there are only finitely many possible values of a(X, L) in the range (ǫ, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The Fujita invariant is most useful for analyzing pairs satisfying an additional assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let X be a smooth projective variety and let L be a big and nef Q-divisor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We say that (X, L) is adjoint rigid if KX + a(X, L)L has Iitaka dimension 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If X is singular and L is a big and nef Q-Cartier divisor, we say that (X, L) is adjoint rigid if (X′, φ∗L) is adjoint rigid for some resolution of singularities φ : X′ → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This definition does not depend on the choice of resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Slope stability for smooth projective varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The notion of slope stability with respect to movable curve classes was developed by [CP11], [GKP14], [GKP16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let X be a smooth projective variety and let α ∈ Nef1(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For any torsion-free sheaf E on X, we define µα(E) = c1(E) · α rk(E) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We say that E is α-semistable if for every non-zero torsion-free subsheaf F ⊂ E we have µα(F) ≤ µα(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Every torsion free sheaf admits a maximal destabilizing subsheaf with respect to this slope function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus we get a theory of α-Harder-Narasimhan filtrations for torsion-free sheaves on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The following definition captures the slopes of the pieces of the Harder-Narasimhan filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let X be a smooth projective variety and let α ∈ Nef1(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that E is a torsion-free sheaf of rank r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Write 0 = F0 ⊂ F1 ⊂ F2 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' ⊂ Fk = E for the α-Harder-Narasimhan filtration of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The slope panel SPα(E) is the r-tuple of rational numbers obtained by combining for every index i the list of rk(Fi/Fi−1) copies of µα(Fi/Fi−1) (arranged in non-increasing order): SPα(E) = (µα(F1/F0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' � �� � rk(F1/F0) copies , µα(F2/F1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' � �� � rk(F2/F1) copies , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' , µα(Fk/Fk−1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' � �� � rk(Fk/Fk−1) copies ) We denote by µmax α (E) the maximal slope of any torsion-free subsheaf, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=', µmax α (E) = µα(F1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We denote by µmin α (E) the minimal slope of any torsion-free quotient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=', µmin α (E) = µα(E/Fk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' When discussing slope panels in the case when X is a curve, we will always let α be an ample class of degree 1 and thus we will simply write SP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Variations of the next result have been proved many times in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' theo:HNisfoliation Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='13 ([Pan15, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let X be a smooth projective variety and let α ∈ Nef1(X) be a nef curve class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Denote the α-Harder-Narasimhan filtration of TX by 0 = F0 ⊂ F1 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' ⊂ Fk = TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then every term Fi such that µmin α (Fi) > 0 defines a foliation on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 12 Suppose that f : X ��� Y is a rational map from a smooth projective variety X to a normal projective variety Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let U be the open locus where f is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is a unique foliation on X whose restriction to U is the saturation in TU of the kernel of TU → f ∗TY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We call this the foliation induced by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that if f ′ : X ��� Y ′ is a rational map birationally equivalent to f then f and f ′ induce the same foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Numerical equivalence on Fano fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose π : X → B is a Fano fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In this section we give some reminders about the basic properties of numerical equivalence and the cone of nef curves on X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' lemm:relativeprops Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a Fano fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then for every smooth fiber F the space N1(F)R has the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Furthermore, if L is a Q-Cartier divisor on X then the following are equivalent: (1) L|F is ample for some smooth Fano fiber F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) L|F is ample for all smooth Fano fibers F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (3) L|Xη is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The analogous statement is true for nefness, for bigness, and for pseudo-effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' It follows from [Kol96, IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5 Corollary] that every smooth fiber is rationally con- nected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then our first assertion follows from standard Hodge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The equivalence of the three conditions for ampleness and nefness follows from [Wi´s09, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For the equivalence of the three conditions for bigness and pseudo-effectiveness, see the paragraph before [dFH11, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ We will also need the following version of the Cone Theorem for nef curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This theorem was proved by [Ara10] conditional on the Borisov-Alexeev-Borisov Conjecture which has subsequently been proved in [Bir21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' theo:conetheorem Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='15 ([Ara10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let X be a normal Q-factorial projective variety and let ∆ be an effective Q-Cartier divisor on X such that (X, ∆) is ǫ-lc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is a constant ζ = ζ(dim(X), ǫ) such that Eff1(X )KX +∆≥0 + Nef1(X ) = Eff1(X )KX +∆≥0 + � i R≥0[Ci] where {Ci} is a countable collection of curves which satisfy 0 < −(KX + ∆) · Ci ≤ ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If ∆ is big, then the set {Ci} is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For Fano fibrations, the Cone Theorem for nef curves allows us to isolate the behavior of vertical curves: theo:nefconetheorem Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a flat morphism from a normal Q-factorial projective variety X to a smooth projective curve B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that ∆ is an effective Q-Cartier divisor on X such that ∆ is π-relatively big and (X , ∆) is ǫ-lc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let F denote a general fiber of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is a positive integer m = m(dim(X ), ǫ) such that we have an equality Eff1(X )KX +∆+mF ≥0 + Nef1(X ) = Eff1(X )KX +∆+mF ≥0 + � i R≥0[Ci] where {Ci} is a finite set of π-vertical moving curves which satisfy 0 < −(KX+∆+mF)·Ci ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since ∆ is effective and π-relatively big, we see that ∆ + δF is big for any δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By choosing δ sufficiently small we may ensure that δ < 1 and that (X , ∆ + δF) is ǫ/2-lc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='15 there is a constant ζ = ζ(dim(X ), ǫ) such that Eff1(X )KX +∆+δF ≥0 + Nef1(X ) = Eff1(X )KX +∆+δF ≥0 + � j [Cj] where the Cj are a finite set of movable curves satisfying 0 ≤ −(KX ′ + ∆ + δF) · Ci ≤ ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Choose a positive integer m > ζ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then (KX + ∆ + mF) · Cj > 0 for every one of our movable curves Cj that dominates B under π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus we have Eff1(X )KX +∆+δF ≥0 + Nef1(X ) = Eff1(X )KX +∆+mF ≥0 + � i R≥0[Ci] where now the Ci are π-vertical and still satisfy 0 ≤ −(KX ′ + ∆ + mF) · Ci ≤ ζ < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Boundedness and the Fujita invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In this section we recall some results of [LST22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Our first construction shows that the family of subvarieties of X which are adjoint rigid and have the same Fujita invariant as X is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' cons:rigidsubvarieties Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let k be a field of characteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let X be a geometrically uniruled geometrically integral smooth projective k-variety and let L be a big and nef Q-Cartier divisor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By [LST22, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='19] there exist a proper closed subset V , finitely many projective varieties Wi ⊂ Hilb(X), proper families pi : Ui → Wi where Ui is a smooth birational model of the universal family U′ i → Wi, and dominant generically finite morphisms si : Ui → X such that over k, a general fiber of pi,k : Ui,k → Wi,k is an integral uniruled projective variety which is mapped birationally by si,k onto the subvariety of Xk parametrized by the corresponding point of Hilb(Xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' a general fiber Z of pi is a smooth projective variety satisfying a(Z, s∗ i L|Z) = a(X, L) and is adjoint rigid with respect to s∗ i L|Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' and for every subvariety Y ⊂ X not contained in B+(L) which satisfies a(Y, L|Y ) ≥ a(X, L) and which is adjoint rigid with respect to L, either Y is contained in V or there is some index i and a smooth fiber of pi that is mapped birationally to Y under the map si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In fact more is true: the next result shows that the morphisms f : Y → X such that Y is adjoint rigid and has the same Fujita invariant as X also form a bounded family up to twisting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' theo:ainvboundedandtwists Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let k be a field of characteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let X be a geometrically uniruled geometrically integral smooth projective k-variety and let L be a big and nef Q-Cartier divisor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Denote by {pi : Ui → Wi} the finite set of families equipped with maps si : Ui → X and by V the closed subset of Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is a closed set R ⊂ X and a finite set of smooth projective varieties Yi,j equipped with dominant morphisms ri,j : Yi,j → Ti,j 14 with connected fibers and dominant morphisms hi,j : Yi,j → Ui forming commuting diagrams Yi,j hi,j � ri,j � Ui pi � Ti,j ti,j � Wi that satisfy the following properties: (1) each map hi,j is generically finite and fi,j = si ◦ hi,j is not birational;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) ti,j is a finite Galois cover and Ti,j is normal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (3) Bir(Yi,j/Ui) = Aut(Yi,j/Ui);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (4) every twist Y σ i,j of Yi,j over Ui admits a morphism rσ i,j : Y σ i,j → T σ i,j which is a twist of ri,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (5) we have a(Yi,j, f ∗ i,jL) = a(X, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (6) suppose that Y is a geometrically integral smooth projective variety and that f : Y → X is a morphism that is generically finite onto its image but not birational such that a(Y, f ∗L) ≥ a(X, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose furthermore that y ∈ Y (k) satisfies f(y) ̸⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then: (a) there are indices i, j and a twist hσ i,j : Y σ i,j → Ui of hi,j such that f(y) ∈ si(hσ i,j(Y σ i,j(k))), and (b) if (Y, f ∗L) is adjoint rigid then furthermore f factors rationally through hσ i,j and f maps Y birationally to a fiber of rσ i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Consider the families pi : Ui → Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By applying [LST22, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3] there exists a Zariski open subset W ◦ i such that each map U◦ i → W ◦ i is a good fibration in the sense of [LST22, Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We may then apply [LST22, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3] to each Ui equipped with s∗ i L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The result is a closed set Di ⊂ Ui and a finite set of smooth projective varieties Yi,j equipped with morphisms ri,j : Yi,j → Ti,j, hi,j : Yi,j → Ui, and ti,j : Ti,j → Wi that have the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' First, since ri,j is constructed as the Stein factorization of Yi,j → Ui → Wi we see that every twist Y σ i,j of Yi,j over Ui admits a morphism rσ i,j that is a twist of ri,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Second, suppose that q : Y → Ui is a generically finite morphism such that a(Y, q∗s∗ i L) = a(X, L) and a general fiber of the Iitaka fibration for KY +a(Y, q∗s∗ i L)q∗s∗ i L maps generically finitely onto a general fiber of Ui → Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose furthermore that y ∈ Y (k) is a rational point such that q(y) is not contained in Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then there is some index j and a twist hσ i,j : Y σ i,j → Ui such that q(y) is in hσ i,j(Y σ i,j(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Furthermore every general fiber of the canonical fibration for KY + a(Y, q∗s∗ i L)q∗s∗ i L is birational to a fiber of rσ i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By [LST22, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='18] there is a proper closed subset V ′ ⊂ X which is the union of all subvarieties Y satisfying a(Y, L|Y ) > a(X, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We let R be the union of V and V ′ with ∪isi(Di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We also enlarge R by adding the images of singular fibers of ri,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We verify each property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (1), (2), (3) follow from [LST22, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We have already verified (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (5) follows from [LST22, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 (i) and (ii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Now we verify (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose f : Y → X is as in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By assumption f(Y ) ̸⊂ V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular this implies that a(Y, f ∗L) = a(f(Y ), L|f(Y )) = a(X, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let F be the closure of a general fiber of the canonical fibration for KY + a(Y, f ∗L)f ∗L so that (F, f ∗L|F) is adjoint rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then by [LST22, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='9] we see that (f(F), L|f(F )) is also adjoint rigid and thus is birational to a fiber of some map pi : Ui → Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This induces a rational map T ��� Hilb(X) 15 where T is the base of the canonical fibration of KY + a(Y, f ∗L)f ∗L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since Ui is birational to the universal family over Wi, we also obtain a rational map Y ��� Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since the desired statement only depends on the birational equivalence class of f : Y → X (and not the choice of birational model of Y ), after blowing up Y we may suppose that Y admits a morphism to Ui such that the general fiber of the canonical fibration on Y maps generically finitely onto a fiber of the map Ui → Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then the desired containment of rational points follows from [LST22, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 (vi)] as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' When (Y, f ∗L) is adjoint rigid, the factoring statement also follows from [LST22, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 (vi)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Sections of good fibrations sect:goodfibration Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We say that a morphism π : Z → B is a good fibration if: (1) Z is a smooth projective variety, (2) B is a smooth projective curve, and (3) π is flat and has connected fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that π : Z → B is a good fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We let Sec(Z/B) denote the open subset of the Hilbert scheme parametrizing sections of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If M ⊂ Sec(Z/B) is an irreducible component, the expected dimension of M is χ(TZ/B|C) = −KZ/B · C + (dim Z − 1)(1 − g(B)) where C is any section parametrized by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The expected dimension is a lower bound for the dimension of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' An upper bound is dim H0(B, TZ/B|C) = −KZ/B · C + (dim Z − 1)(1 − g(B)) + dim H1(B, TZ/B|C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' One of the basic facts about sections of a good fibration is the Northcott property, which in our setting should be interpreted in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' coro:boundednegativity Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2 ([LT21a, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : Z → B be a good fibration and let L be a generically relatively ample Q-Cartier divisor on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If we fix a constant Q, then there are only finitely many components of Sec(Z/B) parametrizing sections C satisfying L · C ≤ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Relatively free sections and general points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose π : Z → B is a good fi- bration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix points q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' , qm ∈ Z which are contained in different fibers of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We let Sec(Z/B, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' , qm) denote the sublocus of Sec(Z/B) parametrizing sections containing the points q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' , qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular, if M ⊂ Sec(Z/B, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' , qm) is an irreducible component then the expected dimension of M is χ(TZ/B|C(−q1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' − qm)) = −KZ/B · C + (dim Z − 1)(1 − g(B)) − m(dim(Z) − 1) where C is any section parametrized by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The expected dimension is a lower bound for the dimension of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' An upper bound is dim H0(B, TZ/B|C(−q1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' − qm)) = −KZ/B · C + (dim Z − 1)(1 − g(B)) − m(dim(Z) − 1) + dim H1(C, TZ/B|C(−q1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' − qm)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The following result describes how the normal bundle of a section C controls the number of general points contained in deformations of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 16 prop:deffixpoints Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 ([LT21a] Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : Z → B be a good fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix points q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' , qm of Z contained in different fibers of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let M denote an irreducible component of Sec(Z/B, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' , qm) and suppose that the sections parametrized by M dominate Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then for a general section C parametrized by M and for a general point p ∈ B we have that H0(C, TZ/B|C(−q1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' − qm)) → TZ/B|C|p is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Conversely, suppose we fix a section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' , qm are distinct points of C such that H1(C, TZ/B|C(−q1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' − qm)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let M ⊂ Sec(Z/B, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' , qm) denote the unique irreducible component containing C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If for a general point p ∈ C we have that H0(C, TZ/B|C(−q1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' − qm)) → TZ/B|C|p is surjective, then M parametrizes a dominant family of sections on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' coro:domfamilyexpdim Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : Z → B be a good fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that M is an irreducible component of Sec(Z/B) parametrizing a dominant family of sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Letting C denote a general section parametrized by M, we have −KZ/B · C + (dim Z − 1)(1 − g(B)) ≤ dim(M) ≤ −KZ/B · C + dim Z − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 the bundle TZ/B|C is generically globally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus we have h1(C, TZ/B|C) ≤ g(B)(dim(Z) − 1) by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The desired statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ Recall that a section C is relatively free if H1(C, TZ/B|C) = 0 and TZ/B|C is globally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 shows that any relatively free section deforms in a dominant family on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' It is easiest to work with relatively free sections when we impose further conditions on the positivity of the terms of the Harder-Narasimhan filtration of TZ/B|C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : Z → B be a good fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We say that a section C is HN-free if µmin(TZ/B|C) ≥ 2g(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The following result summarizes the key properties of HN-free sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' lemma:hnfreecurves Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : Z → B be a good fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that C is a HN-free section of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then: (1) H1(C, TZ/B|C) = 0 and for any closed point p ∈ B we have H1(C, TZ/B|C(−p)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) TZ/B|C is globally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (3) C is relatively free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (4) Let b = µmin(TZ/B|C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then deformations of C can pass through at least b−2g(B)+1 general points of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (1) and (2) follow from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7 and (3) follows from (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' To see (4) we apply Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7 to see that for any points q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' , qm on C the twist TZ/B|C(−q1−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='−qm) is globally generated and has vanishing H1 so long as m ≤ b−2g(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The desired statement follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ The next proposition shows that sections through sufficiently many general points must be HN-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' prop:generalimplieshnfree Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : Z → B be a good fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let M be an irreducible component of Sec(Z/B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that the sections parametrized by M pass through ≥ 2g(B) + 1 general points of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then the general section parametrized by M is HN-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If we fix a general section C parametrized by M and a set of 2g(B) general points {qi}2g(B) i=1 on C then Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 shows that TZ/B|C(−q1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' − q2g(B)) is generically globally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5 shows that µmin(TZ/B|C(−q1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' − q2g(B))) ≥ 0 and we conclude that µmin(TZ/B|C) ≥ 2g(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ We will also need to know the following avoidance property of HN-free sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' lemm:HNavoidscodim2 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : Z → B be a good fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that C is a HN-free section of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then for any codimension 2 closed subset W ⊂ Z there is a deformation of C which is HN-free and avoids W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume for a contradiction that every deformation of C meets with W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then there exists p ∈ W such that a general deformation of C containing p is HN-free and the dimension of the family parametrizing such deformations is greater than or equal to −KZ/B · C + (dim Z − 1)(1 − g(B)) − dim W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that this is larger than the expected dimension −KZ/B · C − (dim Z − 1)g(B) for the parameter space of sections through p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' But this contradicts with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6 which shows that H1(C, TZ/B|C(−p)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Grauert-Mulich sect:gm For a good fibration π : Z → B the deformation theory of a section C is controlled by the Harder-Narasimhan filtration of the restriction TZ/B|C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In this section, we show that (under certain hypotheses) the Harder-Narasimhan filtration of TZ/B|C is “approximately” the restriction of the [C]-Harder-Narasimhan filtration of TZ/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Due to the similarity to the Grauert-Mulich theorem ([GM75]) describing the restriction of semistable bundles to lines in Pn, we will refer to such statements as “Grauert-Mulich” results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The material in this section is motivated by [PRT20, Section 3] and by [OSS80, Chapter II, Section 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that Z is a smooth projective variety and W is a variety parametrizing a family of maps s : C → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let E be a torsion-free sheaf on Z and let F be a term in the relative Harder-Narasimhan filtration of E pulled back to the universal family over an open subset of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We would like to determine when F is the pullback of a sheaf FZ from Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In this case we can expect FZ to be a term in the Harder-Narasimhan filtration of E with respect to the numerical class of the curves s∗C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 we prove a general criterion for determining when a torsion-free sheaf on a variety is isomorphic to a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We apply this result in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2 to show that our ability to descend F to Z is controlled by the comparison between several invariants of the normal sheaf of s and the “gaps” in slope between F and adjacent terms of the relative Harder-Narasimhan filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Finally, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 we show that for sections of a good fibration π : Z → B these invariants of the normal sheaf are bounded by functions of dim(Z) and g(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' sect:descent 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Descending sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The first step is to develop a criterion for identifying when a sheaf is pulled back from the base of a morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' lemm:rigidity Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose we have morphisms f : U → V , g : U → G satisfying the following properties: 18 (1) U, V, G are smooth varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) f is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (3) Every fiber of f is contracted to a point by g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then there is some open set V ◦ ⊂ V such that g|f−1(V ◦) factors through f|f−1(V ◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Consider the induced map (f, g) : U → V × G and let Γ denote the closure of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that Γ is still irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For a general point v ∈ V there is a unique point in (f, g)(U)∩π−1 1 (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Taking closures, we see that the general fiber of Γ → V is set-theoretically a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since we are in characteristic 0, by generic smoothness we see that the general fiber is scheme-theoretically a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We deduce that Γ → V is birational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If we let V ◦ denote an open subset where Γ → V is an isomorphism, then the desired statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ Recall that for a coherent sheaf F on a variety we denote by Ftors the torsion subsheaf and by Ftf the quotient of F by its torsion subsheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' lemm:descent Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2 (Descent Lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let U and Z be smooth varieties with a dominant flat mor- phism ev : U → Z such that the general fiber of ev is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let E be a locally free sheaf on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that ev∗E → Q is a surjection onto a locally free sheaf and let S denote the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If Hom(Hom(Q, S), (ΩU/Z)tf) = 0 then there is a subsheaf SZ ⊂ E such that ev∗SZ = S as subsheaves of ev∗E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that the surjection ev∗E → Q corresponds to a map φ : U → G(E, k) = G, where k is the rank of Q and G(E, k) is the relative Grassmannian of rank k quotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The map φ is a map of Z-schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We first show that after replacing U by an open subset the map φ factors through ev : U → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The map φ induces a map (dφ)∗ : φ∗ΩG/Z → ΩU/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since φ∗ΩG/Z = Hom(Q, S), our assumption implies that (dφ)∗ is the 0 map on the com- plement of the support of the torsion subsheaf of ΩU/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We denote this open subset by �U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a general point z ∈ Z and consider the map of fibers �Uz → Gz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Using compatibility of cotangent bundles with base change, we see that φ|∗ �UzΩGz → Ω �Uz must be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Dually, the map T �Uz → φ|∗ �UzTGz is zero, and thus if we precompose by the the locally closed embedding ( �Uz,red)smooth → �Uz,red → �Uz the induced map on tangent sheaves still vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By generic smoothness, it follows that φ must contract ( �Uz,red)smooth to a point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' taking closures, it also contracts each fiber �Uz to a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is an open subset Z◦ ⊂ Z with preimage �U◦ = ev−1(Z◦) ∩ �U such that ev| �U◦ has connected fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1, after possibly shrinking Z◦ and �U◦ we can ensure that φ| �U◦ factors through ev| �U◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Hence, S|U◦ must be the pullback of a locally free sheaf R ⊂ E|Z◦ on Z◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let SZ denote the unique torsion-free saturated subsheaf of E whose restriction to Z◦ is R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since ev is flat, ev∗SZ is a torsion-free saturated subsheaf of ev∗E whose restriction to U◦ agrees with S|U◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This implies that ev∗SZ = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ 19 sect:slopecomputation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Slope computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The key to using the descent lemma is to understand homo- morphisms into ΩU/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' When U is a family of curves mapping to Z, we will control the existence of such homomorphisms using slope calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The next step is to show that in this situation the slope of ΩU/Z is controlled by Lazarsfeld bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Y be a variety and E be a globally generated vector bundle on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The Lazarsfeld bundle ME is the kernel of the evaluation map OY ⊗ H0(Y, E) → E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Given a morphism s : C → Z, we will denote by Ns the normal sheaf of s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' the cokernel of TC → s∗TZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We will also let Mg,0(Z) denote the Kontsevich moduli stack of maps from genus g curves to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' lem-LazMukHoms Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Z be a smooth projective variety, let W be a variety equipped with a gener- ically finite morphism W → Mg,0(Z) and let p : UW → W be the universal family over W equipped with the evaluation map evW : UW → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that a general fiber of p is smooth and irreducible and that evW is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let C denote a general fiber of UW → W equipped with the induced morphism s : C → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let t be the length of the torsion part of Ns, let G be the subsheaf of (Ns)tf generated by global sections, and let V be the tangent space to W at s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let q be the dimension of the cokernel of the composition V → TMg,0(Z),s = H0(C, Ns) → H0(C, (Ns)tf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then µmax((ΩUW /Z|C)tf) ≤ (q + 1)µmax(M∨ G ) + t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since the conclusion only involves a general curve, we may shrink W and thus assume that W is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' After perhaps shrinking W further we may assume that p is smooth, and thus UW is a smooth variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We denote by h the map (p, evW) : UW → W × Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix s : C → Z as in the statement of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let K1 denote the kernel of s∗ΩZ → ΩC and let K2 denote the kernel of h∗ΩW ×Z → ΩUW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that K1 is isomorphic to the dual of (Ns)tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We claim that the following diagram has exact rows and columns: 0 � 0 � Odim(W ) C = � � Odim(W ) C � 0 � K2|C � � h∗ΩW ×Z|C � � Ωim UW |C � � 0 0 � K1 � s∗ΩZ � � Ωim C � � 0 0 0 where Ωim UW is the image of h∗ΩW ×Z → ΩUW and Ωim C is the image of s∗ΩZ → ΩC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The bottom row is exact by definition and the middle row is the restriction of an exact sequence 20 to a general fiber of p and thus remains exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The middle column is exact since C is vertical for the map p : UW → W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By comparing the rightmost column against the middle it is clear that Ωim UW |C maps surjectively onto Ωim C and that the kernel contains p∗ΩW|C ∼= Odim(W ) C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' On the other hand the kernel must be contained in the kernel of ΩUW |C → ΩC which is also isomorphic to p∗ΩW|C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' So the rightmost column is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Finally, by the nine lemma we deduce that K2|C ∼= K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let K3 denote the kernel of the map p∗ΩW → ΩUW/Z and let Ωim UW/Z denote the image of this map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We can make a further comparison via the following diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 0 � 0 � s∗ΩZ = � � s∗ΩZ � 0 � K2|C � � h∗ΩW ×Z|C � � Ωim UW |C � � 0 0 � K3|C � p∗ΩW|C � � Ωim UW/Z|C � � 0 0 0 We claim that the rows and columns are exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since C is general, an exact sequence of torsion-free sheaves on UW will remain exact upon restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus it suffices to show that the rightmost column is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since the map evW : UW → Z is dominant, the map ev∗ WΩZ → ΩUW is generically injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since ΩZ is locally free the map must be injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Restricting to the curve C, we see that s∗ΩZ → ΩUW |C is injective and it is clear that its image is contained in Ωim UW |C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This shows the rightmost column is left-exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Furthermore, we see that the composed map h∗ΩW ×Z → p∗ΩW → Ωim UW/Z is surjective, showing that the map Ωim UW → Ωim UW/Z is also surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Finally, the exact sequence 0 → ev∗ WΩZ → ΩUW → ΩUW /Z → 0 implies that 0 → s∗ΩZ → ΩUW |C → ΩUW /Z|C → 0 is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus the rightmost column must be exact at the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By the nine-lemma, we conclude that K3|C ∼= K2|C ∼= K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Recall that V denotes the tangent space to W at s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let ζ denote the composition V → H0(C, Ns) → H0(C, (Ns)tf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then the map K1 ∼= K3|C → p∗ΩW|C is the dual of the composition V ⊗ OC ζ−→ H0(C, (Ns)tf) ⊗ OC → (Ns)tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let G denote the subsheaf of (Ns)tf that is generated by its global sections, so we have an exact sequence of locally free sheaves 0 → MG → H0(C, (Ns)tf) ⊗ OC → G → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 21 Since C deforms in a dominant family, Ns is generically globally generated and thus the inclusion G → (Ns)tf is generically surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Taking duals, we see that G∨ is the saturation of K1 inside of H0(C, (Ns)tf)∨ ⊗ OC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In other words, if we let S denote the cokernel of the map K1 → H0(C, (Ns)tf)∨ ⊗ OC then the torsion free part of S is isomorphic to M∨ G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Consider the following diagram of short exact sequences 0 � K1 � = � H0(C, (Ns)tf)∨ ⊗ OC � ζ∨ � S � h � 0 0 � K1 � p∗ΩW|C � Ωim UW /Z|C � 0 By the snake lemma we obtain an exact sequence 0 → O⊕q C → S → Ωim UW /Z|C → O⊕e C → 0 where q, e are respectively the dimensions of the cokernel and kernel of ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Recall that Stf ∼= M∨ G and note that the torsion part of S injects into the torsion part of Ωim UW /Z|C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We will denote by R the saturation of O⊕q C in M∨ G so that we obtain an exact sequence 0 → R → M∨ G → (Ωim UW /Z|C)tf → O⊕e C → 0 Let F denote the maximal destabilizing subsheaf of (Ωim UW /Z|C)tf, let F ′ = F ∩ im(M∨ G ), and let Q denote the preimage of F ′ in M∨ G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Our next goal is to prove an upper bound on µ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' First suppose that µ(F) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then the image of F in O⊕e C is 0 and so F = F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This means we have an exact sequence 0 → R → Q → F → 0 and thus µ(F) = deg(Q) − deg(R) rk(Q) − rk(R) ≤ deg(Q) rk(Q) − q ≤ (q + 1)µ(Q) where the final line follows from the elementary inequality 1 b−a ≤ a+1 b when b − 1 ≥ a ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus in this case we see that µmax(Ωim UW /Z|C) ≤ (q + 1)µmax(M∨ G ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If µ(F) ≤ 0, then the same inequality still holds: the right-hand side is a non-negative number since M∨ G is globally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 22 Consider the following diagram 0 0 0 0 � Ωim C � � ΩC � � T � � 0 0 � s∗ΩZ � � ΩUW |C � � ΩUW /Z|C � � 0 0 � K3|C � � p∗ΩW|C � � Ωim UW /Z|C � � 0 0 � 0 � 0 � where Ωim C is the image and T is the cokernel of s∗ΩZ → ΩC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Every row and column is exact except possibly the rightmost column, thus by nine-lemma we see the rightmost column is also exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We have len(T ) = len(cok(Ωim C → ΩC)) = len(cok(TC → T sat C )) = t where T sat C is the saturation of TC in s∗TZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' It follows that µmax((ΩUW /Z|C)tf) ≤ µmax((Ωim UW /Z|C)tf) + t ≤ (q + 1)µmax(M∨ G ) + t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ By combining this analysis with the descent theorem, we obtain: theo:gmtheorem Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Z be a smooth projective variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let W be a variety equipped with a generically finite morphism W → Mg,0(Z) and let p : UW → W denote the universal family over W with evaluation map evW : UW → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume that a general map parametrized by W has smooth irreducible domain, that evW is dominant, that the general fiber of the composition of the normalization map for UW with evW is connected, and that a general fiber of p is contained in the locus where evW is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that E is a torsion-free sheaf on Z that is semistable with respect to a general curve s : C → Z parametrized by W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Write 0 = F0 ⊂ F1 ⊂ F2 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' ⊂ Fk = s∗E for the Harder-Narasimhan filtration of s∗E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let t be the length of the torsion part of Ns, let G be the subsheaf of (Ns)tf generated by global sections, and let V be the tangent space to W at s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let q be the dimension of the cokernel of the composition V → TMg,0(Z),s = H0(C, Ns) → H0(C, (Ns)tf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then for every index 1 ≤ i ≤ k − 1 we have µ(Fi/Fi−1) − µ(Fi+1/Fi) ≤ (q + 1)µmax(M∨ G ) + t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 23 Note that by the flatness assumption we may ensure that the image of a general map s parametrized by W will avoid any codimension 2 locus on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular this implies that s∗E will be a locally free sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since we are assuming that the general fiber of the composition of the normalization map for UW with evW is connected, if we replace W by a smaller open subset then the general fiber of the evaluation map will still have connected fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus to prove the statement, we may replace W by a smaller open subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (For ease of notation we continue to call the smaller subset W and its p-preimage by UW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=') Thus we may suppose that W and UW are smooth and that evW|UW is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' After possibly shrinking W further, by [HL97, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2] we may suppose that for every index 1 ≤ i ≤ k − 1 there is a torsion-free sheaf Si ⊂ ev∗ WE obtained from the relative Harder-Narasimhan filtration of ev∗ WE over W such that Si|C ∼= Fi for every fiber C over W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since torsion-free sheaves are locally free on the complement of a codimension 2 subset, after perhaps shrinking W again we may suppose that each Si is locally free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose for a contradiction that there is an index i such that µ(Fi/Fi−1) − µ(Fi+1/Fi) > (q + 1)µmax(M∨ G ) + t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If there were a non-zero homomorphism Hom(ev∗ W E/Si, Si) → (ΩUW /Z)tf, then its restriction to a general fiber C of p would yield a map that is non-zero on the generic point of C, and thus would induce a non-zero map Hom(ev∗ WE/Si|C, Si|C) → ((ΩUW /Z)|C)tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' But then µmin(Hom(ev∗ WE/Si|C, Si|C)) = µmin(Si|C) − µmax(ev∗ WE/Si|C) > (q + 1)µmax(M∨ G ) + t ≥ µmax((ΩUW /Z|C)tf) where the last inequality follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We conclude that there is no non-zero homomorphism Hom(ev∗ WE/Si, Si) → (ΩUW /Z)tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2, we see that there is a sheaf SZ on Z such that ev∗ WSZ = Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' But such a sheaf would destabilize E: we have µ[s∗C](SZ) = µ(Si|C) > µ(ev∗ WE|C) = µ[s∗C](E) where the equalities follow from the flatness of the evaluation map and the inequality follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This gives a contradiction and we conclude the desired inequalities for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ It will be helpful to have a version of the previous theorem that holds for non-semistable sheaves as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The next theorem controls the difference between the Harder-Narasimhan filtration of E|C and the restriction to C of the [C]-Harder-Narasimhan filtration of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' It is a formal consequence of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' coro:hnfversion Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Z be a smooth projective variety and let E be a torsion free sheaf on Z of rank r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let W be a variety equipped with a generically finite morphism W → Mg,0(Z) and let p : UW → W denote the universal family over W with evaluation map evW : UW → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume that a general map parametrized by W has smooth irreducible domain, that evW is dominant, that the general fiber of the composition of the normalization map for UW with evW is connected, and that a general fiber of p is contained in the locus where evW is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let C denote a general fiber of UW → W equipped with the induced morphism s : C → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let t be the length of the torsion part of Ns, let G be the subsheaf of (Ns)tf generated by global 24 sections, and let V be the tangent space to W at s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let q be the dimension of the cokernel of the composition V → TMg,0(Z),s = H0(C, Ns) → H0(C, (Ns)tf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let ∥ − ∥ denote the sup norm on Q⊕r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then we have ∥ SPZ,[C](E) − SPC(s∗E)∥ ≤ 1 2 � (q + 1)µmax(M∨ G ) + t � rk(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Set δ = (q + 1)µmax(M∨ G ) + t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Given two r-tuples of real numbers a•, b•, we write a• ≥ b• if for every index i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' , r we have ai ≥ bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Write 0 = F0 ⊂ F1 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' ⊂ Fk = E for the [C]-Harder-Narasimhan filtration of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We prove this statement by induction on the length k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We start with the base case k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5 the slopes of successive quotients of the Harder-Narasimhan filtration of s∗E differ by at most δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We conclude the desired statement by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7 (where the pairs (ai, bi) record the degrees and ranks of the various successive quotients in the Harder-Narasimhan filtration of s∗E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For the induction step, write 0 = G0 ⊂ G1 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' ⊂ Gt = s∗E for the Harder-Narasimhan filtration of s∗E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For convenience we define three tuples: c• = (c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' , cr) to be SPC(s∗E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' g• = (g1, g2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' , gr) defined as follows: we start with the tuple SPC(s∗Fk−1) (of length rk(Fk−1)) and then replace every entry that is below µ[C](E/Fk−1) + δ 2 rk(E) by this number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We then append entries equal to µ[C](E/Fk−1) + δ 2 rk(E) to the end so that g• has length equal to rk(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' h• = (h1, h2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' , hr) defined as follows: we start with the tuple SPC(s∗(E/F1)) (of length rk(E/F1)) and then replace every entry that is above µ[C](F1) − δ 2 rk(E) by this number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We then insert entries equal to µ[C](F1) − δ 2 rk(E) at the beginning so that h• has length equal to rk(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since C is a general member of a flat family every term Fi is locally free along C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus we have an exact sequence 0 → s∗Fk−1 → s∗E → s∗(E/Fk−1) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix an index 1 ≤ j ≤ t and suppose that µmin(Gj) > µ[C](E/Fk−1) + δ 2 rk(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By the base case of our result we see that µmin(Gj) > µmax(s∗(E/Fk−1)) so that Gj ⊂ s∗Fk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus we have g• ≥ c•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Similarly, consider the exact sequence 0 → s∗F1 → s∗E → s∗(E/F1) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix an index 1 ≤ j ≤ t and suppose that µmax(s∗E/Gj) < µ[C](F1) − δ 2 rk(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By the base case of our result we see that µmax(s∗E/Gj) < µmin(s∗F1) so that there can be no non-zero homomorphism from s∗F1 to s∗E/Gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We conclude that s∗F1 ⊂ Gj and thus s∗E/Gj is a quotient of s∗(E/F1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus we have c• ≥ h•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Altogether, suppose we define q+ to be SPZ,C(E) + ( δ 2 rk(E), δ 2 rk(E), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' , δ 2 rk(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' q− to be SPZ,C(E) − ( δ 2 rk(E), δ 2 rk(E), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' , δ 2 rk(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 25 The induction assumption for Fk−1 shows that q+ ≥ g• and the induction assumption for E/F1 shows that h• ≥ q− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By the argument above we conclude that q+ ≥ g• ≥ c• ≥ h• ≥ q− yielding the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ lemm:weightedAverage Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let {(ai, bi)}k i=1 be pairs of integers with bi > 0 such that the fractions ai bi are nonincreasing as i gets larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let m denote the mediant of fractions ai bi and let δ denote the maximum of the successive differences ai bi − ai+1 bi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then for every i we have ���� ai bi − m ���� ≤ δ 2 r � i=1 bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' It suffices to prove the inequality for the extremal fractions a1 b1 , ak bk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Replacing each ai by −ai, we see that the latter case is implied by the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' So it suffices to prove the statement for a1 b1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since a1 b1 − ai bi ≤ (i − 1)δ, we have k � i=1 (a1bi − aib1) ≤ k � i=1 b1biδ(i − 1) ≤ b1δ � k � i=1 bi � i−1 � j=1 bj �� = b1δ 2 �� i̸=j bibj � ≤ b1δ 2 � k � i=1 bi �2 Dividing by b1 ��k i=1 bi � gives us the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ sect:genusanddimbounds 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Genus and dimension bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that the Hilbert scheme of sections admits an embedding into the stack Mg,0(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' To apply Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6 to moduli spaces of sections one needs to be able to bound the quantities q, t, and µmax(M∨ G ) appearing in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We will show how to bound these quantities for sections of a good fibration π : Z → B using the genus of B and the dimension of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that since sections are always smooth the quantity t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We first discuss the slope of Lazarsfeld bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' These can be bounded using only the genus of B using the following result of Butler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' theo:butler Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8 ([But94]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let E be a globally generated locally free sheaf on a curve C of genus g and let ME be its Lazarsfeld bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (1) If µmin(E) ≥ 2g then µmin(ME) ≥ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) If µmin(E) < 2g then µmin(ME) ≥ −2g rk(E) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Statement (1) follows from [But94, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 Corollary] except when C is a rational curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' When C is rational ME is a direct sum of O(−1)’s (see for example [PRT20, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10]) and thus the statement still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 26 For statement (2), first suppose that µmax(E) ≥ 2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let F denote the maximal destabi- lizing subsheaf of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Our degree assumption implies that F is globally generated and that h1(C, F) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By the nine lemma we obtain an exact sequence 0 → MF → ME → ME/F → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This implies that µmin(ME) ≥ min{µmin(MF), µmin(ME/F)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By (1) the first quantity is at least −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Arguing by induction on the rank we reduce to the case when µmax(E) < 2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' When µmax(E) < 2g, [But94, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5 Proposition] proves a statement stronger than (2) except in two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The first is when g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In this case, since ME is a subsheaf of H0(C, E) ⊗ OC every subsheaf of ME has non-positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since deg(ME) = − deg(E), we conclude that every quotient of ME has degree at least − deg(E), which implies that it has slope at least − deg(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since we are in the case where deg(E)/ rk(E) < 2 we conclude µmin(ME) ≥ −2 rk(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The second is when g ≥ 2 and E has a trivial summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Write E = E′ ⊕ O⊕k C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that ME = ME′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since we still have µmax(E′) < 2g we can apply [But94, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5 Proposition] to E′ to obtain the desired lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ Next we discuss the quantity q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' lemm:sbound Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : Z → B be a good fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that M ⊂ Sec(Z/B) is an irreducible component parametrizing a dominant family of sections on Z and let W = Mred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For a general section C parametrized by M let V ⊂ H0(B, TZ/B|C) denote the tangent space to W at C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then the codimension of V in H0(B, TZ/B|C) is at most g(B)(dim(Z) − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We have h0(B, TZ/B|C)−dim(V ) ≤ h0(B, TZ/B|C)−dim(M) and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4 shows that this latter quantity is bounded above by g(B)(dim(Z) − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ Putting these results together, we obtain a version of the Grauert-Mulich theorem for sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' theo:hnfforsections Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : Z → B be a good fibration and let E be a torsion-free sheaf on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let M be an irreducible component of Sec(Z/B) parameterizing a dominant family of sections of π and let p : Uν → Mred be the normalization of the universal family over Mred with evaluation map ev : Uν → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume that ev has connected fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let C be a general section parametrized by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then: (1) Suppose there is an open subset M◦ red ⊂ Mred such that if we define Uν,◦ = p−1M◦ red then ev|Uν,◦ is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then we have ∥ SPZ,[C](E) − SPC(E|C)∥ ≤ (g(B) dim(Z) − g(B) + 1)2 rk(E) where ∥ − ∥ denotes the sup norm on Q⊕r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) Suppose that the general curve C is HN-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then we have ∥ SPZ,[C](E) − SPC(E|C)∥ ≤ rk(E) where ∥ − ∥ denotes the sup norm on Q⊕r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (1) Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8 shows that µmax(M∨ G ) ≤ 2g(B)(dim(Z) − 1) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='9 we have q ≤ g(B)(dim(Z) − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We then apply Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6 with t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) When C is HN-free then M is generically smooth and TZ/B|C′ is globally generated for a general deformation C′ of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Furthermore, there is an open subset of M over which the evaluation map is smooth and thus flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8 shows that µmax(M∨ G ) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We then apply Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6 with q = t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ 27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Sections through general points sect:genpoints Suppose that π : Z → B is a good fibration and M is an irreducible component of Sec(Z/B) parametrizing a dominant family of sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let C be a general section parametrized by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 we can identify a lower bound on the slopes in the Harder- Narasimhan filtration of TZ/B|C by computing how many general points of Z we can impose on the sections parametrized by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Even when C has very large anticanonical degree, deformations of C do not need to go through many general points of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In this section we construct a dominant family of subvarieties Y ⊂ Z such that deformations of C go through many general points in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Results of this type were used earlier in [She12], [LT22], and [LT21a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Here is the idea behind the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that M parametrizes a family of sections C which have large degree but do not go through many points of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This implies that the Harder-Narasimhan filtration of TZ/B|C has a large gap in the slopes between two consecutive terms Gk ⊂ Gk+1 for some index k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' When M satisfies the conditions of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6, we can deduce that there is a foliation F on X that restricts to Gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Appealing to the results developed in the sequence of papers [BM16], [KSCT07], [CP19], the foliation is induced by a rational map φ : Z ��� W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We can expect that there will be many deformations of C in directions tangent to the fibers of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular, deformations of C should go through many general points of the main component Y of φ−1(φ(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' sect:genpointfoliations 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' General points and foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We will need the following construction describ- ing the relationship between foliations and relative tangent bundles which is adapted from [KSCT07, Remark 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' cons:foliationrestriction Construction 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : Z → B be a good fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that F is a foliation on Z that is contained in the relative tangent bundle TZ/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume that F is induced by a rational map φ : Z ��� W that has connected fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that φ must be a rational map over B and we may assume that W is a projective B-variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that C is a section of π that is contained in the regular locus of F and goes through a general point of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since C is transversal to F, the Frobenius theorem shows that there is an irreducible analytic submanifold W ⊂ Z containing C such that the fibers of π|W are smooth analytic manifolds which are open subsets of the leaves of φ with the property NC/W ∼= F|C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Y denote the main component of φ−1(φ(C)) (that is, the unique irreducible component that dominates φ(C) under φ) and let Y′ denote its normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Using the universal property of normalization, we see that W admits an embedding into Y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular, Y′ admits a section C′ in its smooth locus that maps to C and has normal bundle NC′/Y′ ∼= F|C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus we can choose a resolution �Y of Y that admits a section �C which maps to C and satisfies T �Y/B| � C ∼= F|C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Grauert-Mulich only applies when a general curve is contained in the flat locus of the eval- uation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This can always be achieved after a birational modification as in the following construction: cons:flatteningfamilyofcurves Construction 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Z be a smooth projective variety and let W be a variety admitting a morphism W → Mg,0(Z) that is generically finite onto its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let UW denote the 28 universal family over W and let Uν W denote the normalization of UW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then Uν W is equipped with a map p : Uν W → W and an evaluation map evW : Uν W → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume that a general fiber of p is a smooth projective curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We claim there is a birational map φ : Z′ → Z from a smooth variety Z′ and an open subset W ◦ ⊂ W such that the preimage Uν,◦ W := p−1W ◦ admits a flat morphism ev′ : Uν,◦ W → Z′ satisfying evW|Uν,◦ W = φ◦ev′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Indeed, suppose we take a flattening of ev, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' a diagram V � ev � ψ � �Z ψZ � Uν W evW � Z where V and �Z are varieties, ψ and ψZ are projective birational morphisms, and �ev is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let ρ : Z′ → �Z be a resolution of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since �ev is flat, V′ := V × �Z Z′ is also a variety and the projection map ev′ : V′ → Z′ is still flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The induced map ψ′ : V′ → Uν W is still birational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since p defines a family of curves, there is an open subset W ◦ ⊂ W such that p−1W ◦ is disjoint from every ψ′-exceptional center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then W ◦ has the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We are now ready to state the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' theo:betterpointsandfoliations Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : Z → B be a good fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive integer J ≥ 2g(B) + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose M is an irreducible component of Sec(Z/B) and let ev : Uν → Z denote the evaluation map for the normalization of the universal family over Mred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume that ev is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let g : S → Z denote the finite part of the Stein factorization of ev and let N denote the family of sections on S corresponding to general members of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let ρ : S′ → S be a birational map from a smooth projective variety that flattens the evaluation map for the normalization of the universal family over N as in Construction 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let C′ denote the strict transform on S′ of a general section on S parametrized by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that S′ is equipped with a dominant rational map ψ : S′ ��� T over B where T is a normal projective B-variety and ψ has connected fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let G denote the foliation induced by ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Furthermore assume that µmax [C′] (G) ≥ (J + 2g(B) + γ − 1) > µmax [C′] (TS′/G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' where we define γ = (g(B) dim(Z) − g(B) + 1)2(dim(Z) − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then either: (1) the deformations of C′ in the main component P of ψ−1(ψ(C′)) contain at least J general points of P, or (2) there is a dominant rational map φ : S′ ��� W over B to a normal projective B- variety W such that ψ factors rationally through φ and the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let C′ be a general section in our family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Y denote the main component of φ−1(φ(C′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then there is a resolution �Y of Y and a section �C on �Y that maps to C′ such that: (a) The deformations of �C in �Y contain at least J general points of �Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (b) The space of deformations of C′ in Y has codimension at most (dim(P)−1)(J + 2g(B) + γ) in the space of deformations of C′ in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (c) Letting H denote the foliation induced by φ, we have µmax [C′] (TS′/H) < J +2g(B)+ γ − 1 ≤ µmin [C′] (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 29 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let us assume that deformations of C′ do not go through J general points of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since C′ deforms in a flat family on S′, a general section C′ in the family will be contained in the smooth locus of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus, if take a resolution �P and consider the strict transform C† then Construction 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 shows that the normal bundle of C† in �P is isomorphic to G|C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6 our deformation assumption implies that µmin(G|C′) < J + 2g(B) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Applying Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10 we obtain µmin [C′] (G) ≤ µmin(G|C′) + γ < J + 2g(B) + γ − 1 eq:minslopegbound eq:minslopegbound (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1) Write the Harder-Narasimhan filtration of G with respect to [C′] as 0 = F0 ⊂ F1 ⊂ · · · ⊂ Fk = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By assumption µmax [C′] (G) ≥ J + 2g(B) + γ − 1 so there is some index i ≥ 1 such that we have µmin [C′] (Fi) ≥ J + 2g(B) + γ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let i be the maximum index for which this inequality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' On the one hand, since Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1) shows that µmin [C′] (G) < J + 2g(B) + γ − 1 we must have i < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' On the other hand, since i was selected to be as large as possible we must have µmax [C′] (G/Fi) < J + 2g(B) + γ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We claim that Fi is a foliation on S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='13 it suffices to check that Fi is a term in the Harder-Narasimhan filtration of TS′ with respect to [C′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By assumption µmax [C′] (TS′/G) < J + 2g(B) + γ − 1 ≤ µmin [C′] (Fi) and thus the Harder-Narasimhan filtration of TS′ agrees with the Harder-Narasimhan filtration of G up to the ith entry, proving the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By [CP19, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1] the foliation Fi is induced by a rational map φ : S′ ��� W over B that has connected fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since i < k this rational map is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By our flatness assumption a general section C′ will be contained in the regular locus of Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let �Y denote a resolution of the main component of φ−1(φ(C′)) and let �C denote the section chosen as in Construction 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular we have T �Y/B| � C ∼= Fi|C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10 implies that µmin(Fi|C′) ≥ µmin [C′] (Fi) − γ ≥ J + 2g(B) − 1 and so by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6 we see that �C can go through at least J general points of �Y verifying (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' To prove (b), let NP denote the space of deformations of the strict transform of C′ in �P and let NY denote the space of deformations of �C in �Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Appealing to Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4, we see that dim(NP) − dim(NY) ≤ (c1(G) · C′ + (dim( �P) − 1)) − (c1(Fi) · C′ + (dim( �Y) − 1)(1 − g(B))) = c1(G/Fi) · C′ + (dim(P) − dim(Y)) + g(B)(dim( �Y) − 1) < (dim(P) − dim(Y))(J + 2g(B) + γ) + g(B)(dim( �Y) − 1) ≤ (dim(P) − 1)(J + 2g(B) + γ) 30 Since the dimension of the space of sections is birationally invariant, we obtain (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Finally, by construction µmin [C′] (Fi) ≥ J + 2g(B) + γ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' On the other hand we have already seen that both µmax [C′] (G/Fi) and µmax [C′] (TS′/G) are strictly less than J +2g(B)+γ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This implies (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Twists over function fields of complex curves sect:twists Let B be a smooth projective curve over an algebraically closed field k of characteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose we have a dominant generically finite morphism fη : Yη → Xη between normal projective K(B)-varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In this section we study the set of twists of fη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Recall that a twist of fη is a generically finite K(B)-morphism f ′ η : Y′ η → Xη such that there is an Xη-isomorphism between Yη and Y′ η (where the subscript η denotes the base change to Spec K(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 we discuss the Hurwitz space as described by [Wew98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Using this con- struction, we show in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2 that the set of twists of a dominant generically finite map fη : Yη → Xη can be parametrized by a scheme with countably many components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We will not construct a universal stack, since there are some steps in the construction which might not be valid in the setting of stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Instead, we will construct a morphism of schemes such that every twist of fη is the fiber over some closed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The remainder of the section is devoted to analyzing the canonical divisor for twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3, we prove a local-to-global principle (Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5) for the Galois cohomology group parametrizing twists of fη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular the local invariant gives us a convenient way to identify the places of K(B) where two twists are “the same” locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4 we apply Hensel’s Lemma to give a geometric criterion that will guarantee the vanishing of the local invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Finally, in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5 we analyze how the canonical divisor changes as we choose different twists of fη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The key point is that its positivity is controlled by the places of K(B) where the local invariant does not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular, we show that if we bound the positivity of the canonical divisor then the parameter space of twists has finite type (Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' sect:hurwitzspace 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Hurwitz space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The starting point is the following version of the Hurwitz space: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 ([Wew98]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let B be a smooth projective curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive integer r and a finite group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is a smooth Deligne-Mumford stack H(G, r, B) parametrizing pairs (q, ψ) where q : C → B is a Galois morphism from a smooth projective curve C that has r branch points and ψ is an isomorphism ψ : Aut(C/B) → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose we fix a finite group G and set H(G, B) = ⊔rH(G, r, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then we can think of H(G, B) as a parameter space for pairs (C/B, ψ) where C/B is a finite Galois cover and ψ : Gal(K(C)/K(B)) → G is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We denote the universal family over H(G, B) by U(G, B) → H(G, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This means that there is a morphism U(G, B) → H(G, B) × B which over every point of the form Spec(k) → H(G, B) is the corresponding cover C → B with an isomorphism ψ : Gal(K(C)/K(B)) → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since our parameter space includes the data of an isomorphism ψ : Aut(C/B) → G, the fiber of G := G × H(G, B) over (C/B, ψ) ∈ H(G, B) can be canonically identified with the Galois group Gal(K(C)/K(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 31 sect:familyoftwists 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The space of twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We fix a dominant generically finite map fη : Yη → Xη of normal geometrically integral projective K(B)-schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let G be a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' To avoid some stacky issues in our constructions, we will fix an ´etale cover HG → H(G, B) from a scheme HG whose irreducible components are varieties of finite type over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We denote the pullback of the universal family by UHG → HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We will use (C/B, ψ) to denote any closed point of HG such that the corresponding fiber of UHG → HG is the map C → B equipped with the isomorphism ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We let GHG → HG denote the morphism whose fiber over (C/B, ψ) ∈ HG is the corresponding Galois group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=', GHG = G×HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We let K(Yη/Xη)HG := K(Yη/Xη)×HG denote the trivial group scheme over HG associated to K(Yη/Xη) = Aut(Yη/Xη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We consider the universal family UHG → HG×B and its base change U∗ HG → HG×Spec K(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since Yη ×Spec K(B) U∗ HG is flat and projective over HG × Spec K(B) and Xη ×Spec K(B) U∗ HG is projective over HG × Spec K(B), by [Kol96, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10 Theorem] we can define the relative automorphism scheme �K(Yη/Xη)HG := AutHG×Spec K(B)(Yη ×Spec K(B) U∗ HG/Xη ×Spec K(B) U∗ HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This is a quasi-finite group scheme over HG × Spec K(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since �K(Yη/Xη)HG can be embedded into K(Yη/Xη)HG ×Spec K(B) as a HG×Spec K(B)- scheme, we conclude that �K(Yη/Xη)HG is quasi-affine over HG × Spec K(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Using the functoriality of the AutHG×Spec K(B)-functor we can construct descent data for the quasi-finite group scheme �K(Yη/Xη)HG → HG ×Spec K(B) with respect to the map HG ×Spec K(B) → HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Indeed, we denote by p1, p2 the two projections HG × Spec K(B) × Spec K(B) → HG × Spec K(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then both p∗ 1 �K(Yη/Xη)HG and p∗ 2 �K(Yη/Xη)HG are canonically isomorphic to AutHG×Spec K(B)×Spec K(B)(Yη ×Spec K(B) U∗ HG × Spec K(B)/Xη ×Spec K(B) U∗ HG × Spec K(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This defines the canonical descent data p∗ 1 �K(Yη/Xη)HG → p∗ 2 �K(Yη/Xη)HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then it is easy to check that this data satisfies the gluing condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By the fpqc descent theory for quasi affine schemes as in [Poo17, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5(ii)] we conclude the existence of a quasi-finite group scheme K(Yη/Xη)HG → HG whose base change to HG × Spec K(B) is isomorphic to �K(Yη/Xη)HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that this is a locally closed subgroup scheme of K(Yη/Xη)HG, so in particular K(Yη/Xη)HG is quasi-affine over HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For (C/B, ψ) ∈ HG consider the Galois action by conjugation φC/B,ψ : Gal(K(C)/K(B)) × (K(Yη/Xη)HG)(K(C)/K(B),ψ) → (K(Yη/Xη)HG)(K(C)/K(B),ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This fiberwise Galois action defines a group scheme action φ : GHG ×HG K(Yη/Xη)HG → K(Yη/Xη)HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Consider the morphism scheme MorHG(GHG, K(Yη/Xη)HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We define the space C1(GHG, K(Yη/Xη)HG) of 1-cocycles as the closed subscheme of MorHG(GHG, K(Yη/Xη)HG) consisting of 1-cocycles (C/B, ψ, σ : G → (K(Yη/Xη)HG)(K(C)/K(B),ψ)) which satisfy the cocycle condition σst = σsφ(C/B,ψ)(s)(σt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 32 Next our goal is to construct a family of twists of (Yη/Xη) over C1(GHG, K(Yη/Xη)HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Define Y′ = Yη ×Spec K(B) U∗ HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then the fiber of the projection Y′ → HG over (C/B, ψ) is isomorphic to Yη ⊗ K(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We consider Y′ ×HG C1(GHG, K(Yη/Xη)HG) → Xη ×Spec K(B) U∗ HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is a group scheme action of GHG on this fiber product by (s, (C/B, ψ)) · (y, σ, (C/B, ψ)) = (σs ◦ (1 ⊗ s)(y), σ, (C/B, ψ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We let �Y denote the quotient of Y′×HG C1(GHG, K(Yη/Xη)HG) by the finite flat group scheme GHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then �Y comes equipped with a map �Y → C1(GHG, K(Yη/Xη)HG) × Xη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' such that the fiber over (σ, (C/B, ψ)) ∈ C1(GHG, K(Yη/Xη)HG) is the map Yσ η → Xη, where Yσ η is the quotient of Yη ⊗ K(C) by the Galois action Gal(K(C)/K(B)) ∋ s �→ σs ◦ 1 ⊗ s ∈ Aut(Yη ⊗ K(C)/Xη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By construction every twist of Yη → Xη is parametrized by the fiber over some point (σ, (C/B, ψ)) ∈ C1(GHG, K(Yη/Xη)HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that the scheme C1(GHG, K(Yη/Xη)HG) constructed above need not have finite type over K(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' However, if we fix certain invariants then the corresponding subscheme will have finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' lemm:twistboundedconditions Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a smooth projective curve B and positive integers d, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose we have a generically finite dominant K(B)-morphism fη : Yη → Xη where Xη and Yη are normal projective varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let S denote the set of twists f σ η such that Yσ η and Yη become isomorphic after a base change by a Galois extension K(C)/K(B) whose degree is ≤ d and whose branch locus consists of at most b points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is a finite type scheme R over C and morphisms ψ : UR → R, g : UR → Xη such that every element Yσ η ∈ S is isomorphic to the fiber of ψ over some closed point t ∈ R and f σ η = g|ψ−1t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There are finitely many isomorphism classes of finite groups G of order ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' As we vary G over all such groups and vary over all r ≤ b, we obtain a finite type Deligne-Mumford stack ⊔H(G, r, B) parametrizing extensions K(C)/K(B) and automorphisms Aut(C/B) → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let HG,r be the preimage of H(G, r, B) via HG → ⊔rH(G, r, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then the space ⊔G,rC1(GHG,r, K(Yη/Xη)HG,r) is a finite type scheme over C where C1(GHG,r, K(Yη/Xη)HG,r) is the base change of C1(GHG, K(Yη/Xη)HG) via HG,r → HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus our assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ 33 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The spaces of twists in families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Here we perform the constructions in the previous section in families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' As before we fix a smooth projective curve B defined over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let X → S×Spec K(B), Y → S×Spec K(B) be flat families of normal projective K(B)-varieties where S is a scheme of finite type over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We also assume that we have a S × Spec K(B)- morphism f : Y → X which is fiberwise dominant and generically finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We fix a finite group G and take an ´etale open cover HG → H(G, B) where HG is a scheme over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' As before we denote the pullback of the universal family by UHG → HG × B and consider its base change U∗ HG → HG × Spec K(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since Y ×Spec K(B) U∗ HG is projective and flat over S × H × Spec K(B) and X ×Spec K(B) U∗ HG is projective over S × HG × Spec K(B), we can define the relative automorphism group �K(Y/X)S×HG = AutS×HG×Spec K(B)(Y ×Spec K(B) U∗ HG/X ×Spec K(B) U∗ HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This is a quasi-finite group scheme over S × HG × Spec K(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since the above relative automorphism group is also separated over S×HG×Spec K(B), �K(Y/X)S×HG is quasi-affine over S×HG×Spec K(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Using fpqc descent theory, �K(Y/X)S×HG → S×HG×Spec K(B) descends to K(Y/X)S×HG → S × HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let GS×HG = G × S × HG and consider the natural conjugation group action φ : GS×HG ×S×HG K(Y/X)S×HG → K(Y/X)S×HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Consider the morphism scheme MorS×HG(GS×HG, K(Y/X)S×HG) and the closed subscheme consisting of the space of 1-cycles C1(GS×HG, K(Y/X)S×HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Define Y′ = Y ×Spec K(B) U∗ HG as a scheme over S × HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Again we have a natural group action of GS×HG on Y′ ×S×HG C1(GS×HG, K(Y/X)S×HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We define �Y to be the quotient of this group action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' It comes equipped with a morphism �Y → C1(GS×HG, K(Y/X)S×HG) ×S X realizing C1(GS×HG, K(Y/X)S×HG) as the parameter space of twists of the maps fs,η : Ys,η → Xs,η for closed points s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Regarding this family we have the following boundedness statement: lemm:twistboundedconditions2 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a smooth projective curve B and positive integers d, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let p : X → S×Spec K(B), q : Y → S×Spec K(B) be flat families of normal projective K(B)-varieties where S is a scheme of finite type over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We also assume that we have a S × Spec K(B)- morphism f : Y → X which is fiberwise dominant and generically finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let A denote the set of twists f σ η,s : Yσ η,s → Xη where s is a closed point of S and Yσ η,s and Yη,s become isomorphic after a base change by a Galois extension K(C)/K(B) whose degree is ≤ d and whose branch locus consists of at most b points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is a finite type scheme R over S and morphisms ψ : UR → R, g : UR → R ×S X such that every element f σ η,s : Yσ η,s → Xη,s ∈ A is isomorphic to the fiber of ψ over some closed point t ∈ R and f σ η,s = g|ψ−1t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' subsec:functoriality 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Functoriality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let X → S × Spec K(B), Y → S × Spec K(B) be flat families of normal projective K(B)-varieties where S is a smooth scheme of finite type over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We also assume that we have a S × Spec K(B)-morphism f : Y → X which is fiberwise dominant and generically finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We further assume that we have flat families W → S × Spec K(B), T → S × Spec K(B) of projective varieties such that Ts is normal for any s ∈ S and we 34 have a commutative diagram Y f � r � X p � T t � W over S × Spec K(B) where p, r are dominant with connected fibers and t is dominant, finite, and fiberwise Galois over S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=', each fiber Ts → Ws is a finite Galois cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We also assume that the Stein factorization of Y → X → W is given by Y → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since a relative automorphism induces a relative automorphism of Stein factorizations, we obtain a homomorphism AutS×HG(Y×Spec K(B)U∗ HG/X×Spec K(B)U∗ HG) → AutS×HG(T×Spec K(B)U∗ HG/W×Spec K(B)U∗ HG), and this induces a morphism C1(GS×HG, K(Y/X)S×HG) → C1(GS×HG, K(T/W)S×HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' sect:localtoglobal 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Local-to-global principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let K(B) be the function field of a smooth projective curve B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let f : Yη → Xη be a dominant generically finite morphism between normal projective varieties Yη and Xη defined over K(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We fix a place ν of K(B) over a place ν of K(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This specifies for every finite cover C → B a point pν,C on C such that for every factoring C g−→ C′ → B we have g(pν,C) = pν,C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Consider the decomposition group Dν = {σ ∈ Gal(K(B)) | σ(ν) = ν}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This is isomorphic to Gal(K(B)ν) ∼= lim ←−(Z/NZ)×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that if we have two places ν, ν′ corresponding to the same place ν on K(B), then Dν and Dν′ are conjugate to each other in Gal(K(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Recall that the Galois group acts on Aut(Yη/Xη) by conjugation, and in this way one can consider Galois cohomology H1(Gal(K(B)), Aut(Yη/Xη)) The injection Gal(K(B)ν) ∼= Dν ⊂ Gal(K(B)) induces a map on Galois cohomology H1(Gal(K(B)), Aut(Yη/Xη)) → H1(Gal(K(B)ν), Aut(Yη/Xη)) which we denote by invν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (Although the choice of isomorphism Gal(K(B)ν) ∼= Dν depends on the choice of ν, the induced map of Galois cohomology only depends on the place ν up to an isomorphism of the pointed set, justifying our mild abuse of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=') For any twist [σ] of Yη/Xη the local invariant invν([σ]) vanishes for all but finitely many places ν of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus we obtain a map H1(Gal(K(B)), Aut(Yη/Xη)) → � ν∈B H1(Gal(K(B)ν), Aut(Yη/Xη)) and we would like to use this map to establish a local-to-global principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that this map does not need to be injective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' for example, there can be twists of Yη/Xη which are trivialized by an ´etale cover of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' However, we will show that the fibers of this map are finite, which is good enough for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 35 lemm:boundingdeganddisc Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let f : Yη → Xη be a dominant generically finite morphism between normal projective varieties over K(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then there exists a positive integer d = d(Yη/Xη) and a fixed finite subset P ⊂ B (depending only on Yη/Xη) such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that [σ] ∈ H1(Gal(K(B)), Aut(Yη/Xη)), is a cohomology class and let Q ⊂ B denote the finite set of places ν ∈ B with invν([σ]) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then there exists a Galois cover C → B of degree at most d and whose branch locus is contained in P ∪ Q such that Yσ η → Xη splits over K(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let K(B′)/K(B) be a fixed Galois extension so that Aut(Yη/Xη) = Aut(Yη ⊗ K(B′)/Xη ⊗ K(B′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We define P to be the set of branch points for B′ → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We can restrict our cocycle σ : Gal(K(B)) → Aut(Yη/Xη) to the subgroup Gal(K(B′)) to get a cocycle σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then σ′ : Gal(K(B′)) → Aut(Yη ⊗ K(B′)/Xη ⊗ K(B′)) is a honest homomorphism because the Galois action of Gal(K(B′)) on Aut(Yη ⊗ K(B′)/Xη ⊗ K(B′)) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The kernel is an open subgroup and thus defines a Galois cover C over B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then the induced cocycle [τ] ∈ H1(Gal(K(C)), Aut(Yη ⊗ K(C)/Xη ⊗ K(C))) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus Yσ η → Xη splits over K(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Now note that the degree of K(C)/K(B′) is bounded by the order of Aut(Yη/Xη) and the degree of K(B′)/K(B) only depends on Yη/Xη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Furthermore for any place ν ∈ B with invν([σ]) = 0 and ν ̸∈ P, we have Dν ⊂ Gal(K(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus C/B′ cannot be ramified over any place ν ∈ B\\(Q ∪ P) and our assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ coro:localtoglobal Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let f : Yη → Xη be a dominant generically finite morphism between normal projective varieties over K(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The fibers of the local invariant map H1(Gal(K(B)), Aut(Yη/Xη)) → � ν∈B H1(Gal(K(B)ν), Aut(Yη/Xη)) are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that ξ is a element of the direct sum and let Q ⊂ B denote the finite set of indices for which the entries of ξ are non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4, there is a fixed integer d and a fixed finite set P ⊂ B such that any twist that lies in the fiber over ξ is split by a Galois cover C → B of degree at most d and whose branch locus is contained in P ∪ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There are only finitely many such maps C → B, and for each such map there are only a finite set of twists trivialized by the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ sect:henselslemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Hensel’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let B be a smooth projective curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a good fibration and f : Y → X be a dominant finite morphism from a normal projective variety such that Yη is geometrically integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In this setting we have the equality Bir(Yη/Xη) = Aut(Yη/Xη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For each twist [σ] ∈ H1(Gal(K(B)), Aut(Yη/Xη)) of Yη/Xη, we can construct an integral model Yσ → B in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that K(C)/K(B) is a Galois extension such that Yη/Xη and Yσ η /Xη are isomorphic after base change to K(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then the cohomology class [σ] is represented by a cocycle σ : Gal(K(C)/K(B)) → Aut(Yη ⊗K(C)/Xη ⊗K(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let �YC be the normalization of Y ×B C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then σ defines a homomorphism from Gal(K(C)/K(B)) to the birational automorphism group of �YC,η over Xη, or equivalently, to the birational 36 automorphism group of �YC over X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By construction a birational automorphism of �YC over X is actually an automorphism, so that Gal(K(C)/K(B)) acts on �YC via σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We let Yσ denote the quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that Yσ is normal and comes equipped with a finite B-morphism f σ : Yσ → X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Here we prove the following birational version of Hensel’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' lemma:birationalHensel Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a good fibration and let Y be a normal projective variety equipped with a dominant morphism Y → B such that Yη is geometrically integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that f : Y → X is a dominant finite B-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a place ν ∈ B and assume that Xν is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We also assume that AutB(Y/X ) → B is flat at ν ∈ B and Yν is reduced and normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that Yσ is an integral model of a twist of fη as constructed above and that there is a birational Xν-map hν : Yν ��� Yσ ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then there is an X ⊗ K(B)ν-isomorphism between Y ⊗η K(B)ν and Yσ ⊗η K(B)ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular invν(σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since Yν is reduced and normal, the automorphism group Bir(Yν/Xν) = Aut(Yν/Xν) is a reduced finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The flatness of AutB(Y/X ) → B at ν implies that the lengths of Aut(Yη/Xη) and Aut(Yν/Xν) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Also note that since Yν is normal and finite over Xν and Yσ ν is also finite over Xν, our birational map hν extends to a birational morphism Yν → Yσ ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let us consider the relative X -birational morphism scheme over Spec( �OB,ν): B = BirMorSpec( � OB,ν)(Y ×B Spec( � OB,ν), Yσ ×B Spec( � OB,ν)) equipped with a morphism B → Spec( �OB,ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (This scheme can be constructed as an open subscheme of the relative Hilbert scheme parametrizing graphs in (Y ×X Yσ)×B Spec( � OB,ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=') Note that B is an AutSpec( � OB,ν)(Y ×B Spec( � OB,ν)/X ×B Spec( �OB,ν))-torsor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Using the fact that AutB(Y/X ) → B is flat at ν ∈ B we conclude that the above relative birational morphism scheme is also flat at ν ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since we have a birational morphism of fibers Yν → Yσ ν , Hensel’s lemma (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=', [Gro67, Th´eor`em 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='17]) implies that we have an X × Spec( � OB,ν)-birational morphism from Y ×B Spec( �OB,ν) to Yσ ×B Spec( �OB,ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since Y ⊗η K(B)ν and Yσ ⊗η K(B)ν are normal and Y ⊗η K(B)ν → X ⊗K(B)ν, Yσ ⊗η K(B)ν → X ⊗K(B)ν are finite, this birational morphism induces an isomorphism of the generic fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ sect:splittingfields 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Splitting fields and ramification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose given an algebraic fiber space π : Y → B with Y a normal projective variety and a dominant finite morphism of smooth projective curves B′ → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We will use the term “normalized base change” to refer to the normalization of Y ×B B′ equipped with the structure morphism to B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that the normalized base change Y′ admits a dominant finite morphism Y′ → Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' lemm:genreducedpreserved Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Y be a normal projective variety equipped with a surjective morphism π : Y → B with connected fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose B′ → B is a dominant finite morphism of smooth projective curves and that π′ : Y′ → B′ is the normalized base change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a closed point t ∈ B and let t′ ∈ B′ be any point mapping to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If Yt is generically reduced, then Y′ t′ is also generically reduced and the morphism Y′ t′ → Yt is birational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 37 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since Y is normal, the open set U ⊂ Yt consisting of points that lie in the smooth locus of Y and the smooth locus of Yt is dense in Yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' It is clear that the fiber of Y ×B B′ over t′ is generically smooth, hence generically reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Furthermore, the preimage of U in Y ×B B′ will be contained in the smooth locus of Y ×B B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus the normalization map restricts to a birational morphism on this fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ corollary:fiberwisebirational Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : Y → B and πσ : Yσ → B be morphisms from normal projective varieties with connected fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that f : Y → X and f σ : Yσ → X are dominant finite B-morphisms whose generic fibers are twists of each other over X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose further that t ∈ B is a closed point such that the fibers Yt, Yσ t are generically reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then the maps ft, f σ t are birationally equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Choose a dominant finite morphism B′ → B such that the normalized base changes f ′ : Y′ → X ′, f ′σ : Y′σ → X ′ are birationally equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since f ′, f ′σ are finite, they are equal to their own Stein factorizations and thus f ′ and f ′σ are isomorphic to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular the maps f ′ t′ : Y′ t′ → Xt and f ′σ t′ : Y′σ t′ → Xt are isomorphic to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' But by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7 these are birationally equivalent to ft and f σ t respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ We next analyze how the canonical divisor changes upon normalized base change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This is well-understood, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=', in the context of semistable reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose h : Y′ → Y is a finite morphism of normal projective varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let U ⊂ Y and U′ ⊂ Y′ denote the smooth loci and set V = U′ ∩ h−1(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that the complement of V in Y′ has codimension ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The Riemann-Hurwitz formula gives us a distinguished effective representative E in the linear equivalence class of KV/U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We define the relative canonical divisor KY′/Y to be the effective Weil divisor obtained by taking the closure of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' lemm:canonicalandnbc Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Y be a normal projective variety equipped with a surjective morphism π : Y → B with connected fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose g : B′ → B is a dominant finite morphism of curves and consider the normalized base change Y′ φ � π′ �■ ■ ■ ■ ■ ■ ■ ■ ■ ■ Y ×B B′ h � � Y π � B′ g � B We define R to be the π′-pullback of the ramification divisor of g and for any point t′ ∈ B′ we let Rt′ denote the intersection of R with the preimage of t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (1) We have KY′/Y ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) If t ∈ B is a closed point such that the fiber Yt is generically reduced then for any t′ ∈ B′ mapping to t we have Rt′ ≤ KY′/Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (1) First note that the support of KY′/Y is contained in the support of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Indeed, the map h is ´etale away from φ(Supp(R)) and thus φ is an isomorphism over this locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose t′ ∈ B′ is a ramification point for g with index e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let t ∈ B be the image of t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We choose local coordinates s′ and s at t′ and t respectively so that g is defined locally by s = s′e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let T ′ be an irreducible component of the fiber π′−1(t′) and let q′ denote its multiplicity in its component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let u′ be a generic local equation of T ′ so that generically the map Y′ → B′ is given by by s′ = u′q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let T ⊂ U be the image of T ′ and let u denote 38 a local equation of T at a generic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We denote the multiplicity of T in π−1(t) by q so that Y is generically defined by s = uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The coefficient of T ′ in R is given by (e − 1)q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Also uq = s = s′e = u′eq′ so the coefficient of T ′ in KY′/Y is given by eq:multiplicityincanonical eq:multiplicityincanonical (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1) eq′ q − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since e > 1, we have (e − 1)q′ ≥ eq′ q − 1 when q > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' When q = 1, q′ = 1 by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7 and so the inequality still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus our assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) As explained above when q = 1 we have q′ = 1 as well by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus we have our assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ prop:ramificationdivisor Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let X → B be a good fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Y, Yσ be normal projective varieties which admit surjective morphisms π : Y → B and πσ : Yσ → B with connected fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose there are dominant finite B-morphisms f : Y → X and f σ : Yσ → X whose generic fibers are twists of each other over K(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Choose a finite Galois morphism g : B′ → B such that the normalized base changes of Y and Yσ over B′ are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We let �Y denote this abstract variety;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' it is equipped with finite morphisms ρ1 : �Y → Y and ρ2 : �Y → Yσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We denote the degree of B′ → B by d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is a Weil divisor E on �Y , which we write as E = E+ − E− where E+, E− are effective with no common divisor in their support, such that K �Y/Y − K �Y/Yσ ≥ E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' we have E+ ≥ � t′ �Yt′ as t′ ∈ B′ varies over closed points whose image t ∈ B satisfies that Yt is normal but the fiber Yσ t is not generically reduced;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' we have E− ≤ d · � t′ �Yt′ as t′ ∈ B′ varies over closed points whose image t ∈ B satisfies that Yt is not normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We compare these divisors along each fiber separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let t′ ∈ B′ be a closed point and let t ∈ B be its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If the fiber Yt is not normal, then it follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10 that (K �Y/Y)t′ − (K �Y/Yσ)t′ ≥ −(e − 1) �Yt′, where e is the ramification index of t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular this difference is ≥ −d �Yt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If Yt is normal, then in particular Yt is irreducible and reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus the fiber �Yt′ is also irreducible and reduced, and we conclude that Yσ t is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let us denote the multiplicity of Yσ t by q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' It follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10 that (K �Y/Y)t′ − (K �Y/Yσ)t′ ≥ 0 and equality holds when q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The two inequalities above together prove the upper bound on E−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' To prove the lower bound on E+, we analyze those fibers such that Yt is normal and which satisfy q > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since �Yt′ is reduced we must have e > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since the multiplicities of Yt and �Yt′ are 1, Equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1) in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10 shows that (K �Y/Y)t′ = (e − 1) �Yt′, (K �Y/Yσ)t′ = � e q − 1 � �Yt′ so that we conclude (K �Y/Y)t′ − (K �Y/Yσ)t′ ≥ e � 1 − 1 q � �Yt′ ≥ �Yt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 39 Thus our assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ prop:curveintandrambound Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let X → B be a good fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Y, Yσ be normal projective varieties which admit surjective morphisms π : Y → B and πσ : Yσ → B with connected fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose there are dominant finite B-morphisms f : Y → X and f σ : Yσ → X whose generic fibers are twists of each other over K(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Choose a finite Galois morphism g : B′ → B such that the normalized base changes of Y and Yσ over B′ are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We let �Y denote this abstract variety;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' it is equipped with finite morphisms ρ1 : �Y → Y and ρ2 : �Y → Yσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We denote the degree of B′ → B by d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Yσ′ be a smooth birational model of Yσ equipped with a birational morphism β : Yσ′ → Yσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume that there exists a section C on Yσ′ and a constant R > 0 such that C corre- sponds to a rational point on the smooth locus of Yσ η and (KYσ′/B − β∗(f σ)∗KX/B) · C ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let r denote the number of closed points t ∈ B such that the fiber Yt is normal but the fiber Yσ t is not generically reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then we have r ≤ dR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We choose smooth models Y′, �Y′ of Y, �Y respectively such that there are birational morphisms α : Y′ → Y, γ : �Y′ → �Y and generically finite morphisms �ρ1 : �Y′ → Y′, �ρ2 : �Y′ → Yσ′ which are birationally equivalent to ρ1, ρ2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We may ensure that γ−1 is well-defined along the smooth locus of Yσ η ⊗ K(B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Our intersection bound implies that KYσ′/X · C ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since by assumption C is not contained in the ρ2-image of the γ-exceptional centers, there is a section C′ of �Y′/B′ such that (�ρ2)∗C′ = dC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then we have �ρ∗ 2KYσ′/X · C′ ≤ dR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that �ρ∗ 2KYσ′/X = K �Y′/X − K �Y′/Yσ′ = �ρ∗ 1KY′/X + K �Y′/Y′ − K �Y′/Yσ′ ≥ K �Y′/Y′ − K �Y′/Yσ′ Let E be the divisor on �Y defined by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='11, so E+ is at least as effective as the sum of the r fibers of �Y corresponding to the r closed points t ∈ B such that the fiber Yt is normal but the fiber Yσ t is not generically reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Taking strict transforms, we see that K �Y′/Y′ −K �Y′/Yσ′ is at least as effective as the sum of the strict transforms of these r fibers of �Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Furthermore every exceptional divisor of β : Yσ′ → Yσ is contracted by f σ ◦ β : Yσ′ → X as well, and thus appears with positive coefficient in the ramification divisor KYσ′/X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We conclude that the support of the effective divisor �ρ∗ 2KYσ′/X contains the r reduced fibers over these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since our section C′ must meet each fiber in a component of multiplicity one, we conclude that r ≤ �ρ∗ 2KYσ′/X · C′ ≤ dR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ coro:boundedintimpliesboundedtwists Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let X → B be a good fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Y, Yσ be normal projective varieties which admit surjective morphisms π : Y → B and πσ : Yσ → B with connected fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 40 Suppose there are dominant finite B-morphisms f : Y → X and f σ : Yσ → X whose generic fibers are twists of each other over K(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let �Yσ be a smooth birational model of Yσ equipped with a birational morphism β : �Yσ → Yσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume that there exists a section C on �Yσ and a constant R > 0 such that C corre- sponds to a rational point on the smooth locus of Yσ η and (K �Yσ/B − β∗(f σ)∗KX/B) · C ≤ R Then there exists constants d = d(Y/X ) and n = n(Y/X , R) such that there exists a finite Galois morphism B′ → B of degree at most d with at most n branch points such that the normalized base changes of Y/B and Yσ/B by B′ → B become X ×B B′-isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular, the set of such twists is a bounded family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' It follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4 that there exists d = d(Y/X ) and a finite Galois morphism �B → B of degree at most d such that the normalizations of Y ×B �B and Yσ ×B �B are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let s = s(Y/X ) be the number of t ∈ B such that Yt is not normal or AutB(Y/X ) → B is not flat at t ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let r be the number of t ∈ B such that Yt is normal but Yσ t is not generically reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='12 we have r ≤ dR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For t ∈ B such that Yt is normal, AutB(Y/X ) → B is flat at t ∈ B and Yσ t is generically reduced, it follows from Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6 that invt(σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus the number of t ∈ B such that invt(σ) ̸= 0 is bounded above by s + dR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus our first assertion follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The final statement then follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fujita invariant and sections sec:fujinv Suppose that π : X → B is a good fibration and L is a generically relatively big and semiample Cartier divisor on X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In this section the goal is to classify the generically finite B-morphisms f : Y → X such that Y carries a family of sections N with the property that f∗N has small codimension in an irreducible component of Sec(X /B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assuming the sections have large L-degree but small degree against f ∗(KX/B + a(Xη, L|Xη)L), we show that the Fujita invariant of Yη must be at least as large as the Fujita invariant of Xη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This puts a strong constraint on the set of morphisms f which have this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' After addressing some preliminaries in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 and Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2, we show the funda- mental result discussed above in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' When working with the Fujita invariant it is often helpful to know that the pair (Yη, f ∗L|Yη) is adjoint rigid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4 we show that if we additionally assume that the sections parametrized by N go through many general points of Y then we can also guarantee adjoint rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' sect:pivertical 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Modifying by π-vertical divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a good fibration and let L be a generically relatively big and semiample Cartier divisor on X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We know that L|Xη is Q-linearly equivalent to a divisor which has smooth support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The following proposition discusses how to reframe this property as a global statement by adding π-vertical divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' prop:generalsurjectiontofiber Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a good fibration, let L be a generically relatively big and semiample Cartier divisor on X , and let a be a positive rational number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let b > a be a positive integer such that bL|Xη defines a basepoint free linear series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is some effective π-vertical Q-Cartier divisor E on X such that the following property holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 41 Suppose ψ : Y → B is a good fibration and f : Y → X is a B-morphism that is generically finite onto its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then there is an effective Q-Cartier divisor D on Y that is Q-linearly equivalent to f ∗(aL + E) such that D|Yη has smooth irreducible support and coefficient a b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular (Yη, D|Yη) is a terminal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' , Tr be a K(B)-basis for |bL|Xη|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that ∩iTi = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We denote by T i the closure of Ti in X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is some effective π-vertical Q-Cartier divisor �E such that for every i there is an effective π-vertical divisor Fi satisfying T i+Fi ∼ b(L+ �E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let f : Y → X be a morphism as in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By construction we have ∩if ∗(T i +Fi) does not intersect Yη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus f ∗(b(L+ �E)) is linearly equivalent to a divisor �D whose restriction to Yη is smooth and irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then D = a b �D and E = a �E have the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ We will use the following definition to capture the effect of the extra divisor E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' defi: invariant_tau Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a good fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that E is an effective π-vertical Q-Cartier divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Define τ(π, E) = sup sections C E · C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that this supremum is achieved by some section C since the intersection number is bounded above by the sum of the coefficients of E and is contained in 1 rZ where r is the least common multiple of the denominators of the coefficients of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' sect:relvsabs 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Relative versus absolute positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We will also need a couple results comparing relative and absolute positivity for a fibration over a curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' lemm:relvsabsmmp Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : Z → B be a good fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that D is an effective Q-Cartier divisor on Z such that (Z, D) is a terminal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (1) Suppose that g(B) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that ρ : Z ��� ˜Z is a rational map obtained by running the (KZ + D)-MMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then ρ is also a run of the relative (KZ + D)-MMP over B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) Suppose that g(B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is a constant m = m(dim(Z)) such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a general fiber F of π and suppose that ρ : Z ��� ˜Z is a birational morphism obtained by running the (KZ + D + mF)-MMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then ρ is also a run of the relative (KZ + D + mF)-MMP over B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular, in case (1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' case (2)) if KZ + D (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' KZ + D + mF) is not pseudo- effective then its restriction to Zη is also not pseudo-effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (1) By [Kaw91] each step of the (KZ + D)-MMP contracts an extremal ray that is spanned by a rational curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This rational curve must be vertical with respect to π because B has genus ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) Since (Z, D) is 1 2-lc, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='16 there is an integer m = m(dim(Z)) such that Nef1(Z) + Eff1(Z)KZ+D+mF ≥0 = Eff1(Z)KZ+D+mF ≥0 + � j [Cj] where the Cj are π-vertical moving curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular, any contraction of a (KZ+D+mF)- negative extremal ray must define a relative contraction over B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Furthermore the analogous equality of cones holds for any birational model of Z obtained by running the MMP (since 42 the 1 2-lc condition is preserved).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus we see that every step of the (KZ + D + mF)-MMP is actually a step of the relative MMP over B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' To see the final statement, suppose we are in case (1) and KZ + D is not pseudo-effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then we can run the (KZ + D)-MMP with scaling of an ample divisor and the outcome will be a Mori fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' But then this Mori fibration must be a relative fibration over B, so that (KZ + D) is not relatively pseudo-effective over B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The same argument applies in case (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ Our next result shows how to turn intersection inequalities into Fujita invariant inequali- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' lemm:terminalsectiontofiber Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : Z → B be a good fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that D is an effective Q-Cartier divisor on Z such that (Zη, D|Zη) is a terminal pair and D|Zη is big and nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (1) Suppose that g(B) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If there is a dominant family of HN-free sections C on Z which satisfy −(KZ + D) · C > 0 then a(Zη, D|Zη) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) Suppose that g(B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is a constant Ξ = Ξ(dim(Z)) such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If there is a dominant family of HN-free sections C on Z which satisfy −(KZ + D) · C > Ξ then a(Zη, D|Zη) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let φ : Z′ → Z be a log resolution of (Z, D) and let D′ be the strict transform of the π-horizontal components of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' After perhaps taking a further blow-up, we may assume that two irreducible components of D′ intersect if and only if their restrictions to the generic fiber intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Along the central fiber we can write KZ′η + D′|Z′η = φ∗(KZη + D|Zη) + Eη where Eη is an effective φ-exceptional divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We conclude that KZ′η + D′|Z′η is pseudo-effective if and only if KZη + D|Zη is pseudo-effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We claim that the pair (Z′, D′) has terminal singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since Supp(D′) is an SNC divisor, by [Kol97, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='11 Lemma] the pair (Z′, D′) will be terminal if and only if when we write D′ = � i diD′ i in terms of irreducible components we have min i {1 − di} > 0 and min i,j|Di∩Dj̸=∅{1 − di − dj} > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Recall that by construction two irreducible components of D′ intersect if and only if their restrictions to the generic fiber intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus this computation can be done on the generic fiber, where the desired inequalities follow from the fact that (Zη, D|Zη) is terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let C′ be the strict transform of a general deformation of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since C is HN-free, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8 we can assume that C′ avoids any codimension 2 locus in Z and thus C′ has vanishing intersection against every φ-exceptional divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We have (KZ′ + D′) · C′ ≤ (KZ′ + φ∗D) · C′ = φ∗(KZ + D) · C′ (1) We are in the case g(B) ≥ 1 and (KZ′ + D′) · C′ ≤ φ∗(KZ + D) · C′ < 0 Since C′ is a movable curve, we see that KZ′ + D′ is not pseudo-effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 we see that (KZ′ + D′)|Z′η is also not pseudo-effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' As demonstrated above this means that (KZ + D)|Zη also fails to be pseudo-effective, showing that a(Zη, D|Zη) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 43 (2) We are in the case g(B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let m = m(dim(Z)) be the constant from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) and set Ξ = m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We have (KZ′ + D′) · C′ ≤ φ∗(KZ + D) · C′ < −Ξ and thus (KZ′+D′+mF)·C′ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since C′ is a movable curve, we see that that KZ′+D′+mF is not pseudo-effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 we see that (KZ′ +D′)|Z′η is also not pseudo-effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' As demonstrated above this means that (KZ+D)|Zη also fails to be pseudo-effective, showing that a(Zη, D|Zη) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ sect:fujinvongenfiber 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fujita invariant along the generic fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In this section we show that the Fujita invariant along the generic fiber controls the expected dimension for families of sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We will use the following easy lemma many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' lemm:easyintcalc Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a good fibration and let L be a generically relatively big and semiample Cartier divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume that Xη is geometrically uniruled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive rational number arel and define a = arela(Xη, L|Xη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a rational number β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive integer T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that ψ : Y → B is a good fibration equipped with a B-morphism f : Y → X that is generically finite onto its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that N is an irreducible component of Sec(Y/B) parametrizing a dominant family of sections C on Y which satisfy f ∗(KX/B +a(Xη, L|Xη)L)· C ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Finally, suppose that eq:dimension eq:dimension (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1) dim(N) ≥ arel(−f ∗KX/B · C + (dim(X ) − 1)(1 − g(B))) − T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then (KY + af ∗L) · C ≤ arelβ + T + arel(dim(X ) − 1)(g(B) − 1) + (dim(X ) − 1) + 2g(B) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let C denote a general section parametrized by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4 dim(N) ≤ −KY/B · C + (dim(Y) − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Combining this equality with Equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1) and rearranging we get (KY/B − arelf ∗KX/B) · C ≤ T + arel(dim(X ) − 1)(g(B) − 1) + (dim(Y) − 1) ≤ T + arel(dim(X ) − 1)(g(B) − 1) + (dim(X ) − 1) eq:stupidinequality eq:stupidinequality (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2) Adding in the fact that f ∗(KX/B + a(Xη, L|Xη)L) · C ≤ β, we see that eq:alternateform eq:alternateform (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3) (KY/B + af ∗L) · C ≤ arelβ + T + arel(dim(X ) − 1)(g(B) − 1) + (dim(X ) − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' or equivalently (KY + af ∗L) · C ≤ arelβ + T + arel(dim(X ) − 1)(g(B) − 1) + (dim(X ) − 1) + 2g(B) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ We can now prove our basic result for controlling the Fujita invariant using pathological families of sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' theo:generalainvsections Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a good fibration and let L be a generically relatively big and semiample Cartier divisor on X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume that Xη is geometrically uniruled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive ra- tional number arel and set a = arela(Xη, L|Xη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a rational number β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive integer T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive integer b > a such that bL|Xη defines a basepoint free linear series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Use b to construct an effective π-vertical Q-Cartier divisor E satisfying the conclusion of Proposition 44 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 with respect to aL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is some constant ξ = ξ(dim(X ), g(B), τ(π, E), arel, a, T, β, b) with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that ψ : Y → B is a good fibration equipped with a B-morphism f : Y → X that is generically finite onto its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that N is an irreducible component of Sec(Y/B) parametrizing a dominant family of sections C on Y which satisfy f ∗L · C ≥ ξ and f ∗(KX/B + a(Xη, L|Xη)L) · C ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Finally, suppose that eq:eh eq:eh (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4) dim(N) ≥ arel(−f ∗KX/B · C + (dim(X ) − 1)(1 − g(B))) − T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then a(Yη, f ∗L|Yη) ≥ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We first prove the statement when the general section C parametrized by N is HN-free on Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='9 there is a rational number ǫ > 0 depending only on a and dim(X ) such that no smooth variety of dimension ≤ dim X − 1 has Fujita invariant in the range [(1 − ǫ)a, a) with respect to any big and nef Cartier divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Define Ξ as: Ξ = 0, if g(B) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Ξ is the supremum of the constants obtained by applying Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4 to all dimensions ≤ dim(X ), if g(B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We define ξHN(dim(X ), g(B), τ(π, E), arel, a, T, β, b) to be 1 aǫ ((1 − ǫ)τ(π, E) + arelβ + T + arel(dim(X ) − 1)(g(B) − 1) + (dim(X ) − 1) + 2g(B) − 2 + Ξ) + 1 and assume that our sections C satisfy f ∗L · C ≥ ξHN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let C denote a general section parametrized by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since we are assuming C moves in a dominant family on Y Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5 shows that (KY + af ∗L) · C ≤ arelβ + T + arel(dim(X ) − 1)(g(B) − 1) + (dim(X ) − 1) + 2g(B) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Adding in E, we get (KY+af ∗L+f ∗E)·C ≤ arelβ+T+arel(dim(X )−1)(g(B)−1)+(dim(X )−1)+2g(B)−2+τ(π, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' When the section C has degree ≥ ξHN then the inequality simplifies to (KY + a(1 − ǫ)f ∗L + (1 − ǫ)f ∗E) · C < −Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since E satisfies the conclusion of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 with respect to aL the pullback f ∗(aL + E) is Q-linearly equivalent to an effective Q-divisor D such that (Yη, D|Yη) has terminal singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Of course (Yη, (1−ǫ)D|Yη) also has terminal singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Applying Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4, we deduce that a(Yη, f ∗L|Yη) ≥ (1−ǫ)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By construction this implies that a(Yη, f ∗L|Yη) ≥ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Next we prove the statement when C is not necessarily HN-free on Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Our strategy is to reduce to the HN-free case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Define equation:generalainvsections equation:generalainvsections (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5) ξ = sup � ξHN(dim(X ), g(B), τ(π, E), arel, a, T + (dim(X ) − 1)(4g(B) + 3 + γ), β, b), 1 a((dim(X ) − 1)(5g(B) + 3 + γ) + arelβ + T + arel(dim(X ) − 1)(g(B) − 1)) � 45 where γ = (g(B) dim(X )−g(B)+1)2(dim(X )−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Using the second term as a lower bound on ξ and appealing to Equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3) in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5, we have −KY/B · C ≥ af ∗L · C − arelβ − T − arel(dim(X ) − 1)(g(B) − 1) − (dim(X ) − 1) ≥ (dim(X ) − 1)(5g(B) + 2 + γ) ≥ (dim(Y) − 1)(5g(B) + 2 + γY) where γY = (g(B) dim(Y) − g(B) + 1)2(dim(Y) − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let S′ be a smooth birational model of the finite part of the Stein factorization of the evaluation map for the normalization of the universal family over N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since the dimension of N is the same as the dimension of the corresponding family of sections C′ on S′, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4 shows that −KS′/B · C′ ≥ −KY/B · C − g(B)(dim(S′) − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular we see that µmax [C] (TS′/B) ≥ µ[C](TS′/B) ≥ (4g(B) + 2 + γY).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' On the other hand it is clear that (4g(B) + 2 + γY) > 2 ≥ µmax [C] (π∗TB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus we can apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 to Y with J = 2g(B) + 3, T = B, and G = TS′/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Consider the dominant family of subvarieties W on Y obtained by taking images of the subvarieties constructed on S′ by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' These subvarieties W have the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' First, if we take the strict transform of C in a resolution � W of W then de- formations go through ≥ 2g(B) + 3 general points of � W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7 the strict transform of a general C in � W is HN-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Second, the codimension in N of the space of sections on � W can only increase by at most (dim(Y) − 1)(4g(B) + 3 + γY) ≤ (dim(X ) − 1)(4g(B) + 3 + γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Applying the HN-free version of the desired statement to � W with the constant Tnew = T + (dim(X ) − 1)(4g(B) + 3 + γ), we see that a(Wη, f ∗L|Wη) ≥ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since such W move in a dominant family on Y, [LST22, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8] shows that the generic Fujita invariant of Y is at least as large as that of W so that a(Yη, f ∗L|Yη) ≥ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let M denote the component of Sec(X /B) containing the pushforward of the sections parametrized by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then the right hand side of Equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4) is arel·expdim(M)− T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since the expected dimension is a lower bound on dim(M), we can replace Equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4) by the stronger assumption dim(N) ≥ arel · dim(M) − T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular, when arel = 1 then T should be thought of as the codimension of N in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The same remark holds for later theorems as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' sect:adjrigid 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Adjoint rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Our next goal is to establish a strengthening of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6 that allows us to conclude adjoint rigidity at the cost of increasing the constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' lemm:boundinggeneralpoints Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : Z → B be a good fibration and let M be an irreducible component of Sec(Z/B) parametrizing sections C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that H is a Cartier divisor on Z satisfying H · C + 1 < h0(Z, OZ(H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then the sections parametrized by M go through at most H · C + 1 general points of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 46 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Set Q = H ·C +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since general points impose codimension 1 conditions on the linear series |H| we see that for any set of Q points in Z there is a (possibly reducible) divisor D ∈ |H| containing all Q points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose for a contradiction that the sections parametrized by M can go through Q + 1 general points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This means that the space of sections through Q general points of Z forms a dominant family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular, if we fix Q general points and a divisor D ∈ |H| containing those points, then we can find a section C parametrized by M that contains all the points but is not contained in Supp(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus D · C ≥ Q > H · C, yielding a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ lemm:h0growthbound Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Z be a smooth projective variety of dimension n and let H be a Cartier divisor on Z such that |H| defines a birational morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then for any non-negative integer m we have h0(Z, OZ(mH)) ≥ �n + m n � Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The map |H| defines a morphism g : Z → PN for some N ≥ n such that OZ(H) = g∗O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By composing with a generic projection, we obtain a morphism h : Z → Pn such that OZ(H) = h∗O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus we have h0(Z, OZ(mH)) ≥ h0(Pn, O(m)) = �n + m n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ We can now prove the criterion for adjoint rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' theo:adjointrigidcriterion Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a good fibration and let L be a generically relatively big and semiample Cartier divisor on X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume that Xη is geometrically uniruled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive rational number arel and set a = arela(Xη, L|Xη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a rational number β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive integer T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive integer b > a such that bL|Xη defines a base- point free linear series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Use b to construct an effective π-vertical Q-Cartier divisor E satisfying the conclusion of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 with respect to aL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is some constant Γ = Γ(dim(X ), g(B), τ(π, E), arel, a, T, β, b) with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that ψ : Y → B is a good fibration equipped with a B-morphism f : Y → X that is generically finite onto its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that N is an irreducible component of Sec(Y/B) parametrizing a dominant family of sections C on Y which satisfy f ∗(KX/B +a(Xη, L|Xη)L)· C ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that dim(N) ≥ arel(−f ∗KX/B · C + (dim(X ) − 1)(1 − g(B))) − T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that a(Yη, −f ∗L|Yη) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then either: (1) (Yη, −f ∗L|Yη) is adjoint rigid, or (2) deformations of C go through at most Γ general points of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume that (Yη, f ∗L|Yη) is not adjoint rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We may assume that the general section C is HN-free in Y, since otherwise by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7 the sections parametrized by N can go through at most 2g(B) general points of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Applying Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5 we see that (KY + af ∗L) · C ≤ arelβ + T + arel(dim(X ) − 1)(g(B) − 1) + (dim(X ) − 1) + 2g(B) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 47 Adding in E and rearranging slightly, we obtain (KY+af ∗L+f ∗E)·C ≤ arelβ+T+arel(dim(X )−1)(g(B)−1)+τ(π, E)+(dim(X )−1)+2g(B)−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We denote the right hand side of this equation by R = R(dim(X ), g(B), τ(π, E), arel, a, T, β, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since E satisfies the conclusion of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 with respect to aL, the pullback f ∗(aL+ E) is Q-linearly equivalent to an effective Q-divisor D such that D|Yη has smooth irreducible support and coefficient a b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let φ : Y′ → Y be a log resolution of (Y, D) and let D′ denote the strict transform of the π-horizontal components of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We may ensure that φ is an isomorphism on an open neighborhood of Yη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular D′ is still generically relatively big and nef and is irreducible with coefficient a b, so (Y′, D′) has terminal singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since we are assuming the sections are HN-free, the strict transform C′ of a general deformation of C avoids any φ-exceptional divisor and thus satisfies (KY′ + D′) · C′ ≤ (KY′ + φ∗D) · C′ = φ∗(KY + D) · C′ ≤ R Let F ′ denote a general fiber of Y′ → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3, there is some integer m only depending on dim(X ) such that the (KY′ + D′ + mF ′)-MMP is the same as a relative MMP over B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By assumption on the Fujita invariants KY′ + D′ + mF ′ is on the boundary of the relative pseudo-effective cone over B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus the result of the MMP will be a relative Iitaka fibration ψ : Y′ ��� Z for this divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Furthermore since we are assuming (Yη, f ∗L|Yη) is not adjoint rigid we know that dim(Z) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since D′ is relatively big over B, it is also relatively big over Z in the sense of [HX15, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By [HX15, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4], there is a positive integer k only depending on dim(X ) and a b such that |k(KY′ + D′ + mF ′)| defines a rational map birational to the Iitaka fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Applying the canonical bundle formula as in [FM00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Section 4],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' there is a birational model ρ : W → Y′ and a morphism ψW : W → ZW birationally equivalent to ψ such that: W and ZW are smooth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' there is an effective Q-Cartier divisor BW and a nef Q-Cartier divisor MW such that (ZW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' BW) is klt and k(KZW + BW + MW) is Cartier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' KZW + BW + MW is big,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' and for every integer p divisible by k we have that h0(Y′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' OY′(p(KY′ + D′ + mF ′))) = h0(ZW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' OZW (p(KZW + BW + MW))) and the linear series |p(KZW + BW + MW)| defines a birational map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We claim that there is an integer Q = Q(dim(X ), g(B), τ(π, E), arel, a, T, β, b) such that Q(KZW + BW + MW) is Cartier and h0(ZW, OZW (Q(KZW + BW + MW))) > Q(R + m) + 1 Indeed, consider the birational map ZW ��� V defined by |k(KZW + BW + MW)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let µ : Z′ → ZW be a smooth model resolving the map and let H denote the basepoint free part of µ∗(k(KZW + BW + MW)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There are only finitely many possible values of dim(ZW) which satisfy dim(X ) ≥ dim(ZW) ≥ 2, and thus Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='9 gives a quadratic lower bound (that depends only on dim(X )) on the growth rate of sections of multiples of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular 48 there is a constant Q′ = Q′(dim(X ), g(B), τ(π, E), arel, T, β, b) such that h0(Z′, OZ′(Q′H)) > (Q′)k(R + m) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then we have h0(Z′, OZ′(Q′H)) ≤ h0(Z′, OZ′(Q′µ∗(k(KZW + BW + MW)))) = h0(ZW, OZW (Q′k(KZW + BW + MW))) finishing the proof of the claim with Q = Q′k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Using the comparison of spaces of sections above, we conclude that also h0(Y′, OY′(Q(KY′ + D′ + mF ′))) > Q(R + m) + 1 ≥ Q(KY′ + D′ + mF ′) · C′ + 1 By applying Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8 to the divisor Q(KY′ + D′ + mF ′) we obtain an upper bound Γ(dim(X ), g(B), τ(π, E), arel, a, T, β, b) = Q(R + m) + 1 on the number of general points that can be contained in deformations of the sections C′ on Y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' But this also implies an upper bound Γ on the number of general points that can be contained in deformations of the sections C on Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ Suppose that Y carries a family of sections which have large L-degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Although we cannot necessarily use Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10 to show that (Yη, −f ∗L|Yη) is adjoint rigid, by combining with the results of Section 5 we can at least find a covering family of subvarieties of Y whose generic fibers are adjoint rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' coro:adjointrigidcodim Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a good fibration and let L be a generically relatively big and semiample Cartier divisor on X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume that Xη is geometrically uniruled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive rational number arel and set a = arela(Xη, L|Xη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a rational number β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive integer T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive integer b > a such that bL|Xη defines a base- point free linear series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Use b to construct an effective π-vertical Q-Cartier divisor E sat- isfying the conclusion of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 with respect to aL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There are constants ξ+ = ξ+(dim(X ), g(B), τ(π, E), arel, a, T, β, b) and T + = T +(dim(X ), g(B), τ(π, E), arel, a, T, β, b) with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that ψ : Y → B is a good fibration equipped with a B-morphism f : Y → X that is generically finite onto its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that N is an irreducible component of Sec(Y/B) parametrizing a dominant family of sections C on Y which satisfy f ∗L · C ≥ ξ+ and f ∗(KX/B + a(Xη, L|Xη)L) · C ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that dim(N) ≥ arel(−f ∗KX/B · C + (dim(X ) − 1)(1 − g(B))) − T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that a(Yη, −f ∗L|Yη) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let g : S → Y denote the finite part of the Stein factorization of the evaluation map for the normalization of the universal family over N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then there is a dominant rational B-map φ : S ��� T to a normal projective B-variety such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For a general section C† on S parametrized by N let W denote the main component of the closure of φ−1(φ(C†)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then: (1) We have a(Wη, g∗f ∗L|Wη) = a and the pair (Wη, g∗f ∗L|Wη) is adjoint rigid, 49 (2) W is swept out by the sections parametrized by a sublocus NW ⊂ N whose closure has codimension ≤ T + in N, and (3) there is a resolution of W such that the strict transform of a general section in NW to the resolution goes through ≥ 2g(B) + 1 general points and is HN-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Define d = dim(Y) and set γ = (dg(B) − g(B) + 1)2(d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We also define constants Tk and Γk for 2 ≤ k ≤ d as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We first set Td = 0 and Γd = sup{2g(B) + 3, Γ(dim(X ), g(B), τ(π, E), arel, a, T, β, b) + 1} where Γ is the constant defined in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then for 2 ≤ k < d we define via a downward induction Tk = k(Γk+1 + 2g(B) + γ) + Tk+1 and Γk = sup{2g(B) + 3, Γk+1, Γ(dim(X ), g(B), τ(π, E), arel, a, T + Tk, β, b) + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Finally, we set T + = T + supk=2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=',d Tk, Γ+ = supk=2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=',d Γk and ξ+ to be the maximum of the constant ξ(dim(X ), g(B), τ(π, E), arel, a, T +, β, b) as in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6 and of 1 a(dim(X )(Γ+ + 2g(B) + γ + 1) + arelβ + T + + (arel + 1)(dim(X ) − 1)(g(B) − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Recall that S denotes the finite part of the Stein factorization of the evaluation map for the normalization of the universal family over N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We let S′ denote a smooth birational model of S that flattens the family of sections on S as in Construction 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We denote the strict transform of a general section in our family on S′ by C′ and denote the family of deformations of C′ by N′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Appealing to Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4 we have −KS′/B · C′ + (dim(S′) − 1) ≥ dim(N′) = dim(N) ≥ −KY/B · C + (dim(Y) − 1)(1 − g(B)) ≥ af ∗L · C − arelβ − T + − (dim(X ) − 1) − (arel dim(X ) + dim(Y) − arel − 1)(g(B) − 1) where the last line follows from Equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3) of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Combining with the bound f ∗L · C ≥ ξ+, we conclude that eq:sprimebound eq:sprimebound (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6) − KS′/B · C′ ≥ dim(S′)(Γ+ + 2g(B) + γ − 1) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We next inductively define foliations Gd, Gd−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' on S′ by repeatedly applying Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 to Y using the constants Γd, Γd−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='. We will also denote by ψi the rational map on S′ induced by the foliation Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We will inductively verify the inequalities µmax [C] (Gi) ≥ Γi + 2g(B) + γ − 1 µmax [C] (TS′/Gi) < Γi + 2g(B) + γ − 1 which show the requirements necessary to inductively apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For the base case we set Gd = TS′/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By construction we have Γd > 2 ≥ µmax [C] (π∗TB) and Equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6) shows that µmax [C] (TS′/B) ≥ µ[C](TS′/B) ≥ Γ+ + 2g(B) + γ − 1 ≥ Γd + 2g(B) + γ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 50 Thus we have verified the two necessary inequalities in the base case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Now suppose inductively that we apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 to Y for the foliation Gi on S′ and the constant Γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There are two possible outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The first possibility is that if we set Pi to be the main component of ψ−1 i (ψi(C′)) for a general section C′ in our family then the deformations of C′ go through at least Γi general points of Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In this case we stop the inductive process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The second possibility is that we obtain a new foliation Gi−1 and a new rational map ψi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (c) shows that µmax [C] (TS′/Gi−1) < Γi + 2g(B) + γ − 1 ≤ Γi−1 + 2g(B) + γ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' On the other hand, letting r denote the rank of Gi−1 we have µmax [C] (Gi−1) ≥ µ[C](Gi−1) = c1(TS′/B) · C − c1(TS′/B/Gi−1) · C r = c1(TS′/B) · C − c1(TS′/Gi−1) · C + 2g(B) − 2 r ≥ dim(S′)(Γ+ + 2g(B) + γ − 1) − (dim(S′) − r)(Γi + 2g(B) + γ − 1) r ≥ Γ+ + 2g(B) + γ − 1 ≥ Γi−1 + 2g(B) + γ − 1 where the third line is a consequence of Equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus we have verified the necessary inequalities for continuing the inductive process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since the dimension of ψ−1 i (ψi(C′)) is always at least 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' this process stops after at most d − 2 steps with either a foliation Gk such that if we set Pk to be the main component of ψ−1 k (ψk(C′)) for a general section C′ in our family then the deformations of C′ go through at least Γk general points of Pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' or a rank 1 foliation Gk such that if we set Pk to be the main component of ψ−1 k (ψk(C′)) for a general section C′ in our family then the deformations of C′ go through at least Γk+1 general points of Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then the foliation Gk induces a rational map S′ ��� T , and hence also a rational map S ��� T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We prove that in either case this map has the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that the subvarieties Pk of S′ are birational to the subvarieties W in the statement of the theorem, and it suffices to prove that Pk has the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By applying Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (b) inductively, we see that the codimension of the space of deformations of C′ in Pk inside of N is at most d � j=k (j − 1)(Γj + 2g(B) + γ) verifying (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that by construction the deformations of C′ in Pk go through either Γk or Γk+1 general points of Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Both quantities are at least 2g(B) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This property is preserved by passing to the strict transform, and curves through this many general points must be HN-free by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7, proving (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Using the lower bound ξ ≤ ξ+, we can apply Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6 to Pk equipped with the family of deformations of C′ to see that a(Pk,η, f ∗L|Pk,η) ≥ a 51 But since the deformations of Pk,η form a dominant family of subvarieties on Yη the equality must be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' To prove (1), it only remains to verify the adjoint rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If Gk has rank 1, Pk is a P1-fibration over B and thus is automatically adjoint rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If Gk has rank > 1, then deformations of C′ on a resolution �Pk of Pk go through at least Γk general points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We then apply Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10 on �Pk to determine adjoint rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This proves (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Boundedness statements sect:boundedness We now turn to proving boundedness statements for the set of morphisms f : Y → X such that Y carries a family of sections which is “large” on X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 we prove several technical statements which combine [Bir22] with our work on twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2 we state and prove our main boundedness result, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' sect:mainboundedresults 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Boundedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Our first statement appeals to the recent results of [Bir22] to prove birational boundedness when the generic fiber is adjoint rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' theo:adjointrigidbounded Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a good fibration and let L be a generically relatively big and semiample Cartier divisor on X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume that Xη is geometrically uniruled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive ra- tional number arel and set a = arela(Xη, L|Xη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a rational number β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive integer T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive integer b > a such that bL|Xη defines a basepoint free linear series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Use b to construct an effective π-vertical Q-Cartier divisor E satisfying the conclusion of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 with respect to aL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is some constant ξ = ξ(dim(X ), g(B), τ(π, E), arel, a, T, β, b) with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that ψ : Y → B is a good fibration equipped with a B-morphism f : Y → X that is generically finite onto its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that a(Yη, f ∗L|Yη) = a and that (Yη, f ∗L|Yη) is adjoint rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that N is an irreducible component of Sec(Y/B) parametrizing a dominant family of HN-free sections C on Y which satisfy f ∗L · C ≥ ξ and f ∗(KX/B + a(Xη, L|Xη)L) · C ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Finally, suppose that (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1) dim(N) ≥ arel(−f ∗KX/B · C + (dim(X ) − 1)(1 − g(B))) − T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then: (1) The set of such projective varieties Y is birationally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) Suppose that L is big and semiample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Choose a positive integer b > a such that |bL| is basepoint free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then there is a constant ℸ = ℸ(dim(X ), g(B), arel, a, T, β, b) such that vol(f ∗L) ≤ ℸ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5 shows that (KY + af ∗L) · C ≤ arelβ + T + arel(dim(X ) − 1)(g(B) − 1) + (dim(X ) − 1) + 2g(B) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 shows that f ∗(aL + E) is Q-linearly equivalent to an effective Q-Cartier divisor D such that (Yη, D|Yη) is a terminal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let φ : Y′ → Y be a log resolution of this pair and let D′ denote the strict transform of the π-horizontal components of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since D′ is irreducible we see that (Y′, D′) is a terminal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 52 Since we are assuming the sections are HN-free, the strict transform C′ of a general deformation of C avoids any φ-exceptional divisor and thus satisfies (KY′ + D′) · C′ ≤ (KY′ + φ∗D) · C′ = φ∗(KY + D) · C′ ≤ arelβ + T + τ(π, E) + arel(dim(X ) − 1)(g(B) − 1) + (dim(X ) − 1) + 2g(B) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Furthermore C′ is HN-free on Y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Run the relative MMP for KY′ + D′ over B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Due to our adjoint rigidity assumption on the generic fiber, the result will be a birational model ρ : Y′ ��� �Y where K �Y + ρ∗D′ is relatively Q-linearly equivalent to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We denote by �ψ the structural map �ψ : �Y → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Write K �Y + ρ∗D′ ∼Q �ψ∗P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since the map ρ is a (KY′ + D′)-negative birational contraction, �ψ∗P · ρ∗C′ = ρ∗(K �Y + ρ∗D′) · C′ ≤ (KY′ + D′) · C′ ≤ arelβ + T + τ(π, E) + arel(dim(X ) − 1)(g(B) − 1) + (dim(X ) − 1) + 2g(B) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then [Bir22, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3] applies with Z = B and A = ⌈arelβ +T +τ(π, E)+arel(dim(X )− 1)(g(B)−1)+(dim(X )−1)+2g(B)−1⌉p for a point p ∈ B, showing that the set of minimal models ( �Y, ρ∗D′) is log bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Now suppose that L is big and semiample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that the effective divisor E = 0 satisfies the conclusion of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 with respect to aL, and we make this choice for E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Recall that the divisor D is constructed by applying Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' However, in our setting we can choose D ∼Q aL where Supp(D) is smooth and irreducible and has coefficient a b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular we can take Y′ = Y and D′ = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Repeating the argument above, we run a relative MMP ρ : Y ��� �Y and see that the resulting pairs ( �Y, ρ∗D) are log bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By construction ρ∗D is irreducible with coefficient a/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This implies that there is some constant ℸ such that vol(f ∗L) = vol(D) adim Y ≤ vol(ρ∗D) adim Y ≤ ℸ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In the setting of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (1) the variety Y can be replaced by a higher birational model and N can be replaced by the strict transform family of curves without affecting the hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus birational boundedness is the best one can hope for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='17 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='18 construct certain families of varieties over K(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We next modify these constructions to apply to integral models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' cons:allsubvarieties Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a good fibration and let L be a big and semiample Cartier divisor on X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume that Xη is geometrically uniruled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Set a = a(Xη, L|Xη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By applying Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='17 to Xη we obtain a proper closed subset Vη ⊂ Xη and a finite collection of families pi,η : Ui,η → Wi,η whose smooth fibers are birational to closed subvarieties of Xη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let pi : Ui → Wi denote any smooth integral model such that the structural morphism Ui,η → Xη extends to Ui and let V denote the closure of Vη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The subvarieties parametrized by pi,η correspond to the K(B)-points of Wi,η, or equivalently, to sections of Wi over B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Wi denote Sec(Wi/B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We let W = ⊔iWi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We first shrink W 53 so that the generic point of every section parametrized by W is contained in the open locus over which ⊔ipi is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We enlarge V by adding the images in X of the loci where the maps pi fail to be smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Consider the universal family Wi × B → Wi with the evaluation map Wi ×B → Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By taking a base change of pi : Ui → Wi over this morphism, we obtain a morphism which we denote by Zi → Wi × B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We let Z = ⊔iZi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that W is a countable union of quasiprojective schemes and Z → W × B is a finite type morphism such that for every closed point w ∈ W the B-scheme Zw → {w} × B has the property that Zw,η is isomorphic to a fiber of pi,η over a K(B)-point of Wi,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By repeatedly stratifying W into locally closed subsets and taking resolutions of components of Z, we may also ensure that the fibers of Z → W are smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We denote the evaluation map by ι : Z → X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' cons:allfamilies Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let ⊔GH(G, B) be the Hurwitz stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix an ´etale covering ⊔GHG → ⊔GH(G, B) from a scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a good fibration and let L be a big and semiample Cartier divisor on X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume that Xη is geometrically uniruled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Set a = a(Xη, L|Xη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Z → W × B be the morphism constructed in Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='18 there is a finite set of smooth projective K(B)-varieties Yi,j,η equipped with dominant generically finite morphisms hi,j,η : Yi,j,η → Ui,η and a closed set Rη ⊂ Xη that has the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that g : Yη → Xη is a generically finite morphism from a geometrically integral smooth projective variety such that g(Yη) is not contained in Rη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose furthermore that a(Yη, g∗L|Yη) = a and that (Yη, g∗L|Yη) is adjoint rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since Yη is geometrically rationally connected by [LTT18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5], [GHS03, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1] and [HT06, Theorem 12] show that Yη carries a dense set of rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='18 shows that there are indices i, j such that the map g factors rationally through a twist hσ i,j,η and Yη maps birationally to a fiber of a morphism Yσ i,j,η → T σ i,j,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Vη be the union of Rη with the generic fiber of the closed set from Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then we enlarge Vη by adding si(Bi,j,η) where Bi,j,η is the union of the irreducible components of the branch locus of hi,j,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We further enlarge Vη by adding the Zariski closure of the union of the images of the fibers of ri,j,η which fail to be smooth, fail to have the same a-invariant as Yi,j, or fail to be adjoint rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If Yi,j,η is a component such that some twist of Yi,j,η admits a K(B)-rational point mapping to Xη \\ Vη, then we replace Yi,j,η by this twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If Yi,j,η is a component such that no twist of Yi,j,η admits a K(B)-rational point mapping to Xη \\ Vη, then we remove Yi,j,η from our set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Set Di,j = ⊔GC1(GHG, K(Yi,j,η/Ui,η)HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By construction Di,j is a countable union of finite type schemes over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' As described in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2, there is a morphism hi,j,η : Yi,j,η → Di,j × Ui,η which parametrizes twists of hi,j,η : Yi,j,η → Ui,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' After perhaps replacing Di,j by a strat- ification into locally closed subsets, we can construct integral models in families to obtain a map Yi,j → Di,j × Ui → Di,j × X whose composition we denote by fi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' After perhaps again replacing Di,j by a stratification by locally closed subsets and taking resolutions of irreducible components of Yi,j, we may ensure that for every closed point d ∈ Di,j the fiber Yi,j,d is a good fibration equipped with a B-morphism fi,j,d : Yi,j,d → X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By construction every twist of hi,j,η has an integral model hσ i,j : Yσ i,j → Ui that is a member of our fam- ily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' After again replacing Di,j by a stratification into locally closed subsets, we may ensure 54 that the Stein factorization of the composition Yi,j → Di,j × Ui → Di,j × Wi induces for every fiber over a closed point in Di,j the Stein factorization of Yσ i,j → Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Denote the Stein factorization of Yi,j → Di,j × Wi by ri,j : Yi,j → Ti,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then ri,j : Yi,j → Ti,j and ti,j : Ti,j → Wi define a family of Stein factorizations rσ i,j : Yσ i,j → T σ i,j, tσ i,j : T σ i,j → Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Due to the functoriality in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2, we see that Ti,j,η → Di,j × Wi,η parametrizes the family of twists of Ti,j,η → Wi,η which are induced by twists of Yi,j,η → Ui,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let V be the closure of Vη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We further enlarge V by adding the Zariski closure of the union of the images of the fibers of ri,j which fail to be smooth, fail to have the same a-invariant as Yi,j, or fail to be adjoint rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We let Bi,j be the closure of Bi,j,η and define B := ∪i,jBi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We also define B′ i,j ⊂ Wi as the union of components of the branch locus of ti,j which dominate B and the closures of the images of loci where fibers of ri,j fail to be smooth, fail to have the same a-invariant as Yi,j, or fail to be adjoint rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We define B′ := ∪i,jB′ i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Recall that we assume that Yi,j,η admits a K(B)-rational point y mapping to Xη \\ Vη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let �hi,j,η : �Yi,j,η → Ui,η be a geometric Galois closure of hi,j,η : Yi,j,η → Ui,η such that Bir( �Yi,j,η/Ui,η) = Aut( �Yi,j,η/Ui,η) and �Yi,j,η admits a K(B)-rational point �y mapping to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Yi,j �hi,j −−→ Pi,j ℓi,j −−→ Ui be the cover corresponding to the normalizer of Aut( �Yi,j,η/Yi,j,η) in Aut( �Yi,j,η/Ui,η) such that Pi,j is normal and ℓi,j is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that every twist Yσ i,j → Ui factors through ℓi,j : Pi,j → Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By taking the Stein factorization, we have a commutative diagram Yi,j �hi,j � ri,j � Pi,j bi,j � ℓi,j � Ui pi � Ti,j ci,j � Si,j ai,j � Wi where Si,j is projective and normal, bi,j has connected fibers, and ai,j is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We also let Mi,j = ⊔GC1(GHG, K(Ti,j,η/Wi,η)HG) denote the parameter space for all twists of Ti,j,η → Wi,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We denote the universal family over Mi,j by T′ i,j → Mi,j × Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then by the functoriality established in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2 we have a natural isomorphism Ti,j → T′ i,j ×Mi,j Di,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We set Y = ⊔i,jYi,j, T = ⊔i,jTi,j, T′ = ⊔i,jT′ i,j, D = ⊔i,jDi,j, and M = ⊔i,jMi,j with morphisms Y → D × X , T → D and T′ → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For each Yσ i,j → T σ i,j, the varieties described by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (b) are parametrized by the K(B)-points of T σ i,j,η, or equivalently, by the closed points of Sec(T σ i,j/B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We consider the relative space of sections S = SecD(T/B) = SecM(T′/B) ×M D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We first shrink S so that the generic point of every section parametrized by S is contained in the locus in T over which Y → T is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then by taking a base change of Y → T over the evaluation map S × B → T, we obtain a morphism F′ → S × B whose fibers are closed subvarieties of the various Yσ i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that we have a morphism S → Sec(⊔iWi/B), and we replace S by the the fiber product S×Sec(⊔iWi/B) W so that we have a compatible morphism S → W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By repeatedly stratifying S into locally closed subsets, throwing away components whose intersection with a fiber Ys does not dominate B, taking resolutions of irreducible 55 components of F′, and taking Stein factorizations, we obtain a commuting diagram F � � Z � S × B � W × B where for every closed point s ∈ S the fiber Fs is a normal projective B-variety such that Fs → B has connected fibers and Fs → Zw is a finite morphism where w denotes the image of s in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' After taking a stratification of S, we may assume that F → S is a flat family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Altogether, we have constructed a family F → S × B whose base is a countable union of finite type schemes and a morphism g : F → S × X such that (1) for every closed point s ∈ S the fiber Fs is a normal projective B-variety such that Fs → B has connected fibers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) for every closed point s ∈ S the map gs : Fs → X is a B-morphism that is generically finite onto its image and the corresponding morphism Fs → Zw is a finite morphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (3) for every closed point s ∈ S we have a(Fs,η, g∗ sL|Fs,η) = a and (Fs,η, g∗ sL|Fs,η) is adjoint rigid, (4) if Y is a good fibration over B and f : Y → X is a generically finite B-morphism such that a(Yη, f ∗L|Yη) = a and (Yη, f ∗L|Yη) is adjoint rigid, either the map f is birationally equivalent to gs for some closed point s in our family or f(Yη) ⊂ Vη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We also have a family Y → D × X parametrizing integral models hσ i,j : Yσ i,j → Ui of twists hσ i,j,η : Yσ i,j,η → Ui,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that all such twists hσ i,j : Yσ i,j → Ui factor through �hi,j : Yσ i,j → Pi,j and that �hi,j : Yσ i,j → Pi,j is Galois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We will also need two additional lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' lemm:deforminghnfreetonearbyfibers Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that Y → S × B is a family of good fibrations over B with S irre- ducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that for some closed point s ∈ S we have an HN-free section C of Ys/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then the deformations of C form a dominant family on Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let M denote the space of deformations of C in Y and for a closed point s′ ∈ S let Ms′ denote the sublocus parametrizing spaces of sections of Ys′/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since H1(C, TYs/B|C) = 0, [Kol96, Theorem I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2)] shows that dim(M) ≥ dim(Ms) + dim(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since C is an HN-free section in Ys, by replacing C by a general deformation we may ensure that it avoids any codimension 2 locus of Ys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We conclude that TY/S×B|C is locally free, and thus the restriction of this sheaf to a general deformation of C is also locally free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' As the universal family over Sec(Y/B) is smooth, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 shows that the minimal slope of a quotient of TY/S×B|C′ is a lower semicontinuous function as we vary C′ in Sec(Y/B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus there is an open subset of M parametrizing sections which are HN-free in their fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For a general section C′ parametrized by M denote by s′ the closed point of S parametrizing the good fibration Ys′ → B containing C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' As explained above C′ is HN-free in Ys′, and in particular for such points s′ we have dim(Ms′) = dim(Ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus the dimension computation shows that sections parametrized by M must dominate Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ Since the next lemma is well-known we will omit the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 56 lemma:uniquenessoftwists Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let k be a field of characteristic 0 and let f : Y → X be a dominant generically finite morphism between normal projective varieties defined over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume that Bir(Y /X) = Aut(Y /X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let X◦ ⊂ X be a Zariski open subset such that f|f−1(X◦) : f −1(X◦) → X◦ is ´etale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let f σ : Y σ → X be a twist of f over X and suppose that there are k-rational points p ∈ Y (k) and pσ ∈ Y σ(k) which define the same geometric point on f −1(X◦)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then f and f σ are isomorphic as X-schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We are now ready to prove our main boundedness theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For these results we will be in the situation arel = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' theo:positivebounded Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a good fibration and let L be a big and semiample Cartier divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume that Xη is geometrically uniruled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Set a = a(Xη, L|Xη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a rational number β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive integer T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive integer b > a such that bL defines a basepoint free linear series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is: a constant ξ† = ξ†(dim(X ), g(B), a, T, β, b), a closed subset V ⊂ X , and a bounded family of smooth projective varieties q : �F → �S equipped with �S-morphisms p : �F → �S × B and g : �F → �S × X which have the following properties: (1) For every closed point s ∈ �S, �Fs → B is a good fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) For every closed point s ∈ �S the morphism gs : �Fs → X is a B-morphism that is generically finite onto its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (3) For every irreducible component �Fi of �F the composition of g|�Fi : �Fi → �S × X with the projection �S × X → X is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (4) For every closed point s ∈ �S we have a(�Fs,η, g∗ sL|�Fs,η) = a(Xη, L|Xη) and (�Fs,η, g∗ sL|�Fs,η) is adjoint rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (5) Suppose that ψ : Y → B is a good fibration equipped with a B-morphism f : Y → X that is generically finite onto its image and satisfies a(Yη, f ∗L|Yη) ≥ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that N is an irreducible component of Sec(Y/B) parametrizing a dominant family of sections C on Y which satisfy f ∗L · C ≥ ξ and f ∗(KX/B + aL) · C ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let M ⊂ Sec(X /B) be the irreducible component containing the pushforward of the sections parametrized by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Finally, suppose that dim(N) ≥ dim(M) − T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For a general section C parametrized by N, either: C is contained in V, or there is an irreducible component �Fi of �F and an irreducible component N′ of Sec(�Fi/B) parametrizing a dominant family of sections on �Fi such that f(C) is the image of a section C′ parametrized by N′ and if �Fi,s denotes the fiber containing C′ then the strict transform of C′ in a resolution of �Fi,s is HN-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We will divide the proof into five steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Step 1 is devoted to some preliminary work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In Step 2, we construct a bounded family of varieties P → Q such that every Y as in Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (5) is a birational to a twist of a fiber over a closed point of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In Step 3, we bound the 57 invariants from Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='13 for the varieties in our family P → Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then in Step 4 we use this corollary to construct a bounded family of twists �F → �S which carry sections of low L-degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Finally, in Step 5 we verify that �F → �S has the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Step 1: Let m be the maximum of the degrees of the morphisms Yi,j → Ui from Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4 and set d = (m+1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='. Let ⊔GH(G, B) be the Hurwitz stack and fix an ´etale covering ⊔G,|G|≤dHG → ⊔G,|G|≤dH(G, B) by a scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We will work over this base for the entire proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that the divisor E = 0 satisfies the condition of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Define ξ = ξ(dim(X ), g(B), 0, 1, a, T, β, b) as in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We then choose ξ+ = ξ+(dim(X ), g(B), 0, 1, a, T, β, b) and T + = T +(dim(X ), g(B), 0, 1, a, T, β, b) as in Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Define ℸ = ℸ(dim(X ), g(B), 1, a, T, β, b) as in Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Finally we define ξ† = sup {ξ, ξ+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since L is big and semiample, there is a closed subvariety V1 ⊂ X such that the family of subvarieties of X that are not contained in V1 and have L-degree ≤ ℸ is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By [LST22, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2)] there is a closed sublocus V2,η ⊂ Xη that contains all subvarieties with larger generic Fujita invariant and we let V2 denote its closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let V3 be the exceptional closed set from Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We start by setting V to be the union of V1, V2, and V3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' we will later enlarge it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Zi → Wi×B be the families in Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then there is a finite-type subscheme Ri ⊂ Wi parametrizing those varieties whose images in X have L-degree ≤ ℸ and are not contained in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We denote by Zi,Ri → Ri the universal family over Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Set R = ⊔iRi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Y → T → D, T′ → M, F → S and g : F → X × B be defined as in Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For any closed point s ∈ S the map gs : Fs → X has image that is birationally equivalent to a fiber Zs of Z → W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By [LST22, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7] we have a(Fs,η, g∗ sL|Fs,η) ≤ a(Zs,η, ι∗ sL|Zs,η) and if equality is achieved then [LST22, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='9] shows that (Zs,η, ι∗ sL|Zs,η) is adjoint rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If we shrink S to remove all maps gs whose image lies in V, then we will always have equality of Fujita invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We let S′ denote the sublocus of S consisting of maps gs whose image is a member of our fixed bounded family ZR → R and denote by F′ → S′ the corresponding family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Step 2: We next claim that there is a morphism Q → S′ ⊂ S such that Q is of finite type over C and for every map gs parametrized by S′ the map gs,η is a twist of the generic fiber of a map parametrized by Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Indeed, recall from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='18 that we have a finite number of smooth projective varieties Yi,j,η equipped with morphisms Yi,j,η hi,j,η � ri,j,η � Ui,η pi,η � Ti,j,η ti,j,η � Wi,η where ti,j,η is Galois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' It follows from our construction that Ri is contained in finitely many irreducible compo- nents of Sec(Wi/B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Sec(Wi/B, B′) denote the space of sections not contained in B′ and 58 define Sec(Si,j/B, a−1 i,j (B′)) analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then ai,j,∗ : Sec(Si,j/B, a−1 i,j (B′)) → Sec(Wi/B, B′) is of finite type, so the fiber product �Ri,j := Sec(Si,j/B, a−1 i,j (B′)) ×Sec(Wi/B,B′) Ri is of finite type over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Moreover for any C ∈ �Ri,j, the fiber b−1 i,j (Cη) is geometrically ra- tionally connected so �Ri,j is in the image of Sec(Pi,j/B, ℓ−1 i,j (B)) → Sec(Si,j/B, a−1 i,j (B′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus there is a finite disjoint union of locally closed subschemes of finite type �Ri,j ⊂ Sec(Pi,j/B, ℓ−1 i,j (B)) with a surjective morphism �Ri,j → �Ri,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We denote the base change of Zi,Ri → Ri over �Ri,j → �Ri,j → Ri by Z�Ri,j → �Ri,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Recall that we are working over ⊔G,|G|≤dHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We claim that every twist of Yi,j,η/Ui,η splits over an extension K(B′)/K(B) of degree ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Indeed, if we denote Aut(Yi,j,η/Ui,η) by G, then Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4 shows that one may use a Galois base change of degree ≤ |G| · #Aut(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular d = (m + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' gives an upper bound on this degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since each rational point of Wi,η not contained in B′ will lift to a unique twist of Ti,j,η and the number of 1-cycles representing the same Galois cohomology class is at most m, the pushforward morphism ti,j,∗ : SecMi,j(T′ i,j/B, t−1 i,j (B′)) → Sec(Wi/B, B′) × (⊔G,|G|≤dHG) is a quasi-finite morphism onto its image of degree at most m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Now for each C ∈ Sec(Pi,j/B, ℓ−1 i,j (B)), �h−1 i,j (C) decomposes into a union of curves which are Galois conjugate to each other over B where �hi,j : Yi,j → Pi,j is the morphism constructed in Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then after taking a stratification of �Ri,j by locally closed subsets and replacing �Ri,j by an ´etale cover, the universal property of the Hurwitz stack yields a morphism ψi,j : �Ri,j → ⊔G,|G|≤dH(G, B) that sends a section C of Pi,j/B to the cover C′ → B obtained by normalizing an irreducible component of �h−1 i,j (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We denote by Ri,j the fiber product �Ri,j ×⊔G,|G|≤dH(G,B) (⊔G,|G|≤dHG) which is of finite type over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus using the morphism Ri,j → Sec(Wi/B, Bi,j)×(⊔G,|G|≤dHG) we define the scheme Q′ i,j = SecMi,j(T′ i,j/B, t−1 i,j (Bi,j)) ×Sec(Wi/B,Bi,j)×(⊔G,|G|≤dHG) Ri,j which is a finite type scheme over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that Q′ i,j parametrizes the sections of T σ i,j/B which map to Ri such that the twist T σ i,j is trivialized by a base change C′ → B coming from �h−1 i,j (C) as constructed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Q′ = ⊔i,jQ′ i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then we have a morphism Q′ → SecM(T′/B) and S′ → SecD(T/B) → SecM(T′/B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We set Q = Q′ ×SecM(T′/B) S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since S′ → SecM(T′/B) is of finite type over each HG, Q is a scheme of finite type over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then we denote the base change of Y → D over Q → D by �Y → Q and we denote the base change of F → S over Q → S′ ֒→ S by P → Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We still must verify that P → Q satisfies the claimed property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let s ∈ S′ and consider the corresponding gs : F′ s → Zr → X with r ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='18, gs,η : F′ s,η → Xη is birationally equivalent to the map to Xη from a fiber of a twist Yσ i,j,η → T σ i,j,η for some 59 i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then since Yσ i,j factors through Pi,j, by the construction we find a point �r ∈ Ri,j mapping to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This point �r specifies a point (B′/B) ∈ ⊔G,|G|≤dHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' On the other hand one can find a twist Yτ i,j,η → T σ i,j,η such that the preimage (�hτ i,j,η)−1(Cη) of the section C ∈ Sec(Pi,j/B) corresponding to �r consists entirely of K(B)-rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Such a twist will be trivialized by the base change B′ → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This means that r is in the image of the map Q = Q′ ×SecM(T′/B) S′ → Sec(⊔iWi/B, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus there is a point q ∈ Q mapping to r such that F′ s,η → Xη is a twist of Pq,η → Xη for the fiber Pq over q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This finishes the verification of the desired property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Step 3: Note that the degree of hσ q,η : Pσ q,η → Zq,η is bounded by the maximum of the degrees of hi,j : Yi,j → Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular the size of Aut(Pσ q,η/Zq,η) is uniformly bounded by the integer m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' It follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4 that for every closed point q ∈ Q the map hσ q,η : Pσ q,η → Zq,η becomes isomorphic to hq,η : Pq,η → Zq,η after a Galois base change �B → B of degree ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Next we define an integer t by using the family P → Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since normality is a constructible property in proper families ([Gro66, Th´eor`eme 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4]), by Noetherian induction there is a positive integer t1 that bounds the number of non-normal fibers of Pq → B as we vary over all q ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since the relative automorphism scheme AutB(Pq/Zq) is quasifinite over Q × B and since flatness is a constructible property, as we vary over all closed points q ∈ Q there is a positive integer t2 that bounds the number of places in B where the restriction of AutB(Pq/Zq) to {q} × B is not flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We set t = t1 + t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Step 4: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 and Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='13 show that as we vary the closed point q ∈ Q the set of twists of hq : Pq → Zq which are trivialized by a base change B′ → B of degree at most d and with at most t + d(T + T +) branch points is parametrized by a bounded family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We denote by �F → �S the bounded subfamily of F′ → S′ parametrizing maps gs : Fs → Zs satisfying these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' After taking smooth resolutions and stratifying the base, we obtain �F → �S such that each fiber is a good fibration over B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We then shrink �S by removing all irreducible components Sj such that the corresponding family �Fj fails to dominate X and we enlarge V by taking the union with the closures of the images of these families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Step 5: We are now ready to verify the desired properties of �F → �S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Properties (1)-(4) follow from the construction, and we only need to check (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose f : Y → X is as in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Applying Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6 with arel = 1 we see that a(Yη, f ∗L|Yη) ≥ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If we have a strict inequality then f(Y) ⊂ V and so the sections on Y are accounted for by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' From now on we assume that f(Y) ̸⊂ V which implies that a(Yη, f ∗L|Yη) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We apply Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='11 to construct subvarieties on the Stein factorization of the eval- uation map over Y and then take images in Y to obtain a dominant family of subvarieties F ⊂ Y satisfying: the codimension in N of the space of deformations of C in F is at most T +, the strict transform of C to a resolution of F is HN-free, (Fη, f ∗L|Fη) is adjoint rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We will show that the conclusion of (5) holds for the sections on the general subvariety F in our family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Consider a general subvariety F in our family and set Z = f(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since it is not possible for Zη to have larger a-invariant (as it is not contained in Vη), we have a(Zη, L|Zη) = a and 60 thus by [LST22, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='9] (Zη, L|Zη) is adjoint rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) shows that Z is birationally equivalent to a smooth Z′ that is parametrized by the bounded family ZR → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular the map µ : Fη → Zη is birationally equivalent to a twist of hq : Pq,η → Zq,η for some closed point q ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Choose a morphism µ′ : F ′ → Z′ birationally equivalent to µ where F ′ is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let NF denote the moduli space of deformations of the strict transforms C′ of C in F ′ and let MZ denote the moduli space of deformations of the image in Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then dim(MZ) − dim(NF) ≤ dim(M) − (dim(N) − T +) ≤ T + T + so that NF has codimension at most T + + T in MZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then we have −KF′/B · C′ + (dim(F ′) − 1)(1 − g(B)) = dim(NF) ≥ dim(MZ) − T + − T ≥ −KZ′/B · µ′ ∗C′ + (dim(Z′) − 1)(1 − g(B)) − T + − T which rearranges to (KF′/B − µ∗KZ′/B) · C′ ≤ T + + T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This intersection bound and Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='13 imply that µ′ : F ′ → Z′ is birationally equivalent to a twist of hq that is trivialized by a base change B′ → B that has degree at most d and has at most t + d(T + T +) branch points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus µ′ is birationally equivalent to one of the maps hs : �Fs → Zs parametrized by our bounded family �F → �S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Consider the strict transform of our family of sections in the fiber �Fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since these sections go through at least 2g(B)+1 general points of �Fs, they are HN-free in this fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5 shows that the sections deform to give a dominant family on the entire irreducible component �Fi containing �Fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since by construction every irreducible component of �F dominates X , we deduce that the family of sections gives a dominant family on X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Furthermore, we see that the general section parametrized by NF is in the image of the map Sec(�Fi/B) → Sec(X /B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus the same property is true for N, proving (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ Our next boundedness statement is closer in spirit to the results of [LST22]: instead of using a bounded family �F → �S such that the fibers �Fs,η are adjoint rigid, one can instead use a bounded family �Y → �S such that the fibers �Ys,η are twists of the finite set of universal families constructed in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' theo:bounded_bigandsemiample Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a good fibration and let L be a big and semiample Cartier divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Assume that Xη is geometrically uniruled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Set a = a(Xη, L|Xη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a constant β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive integer b > a such that bL defines a basepoint free linear series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is a constant ξ† = ξ†(dim(X ), g(B), a, β, b), a proper closed subset V ⊂ X , and a bounded family �Y → �S×B of good fibrations equipped with a �S×B-morphism �f : �Y → �S×X such that: (1) for every closed point s ∈ �S the map �fs is dominant and generically finite but not birational;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) for every closed point s ∈ �S we have a(�Ys,η, − �f ∗ s KX/B|�Ys,η) = a(Xη, −KX/B|Xη);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (3) as we vary over all closed points s ∈ �S the set of birational equivalence classes of the maps { �fs,η : �Ys,η → Xη} obtained by base changing to Spec(K(B)) is finite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 61 (4) if M ⊂ Sec(X /B) is an irreducible component that generically parametrizes non-HN- free sections C with L · C ≥ ξ† and (KX/B + a(Xη, L|Xη)L) · C ≤ β then a general section C parametrized by M satisfies either C ⊂ V or C ∈ �f∗(Sec(�Ys/B)) for some closed point s ∈ �S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We begin by making exactly the same constructions and definitions as in the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' we continue from the end of this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Additionally we set T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let �f : �Y → �S × X be the family of twists of hi,j : Yi,j → Ui which becomes isomorphic to a member of �Y → Q by a finite base change B′ → B of degree ≤ d and with at most t + dT + branch points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 �S has finite type over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Properties (1), (2), (3) follow from the construction and we only need to verify (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose M ⊂ Sec(X /B) is an irreducible component that generically parametrizes non-relatively free sections C with −KX/B · C ≥ ξ and (KX/B + a(Xη, L|Xη)L) · C ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We may assume that M generically parametrizes sections which are not contained in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then M parametrizes a dominant family of sections due to Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Y → X be the finite part of the Stein factorization for the evaluation map for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Applying Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='11, we find a dominant family of subvarieties F ⊂ Y satisfying: the codimension in N of the space of deformations of C in F is at most T +, the strict transform of C is HN-free in a resolution of F, (Fη, f ∗L|Fη) satisfies a(Xη, L) = a(Fη, f ∗L|Fη) and is adjoint rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Using the universal property described in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='18, there exists a twist Yσ i,j → T σ i,j over Ui such that F is birational to the main component F ′ C of the preimage of a section C under the map Yσ i,j → T σ i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then note that Yσ i,j → Ui factors through Pi,j → Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We claim that there is some closed point q ∈ Q such that there is an Xη-isomorphism between Yσ i,j,η and �Yq,η which maps FC,η to the image P′ q,η of Pq,η under the map Pq,η → �Yq,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Indeed, by the defining property of P → Q we know that FC,η is a twist of Pq′,η for some q′ ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The point q′ specifies a twist Yτ i,j → T τ i,j and a point q′′ on Sec(T τ i,j/B, t−1 i,j (B′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We let p denote the rational point on Sec(Wi/B, B′) obtained by taking the image of q′′ under Sec(T τ i,j/B, t−1 i,j (B)) → Sec(Wi/B, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Cp denote the section of Wi → B corresponding to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since tτ i,j : T τ i,j → Wi is Galois, every geometric point in (tτ i,j)−1(Cp,η) is a K(B)-rational point on T τ i,j,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that the geometric fiber corresponding to FC,η will lie over one of these points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' we replace q′′ by this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Moreover by construction the image Zr of FC has L-degree ≤ ℸ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular this point will lift to q′′′ ∈ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus we can define a point q = (q′′′, s′) ∈ Q′ ×SecM(T′/B) S′ = Q where s′ ∈ S′ is a point corresponding to (q′′, d′) with d′ is the image of q′ via Q → D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This point q maps to p and Pq,η is birational to the same geometric fiber of �Yq,η as FC,η in Yσ i,j,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By the construction of the families �F → �S, FC,η and P′ q,η are trivialized by a base change B′ → B of degree ≤ d with the number of branch points ≤ t + dT +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6 Yσ i,j,η and �Yq,η are trivialized by the same base change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus our assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ sect:boundedconsequences 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' General statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The following proposition will allow us to remove the global positivity assumption in Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 62 prop:birmodelposL Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a good fibration and let L be a generically relatively ample Q-Cartier divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is a birational model φ : X + → X that restricts to an isomorphism of generic fibers over B such that X + is smooth and there is a π ◦ φ-vertical effective Q-Cartier divisor G such that φ∗L + G is a big and semiample Q-Cartier divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Choose a positive integer p such that A = pL − KX is generically relatively ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Choose E as in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 applied to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus there is an effective Q-divisor D ∼Q A+E such that D|Xη has SNC support and has positive coefficients < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let ψ : X ′ → X denote a log resolution and let D′ denote the strict transform of the π-horizontal components of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that ψ is an isomorphism over Xη and so D′ and ψ∗D only differ by a π-vertical divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus we can choose some positive integer m such that D′ + mF −ψ∗D is Q-linearly equivalent to an effective Q-Cartier divisor, where F denotes a general fiber of X ′ → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By passing to a relative canonical model, we obtain a birational map ρ : X ′ ��� � X such that ρ∗(KX ′ + D′ + mF) is relatively ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that ρ is an isomorphism along X ′ η since (KX ′ + D′ + mF)|X ′η was already ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By increasing m, we can ensure that ρ∗(KX ′ + D′ + mF) is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let X + denote a birational model admitting morphisms to X and to � X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then the difference between the pullback of 1 pρ∗(KX ′ + D′ + mF) to X + and the pullback of L to X + is Q-linearly equivalent to a π-vertical Q-Cartier divisor G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since we may add any fiber of X + → B to the pullback of 1 pρ∗(KX ′ + D′ + mF) without affecting semiampleness, we may eliminate the negative part of G′ to obtain the desired effective π-vertical Q-Cartier divisor G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ Putting everything together, we obtain the following variant of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (One can easily develop an analogous variant of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8 using a similar argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=') theo:combinedbounded Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a good fibration and let L be a generically relatively ample Q-Cartier divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a constant β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (1) There is a proper closed subset R ⊊ X such that if M ⊂ Sec(X /B) is an irre- ducible component parametrizing a non-dominant family of sections with (KX/B + a(Xη, L|Xη)L) · C ≤ β then the sections parametrized by M are contained in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) There is a constant ξ, a proper closed subset V ⊂ X , and a bounded family of smooth projective B-varieties Y equipped with B-morphisms f : Y → X satisfying: (a) dim(Y) < dim(X ) and f is generically finite onto its image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (b) a(Yη, −f ∗L|Yη) = a(Xη, L|Xη) and the Iitaka dimension of KYη + f ∗L|Yη is 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (c) if M ⊂ Sec(X /B) is an irreducible component that generically parametrizes non- HN-free sections C with L · C ≥ ξ and (KX/B + a(Xη, L|Xη)L) · C ≤ β then for a general section C parametrized by M we have either (i) C ⊂ V, or (ii) for some f : Y → X in our family there is a HN-free section C′ of Y/B such that C = f(C′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let φ : X + → X be a birational morphism and G be a π-vertical effective Q-Cartier divisor as in Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Choose a positive integer k such that kφ∗(L + G) is Cartier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We define L+ = k(φ∗L + G) and β+ = β + τ(π ◦ φ, KX +/X + ka(X + η , L+|X + η )G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Choose a b > a(X + η , L+|X + η ) such that bL+|X + η defines a basepoint free linear series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We then apply the constructions of the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7 to (X +, L+) with our chosen constants and with T = 0 to obtain a constant ξ†, a closed subset V+ ⊂ X +, and a bounded 63 family of normal projective varieties q : �F → �S equipped with �S-morphisms p : �F → �S × B and g : �F → �S × X +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We define V to be the union of φ(V+) with the locus where φ−1 is not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We define ξ = 1 kξ†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose M ⊂ Sec(X /B) parametrizes a family of sections satisfying L · C ≥ ξ and (KX/B+a(Xη, L|Xη)L)·C ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If the locus swept out by the curves parametrized by M meets the locus where φ−1 is defined, by taking strict transforms we obtain a family of sections C+ on X +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' These sections satisfy L+·C+ ≥ ξ†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Furthermore since a(X + η , L+|X + η ) = 1 ka(Xη, L|Xη) we have (KX +/B + a(X + η , L+|X + η )L+) · C+ ≤ β+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' First we prove the statement (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We define R to be the union of V with the images of the (finitely many) non-dominant families of sections satisfying L · C < ξ and (KX/B + a(Xη, L|Xη)L) · C ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If we have a non-dominant family of sections C such that L · C ≥ ξ then it follows from Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (5) applied to (X +, L+) that the sections will be contained in V and thus in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Altogether we see that R has the desired property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Next we prove (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7 the bounded family �F → �S equipped with the composition �F → �S × X + → �S × X satisfies all the properties except possibly Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If M parametrizes a non-dominant family of sections, then as explained above the sections are contained in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' On the other hand, if M parametrizes a dominant family of non-HN-free sections, then the inclusion TX + → φ∗TX is still injective upon restriction to a general section C+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus the family of strict transforms C+ is a dominant family of non-HN-free curves on X +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7 shows that the general section parametrized by M will be the pushforward of an HN-free section on some fiber �Fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fano fibrations sect:fanofib In this section we apply previous results to Fano fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The Υ-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' defi: invariant_upsilon Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a Fano fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive rational number a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By [KMM92, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2] there is a positive integer b = b(dim(X ), a) such that | − bKXη| is very ample and b > a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Define Υa(π) to be the minimal value of τ(π, E) as we vary over all effective π-vertical Q-Cartier divisors E constructed as in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 with respect to our choice of b and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (Note that there is a divisor E achieving this infimum since if a = p q then each τ(π, E) lies in 1 bqZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=') We also define Υ(π) = Υ1(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The invariant Υ(π) measures the “failure” of π to be a trivial fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For example, if X = X × B for some Fano variety X then we have Υ(π) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Applying the results of Section 7 we obtain: theo:ainvariantandsections Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a Fano fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive rational number arel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive integer T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There is some constant ξ = ξ(dim(X ), g(B), Υarel(π), arel, T) with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that ψ : Y → B is a good fibration equipped with a B-morphism f : Y → X that is generically finite onto its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that N is an irreducible component of Sec(Y/B) 64 parametrizing a dominant family of sections C on Y which satisfy −f ∗KX/B ·C ≥ ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Finally, suppose that dim(N) ≥ arel(−KX/B · C + (dim(X ) − 1)(1 − g(B))) − T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then a(Yη, −f ∗KX/B|Yη) ≥ arel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Set L = −KX/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By [KMM92, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2] there is a positive integer b > arel depending only on dim(X ) and arel such that | − bKXη| is very ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We apply Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6 with β = 0, with a = arel, with our choice of b, and with an effective π-vertical Cartier divisor E as in Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 that achieves the bound τ(π, E) = Υarel(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The explicit bound (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5) for ξ is a max of two terms, which simplifies to ξ = 1 arelǫ((1 − ǫ)Υarel(π)+T + (dim(X ) − 1)(5g(B) + 3 + γ) eq:fanofibrationbound eq:fanofibrationbound (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1) + arel(dim(X ) − 1)(g(B) − 1) + 2g(B) − 2 + Ξ) + 1 where ǫ is a rational number chosen so that no smooth projective variety of dimension ≤ dim(X ) − 1 has a Fujita invariant in the range [1 − ǫ, 1) with respect to any big and nef Cartier divisor, γ = (g(B) dim(X ) − g(B) + 1)2(dim(X ) − 1), and Ξ = 0, if g(B) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Ξ is the supremum of the constants obtained by applying Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4 to all dimensions ≤ dim(X ), if g(B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6 immediately implies the desired conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ rema:exceptionalrelativelyfree Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The exceptional set in Geometric Manin’s Conjecture as described in [LST22] can include families of relatively free sections as well as families of non-relatively free sec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' For example, sometimes we must discount the contributions of irreducible com- ponents M ⊂ Sec(X /B) which parametrize relatively free sections when the evaluation map for the universal family over M has disconnected fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let f : Y → X denote the finite part of the Stein factorization of the evaluation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 shows that a(Yη, −f ∗KX/B|Yη) = a(Xη, −KX/B|Xη) so that such sections can be accounted for by the exceptional set of [LST22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proofs of main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We now prove the theorems stated in the introduction (except for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10 which is postponed to Section 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3: (1) Let Y′ be a resolution of Y and let N parametrize the strict transforms on Y′ of the general sections on Y parametrized by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since the Fujita invariant is birationally invariant, the desired statement follows from Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 applied to Y′ and N with arel = 1 and T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) To see the equality of Fujita invariants for Y, we let Y′ denote a resolution of singu- larities and let N denote the family of sections on Y′ such that f∗N is dense in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The desired equality follows from Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 applied to Y′, N, and L = −KX/B with arel = 1 and T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We next construct the rational map φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By [KMM92, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2] there is a positive integer b > arel depending only on dim(X ) and arel such that |−bKXη| is very ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Define ξ+ as in Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='11 applied to Y′, N, L = −KX/B, arel = 1, β = 0, T = 0, with our choice of b, and with an effective π-vertical Cartier divisor E as in Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 that achieves the 65 bound τ(π, E) = Υ(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='11 constructs a rational map φ : Y′ ��� Z over B that has all the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The only thing left to check is that dim(Z) ≥ 2, or in other words, that the rational map φ is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that we have an inclusion TY′/B → f ∗TX/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since C′ deforms in a dominant family on Y′, this map remains injective upon restriction to a general C′ and we conclude that C′ is not an HN-free section on Y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' But then the map Y′ → B does not satisfy Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (3), showing that φ must be non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6: This follows from Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10 applied with L = −KX/B and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Fix a positive integer T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Define the constant ξ(T) (depending also on dim(X ), g(B), and Υ(π)) as in Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 using the constants arel = 1, an effective π-exceptional divisor E as in Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 such that τ(π, E) = Υ(π), and the chosen value of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that M is an irreducible component of Sec(X /B) parametrizing sections satis- fying −KX/B · C ≥ ξ(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that N ⊂ M is a subvariety of codimension ≤ T that parametrizes sections C such that TX/B|C is not generically globally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular this implies that the sections parametrized by N do not dominate X ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' let Y ⊊ X denote the locus swept out by these sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 shows that a(Yη, −KX/B|Yη) ≥ a(Xη, −KX/B|Xη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Furthermore, the explicit description of ξ(T) in Equation (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1) shows that ξ(T) is linear in T with leading coefficient 1/ǫ where ǫ = ǫ(dim(X )) is a positive rational number such that no smooth projective variety of dimension ≤ dim(X ) − 1 has Fujita invariant in [1 − ǫ, 1) with respect to a big and nef Cartier divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By inverting this linear function we obtain the desired function Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='11: This is the special case of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (1) when E is [C]-semistable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='12: This follows from Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 applied to an irreducible component M ⊂ Sec(X /B) using arel = a and the inequality dim(M) ≥ −KX/B · C + (dim(X ) − 1)(1 − g(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='14: Let C be a general section parametrized by � N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4 we have −K �Y/B · C + (dim( �Y) − 1) ≥ dim( �N) ≥ dim(M) − T ≥ − �f ∗KX/B · C + (dim(X ) − 1)(1 − g(B)) − T Rearranging we see that (K �Y/B − �f ∗KX/B) · C ≤ T + g(B)(dim(X ) − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We conclude the desired statement by Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' □ 66 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Examples sect:examples Our first example illustrates how Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 can be used in practice to understand sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' exam:cubichyp Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1 (Cubic hypersurface fibrations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose that π : X → B is a Fano fibra- tion whose general fiber is a smooth cubic hypersurface of dimension n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We will analyze the irreducible components of Sec(X /B) parametrizing non-relatively free sections of large degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (In the special case when X is a smooth cubic hypersurface and B = P1, [CS09] proves a stronger statement for X × P1 by classifying all the irreducible components of Mor(P1, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=') A straightforward argument combining [H¨or10, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 Proposition] with the techniques of [LT19b, Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='1] shows that: If n ≥ 5 then there are no non-birational generically finite morphisms f : Yη → Xη with a(Yη, −f ∗KXη) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' If n = 4 then there are no non-birational generically finite morphisms f : Yη → Xη with a(Yη, −f ∗KXη) ≥ 1 unless Xη contains a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' When Xη contains a plane, the only possibility is that f is the composition of a birational map φ : Yη → P2 η and the inclusion of a plane P2 η ⊂ Xη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let M be a component of Sec(X /B) of sufficiently large anticanonical degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='3 shows: (1) If n ≥ 5 then M will generically parametrize relatively free sections and the evaluation map for its universal family will have connected fibers as in Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (2) If n = 4, then M can only fail to generically parametrize relatively free sections if it parametrizes a family of sections whose intersection with Xη is contained in a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This finishes the classification of irreducible components parametrizing non-free curves of large degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Our second example addresses the non-generically-globally-generated locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let π : X → B be a Fano fibration and let M be an irreducible component of Sec(X /B) of large degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8 shows that the codimension in M of the non-generically-globally-generated locus will grow linearly in degree except possibly when the sections sweep out a subvariety with large Fujita invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' The following example demonstrates that it is possible for the non-generically-globally-generated locus to have constant codimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' exam:largenonfreelocus Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let X be a smooth cubic threefold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Suppose M is a component of Mor(P1, X) parametrizing maps of anticanonical degree ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We will see that M admits a (possibly reducible) codimension 1 sublocus parametrizing multiple covers of non-free lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In partic- ular, the non-free locus in M will always have codimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' [CS09] shows that for any degree d ≥ 2 the moduli stack M0,0(X, d) has two irreducible components: a component Md that generically parametrizes irreducible free curves and a component Rd that parametrizes degree d covers of lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We will be interested in the intersection Td of these two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Inside of the parameter space of lines on X the sublocus parametrizing non-free lines has codimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus the locus Qd ⊂ Rd parametrizing multiple covers of non-free lines also has codimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Note that Td will be contained in Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' On the other hand, since all components of the moduli stack M0,0(X, d) have the expected dimension M0,0(X, d) has 67 only LCI singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Thus Td must have codimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Altogether, we see that Td consists of a (non-empty) union of irreducible components of Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since the general stable map parametrized by Td has irreducible domain, we see that every irreducible component of Mor(P1, X) of degree ≥ 3 will have a codimension 1 sublocus parametrizing non-free morphisms consisting of multiple covers of non-free lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' This result illustrates Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='8 applied to the projection π : X ×P1 → P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let Y ⊂ X denote a subvariety swept out by the curves parametrized by an irreducible component of Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then Y is also swept out by a one-parameter family of non-free lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' in particular we have a(Y, −KX|Y ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Passing to the relative situation, we see that any codimension 1 locus of Md parametrizing sections whose normal bundle is not generically globally generated will sweep out a subvariety Y × P1 which has generic Fujita invariant ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' An arithmetic application sect:application In this section we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We freely use the notations set up in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' First of all, consider the open subscheme of the relative Hilbert scheme that parametrizes sections of anticanonical height ≤ d which are not contained in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' There exists a finite set of places Sd ⊃ S such that this open subset is flat over Spec oF,Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Let ψ : Hd → Spec oF,Sd be the closure of this set inside the relative Hilbert scheme equipped with the reduced structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Then ψ : Hd → Spec oF,Sd is projective and flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' We denote the generic fiber of ψ by Hd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' (1) every irreducible component of Hd parametrizes a dominant family of sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='4 shows that the dimension of such a component is bounded by d + dim Xη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Define Cd := 2(d+dim Xη) � i=0 hi sing(Han d , Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Since the ℓ-adic sheaf Riψ∗Qℓ is constructible in the pro-´etale topology by [BS15, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='2], there exists a pro-´etale open i : U → Spec oF,Sd such that i−1Riψ∗Qℓ is a constant sheaf in the pro-´etale topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' In particular there exists a finite set of places S′ d ⊃ Sd such that we have hi ´et(Hd,v, Qℓ) = hi sing(Han d , Q) for all i and v ̸∈ S′ d where Hd,v is the base change of Hd,v to the algebraic closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' By applying the Grothendieck-Lefschetz trace formula and a version of the Weil conjectures for singular projective varieties ([Del80]), we conclude that N(Xv \\ Rv, −KXv/Bv, d) ≤ #Hd,v(kv) ≤ Cdqd+dim Xη v Thus assuming dǫ > dim Xη, we can conclude that N(Xv \\ Rv, −KXv/Bv, d) qd(1+ǫ) v → 0 as v → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' References [Ach06] J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Rational curves on smooth cubic hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' IMRN, (24):4626–4641, 2009.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Peternell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Reflexive differential forms on singular spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Geometry and cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Math.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' IMRN, (2):536–570, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' [GM75] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Grauert and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' M¨ulich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Vektorb¨undel vom Rang 2 ¨uber dem n-dimensionalen komplex- projektiven Raum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Manuscripta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=', 16(1):75–100, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' [Gro66] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Grothendieck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' ´El´ements de g´eom´etrie alg´ebrique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' ´Etude locale des sch´emas et des mor- phismes de sch´emas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' Hautes ´Etudes Sci.' metadata={'source': 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+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=', 41(5):779– 781, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content=' 71 Department of Mathematics, Boston College, Chestnut Hill, MA 02467 Email address: lehmannb@bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='edu Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, IN 46556 Email address: eriedl@nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='edu Graduate School of Mathematics, Nagoya University, Furocho Chikusa-ku, Nagoya, 464- 8602, Japan Institute for Advanced Research, Nagoya University Email address: sho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='tanimoto@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='nagoya-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} +page_content='jp 72' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NAzT4oBgHgl3EQfuv1g/content/2301.01695v1.pdf'} diff --git a/4NE2T4oBgHgl3EQf6Qhs/content/tmp_files/2301.04198v1.pdf.txt b/4NE2T4oBgHgl3EQf6Qhs/content/tmp_files/2301.04198v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8151d514d43cf4b28aaf24b8b8b65487fa75979e --- /dev/null +++ b/4NE2T4oBgHgl3EQf6Qhs/content/tmp_files/2301.04198v1.pdf.txt @@ -0,0 +1,1829 @@ +arXiv:2301.04198v1 [math.CO] 10 Jan 2023 +Sharp thresholds for spanning regular graphs +Maksim Zhukovskii∗ +Abstract +Let d ≥ 3 be a constant and let F be a d-regular graph on [n] with not too +many symmetries. The expectation threshold for the existence of a spanning +subgraph in G(n, p) isomorphic to F is p∗(n) = (1 + o(1))(e/n)2/d. We give +a tight bound on the edge expansion of F guaranteeing that the probability +threshold for the appearance of a copy of F has the same order of magnitude +as p∗. We also prove that, within a slight strengthening of this bound, the +probability threshold is asymptotically equal to p∗. In particular, it proves +the conjecture of Kahn, Narayanan and Park on a sharp threshold for the +containment of a square of a Hamilton cycle. It also implies that, for d ≥ 4 +and (asymptotically) almost all d-regular graphs F on [n], p(n) = (e/n)2/d is +a sharp threshold for F-containment. +1 +Introduction +Let d ≥ 3 be a fixed constant. Given a d-regular graph Fn on the vertex set [n] := +{1, . . . , n}, what is the threshold probability to contain its isomorphic copy by the +binomial random graph G(n, p) (i.e. the unique p = p(n) such that the probability +that G(n, p) contains an isomorphic copy of Fn equals 1/2)? Note that the threshold +probability exists since the considered property is monotone [7, Chapter 1.5]. +If Fn has a small enough automorphism group, then, by the union bound, the +threshold probability is at least (1 + o(1))(e/n)2/d. Indeed, let Fn be the set of all +isomorphic copies of Fn on [n], and let the number of automorphisms of Fn be eo(n). +Clearly |Fn| = +n! +eo(n). Let X be the number of graphs from Fn that are subgraphs of +G(n, p). We get +EX = |Fn|pdn/2 = +n! +eo(n)pdn/2 → 0 +as n → ∞ +if p < (1 − ε) +� e +n +�2/d, implying that with high probability (whp for brevity) G(n, p) +does not contain any graph from Fn. Let us denote by p∗(n) = (1+o(1))(e/n)2/d the +∗The University of Sheffield; zhukmax@gmail.com +1 + +expectation threshold for the existence of a spanning subgraph in G(n, p) isomorphic +to Fn (i.e. p∗(n) is the unique solution of the equation EX = 1). +On the other hand, from the recently resolved “expectation–threshold” conjec- +ture of Kahn and Kalai [16] it follows that the threshold does not exceed Cp∗(n) log n +for some constant C > 0. For some specific Fn it is known that the logarithmic fac- +tor can be removed, and the threshold probability equals Θ(n−2/d): it is true for +example for powers of a Hamilton cycle [15] and for the square tori T√n×√n [15]. On +the other hand, if Fn has many small subgraphs with a small edge boundary, this +is no longer true. More precisely, assume that, for some constant v, every vertex +of Fn belongs to a subgraph on v vertices with the edge boundary at most d (the +edge boundary of a subgraph ˜F is the number of edges between ˜F and its vertex +complement) or, equivalently, with at least dv +2 − d +2 edges. Then, a polylogarithmic +factor arises since in order to contain a copy of Fn, the random graph should have +every vertex inside a graph with v vertices and at least dv +2 − d +2 edges — see [17]. +We prove that, when the number of automorphisms of Fn is small enough, this +condition on the edge boundary is the only obstacle. +Theorem 1. Let d ≥ 3 and let Fn be a sequence of d-regular graphs on [n], n ∈ N, +such that +• for every ε > 0 and all large enough n the number of automorphisms of Fn is +less then eεn2/d; +• for every ˜F ⊂ Fn with 3 ≤ |V ( ˜F)| ≤ n − 3, the edge boundary of ˜F is at least +d + 1. +Let ε > 0. If p > (1 + ε)dp∗, then whp (assuming that dn is even) G(n, p) contains +a copy of Fn. +It immediately implies that the threshold probability for containing a copy of Fn +equals p(n) = Θ(n−2/d). As we mentioned above, the restriction on edge boundaries +is tight — if we allow subgraphs with edge boundary d instead of d + 1, then the +assertion becomes false. +Note that a bound on the number of symmetries can not be omitted — as soon +as the number of automorphisms of Fn becomes larger, the expectation threshold +p∗ becomes larger as well. In particular, p(n) = (d! log n) +2 +d(d+1) n−2/(d+1) is a sharp +threshold for the existence of a Kd+1-factor [14]. +In [15] Riordan proved a general result that for d-regular graphs can be stated +as follows: p(n) = Θ(n−2/d) is the threshold probability for containing a copy of Fn +if the d-regular graph Fn (the automorphism group should be at most exponential +2 + +in n) satisfies a stronger condition on the edge boundary: for every ˜F ⊂ Fn with +3 ≤ |V ( ˜F)| ≤ n−3, the edge boundary of ˜F is at least 2d. For powers of a Hamilton +cycle, this result implies the following: for every k ≥ 3, the threshold probability +for containing the kth power of a Hamilton cycle equals Θ(n−1/k). However, the +proof of Riordan does not work for k = 2. In [10], K¨uhn and Osthus proved that +n−1/2+o(1) is the threshold probability for containing the second power of a Hamilton +cycle and conjectured that the threshold is actually Θ(n−1/2). In [13], Nenadov and +ˇSkori´c proved the upper bound n−1/2(log n)4, which was improved to n−1/2(log n)3 by +Fischer, ˇSkori´c, Steger and Truji´c in [4], and to n−1/2(log n)2 in an unpublished work +of Montgomery (see [6]). Eventually, the conjecture was solved by Kahn, Narayanan +and Park in [8]. However, they did non settle a right constant in front of n−1/2 and +conjectured that the right constant is √e and that the threshold p(n) = +� +e/n(1 + +o(1)) is sharp (i.e., if p > (1 + ε) +� +e/n, then whp G(n, p) contains the second power +of a Hamilton cycle). In this paper, we prove this conjecture and even more: for +d ≤ 4 the requirement from Theorem 1 guarantees that p(n) = (1 + o(1)) +� +e/n +is even sharp; however, for d ≥ 5 we need to strengthen the bound on the edge +boundary to 2d − 2 (note that this is still better than the condition of Riordan). +Theorem 2. Let d ≥ 3 and let Fn be a sequence of d-regular graphs on [n], n ∈ N, +such that +• for every ε > 0 and all large enough n the number of automorphisms of Fn is +less then eεn2/d; +• either d ∈ {3, 4} and, for every ˜F ⊂ Fn with 3 ≤ |V ( ˜F)| ≤ n − 3, the edge +boundary of ˜F is at least d + 1, +or d ≥ 5 and, for every ˜F ⊂ Fn with 3 ≤ |V ( ˜F)| ≤ n − 3, the edge boundary +of ˜F is at least 2d − 2. +Let ε > 0. If p > (1+ε) +� e +n +�2/d, then whp (assuming that dn is even) G(n, p) contains +a copy of Fn. +Kahn, Narayanan and Park in [8] noted that the crucial fact that can be used +to prove that the threshold for appearance of the second power of a Hamilton cycle +equals Θ(n−1/2) is that the hypergraph of all copies of the second power of a cycle +on [n] is (1 + o(1)) +� +e/n-spread. Actually, they refined the notion of spreadness by +incorporating the count of the number of components in a subhyperedge. This re- +fined notion was distilled by D´ıaz and Person in [3], named superspreadness and used +to generalise the result of Kahn, Narayanan and Park to a wider class of spanning +subgraphs in G(n, p). In particular, they answered a question of Frieze asked in [5] +— they showed that the threshold for appearance of spanning 2-overlapping 4-cycles +(i.e. the copies of C4 are ordered cyclically, two consecutive C4 overlap in exactly +3 + +one edge, whereby each C4 overlaps with two copies of C4 in opposite edges) equals +Θ(n−2/3). Clearly, Theorem 2 implies that p(n) = (e/n)2/3 is a sharp threshold for +appearance of spanning 2-overlapping 4-cycles. +Let us call a sequence of d-regular graphs on [n] satisfying the conditions of +Theorem 2 good. Note that, for every d ≥ 4, almost all d-regular graphs are good +(see [2, 9, 11]). In particular, if d ≥ 5, then whp in a random d-regular graph on +[n] there are no subgraphs with 3 ≤ v ≤ n − 3 vertices and the edge boundary at +most 2d − 2. If d = 4, then whp there are no subgraphs with 3 ≤ v ≤ n − 3 vertices +and the edge boundary at most d + 1. If d = 3, then whp there are no subgraphs +with 3 ≤ v ≤ n − 3 vertices and the edge boundary at most d + 1 other than C3, C4 +and their vertex-complements. Since the edge boundary of C4 is exactly d + 1 = 4, +a random 3-regular graph is good whp under the condition that it does not contain +triangles. +Corollary 1. For every good sequence Fn, p(n) = +� e +n +�2/d is a sharp threshold for +containing a copy of Fn. In particular, +• for every ℓ ≥ 2, p(n) = (e/n)ℓ is a sharp threshold for containing the ℓth power +of a Hamilton cycle; +• p(n) = (e/n)2/3 is a sharp threshold for containing a spanning 2-overlapping +4-cycle; +• for every 3 ≤ m ≤ √n, p(n) = +� +e/n is a sharp threshold for containing +rectangular tori Tm×n/m (assuming that n is divisible by m); +• for every d ≥ 4 and (asymptotically) almost all d-regular graphs Fn on [n], as- +suming that dn is even, p(n) = (e/n)2/d is a sharp threshold for Fn-containment; +• for (asymptotically) almost all triangle-free 3-regular graphs Fn on [n], assum- +ing that n is odd, p(n) = (e/n)2/3 is a sharp threshold for Fn-containment. +Actually we are able to establish the same sharp threshold for almost all 3-regular +graphs — the condition of the absence of triangles is redundant, since the number +of triangles converges in probability to a Poisson random variable [19], and so it is +bounded in probability. In other words, we may allow Fn to have a bounded number +of subgraphs with a smaller edge boundary. However, we do not want to overload +the proof with technical details, and so we formulate Theorem 1 and Theorem 2 as +well as Corollary 1 in their current laconic forms. +We prove Theorem 1 using the “planted trick” that in different forms appears +in many applications — one of them is the well-known and very useful “spread +4 + +lemma” [1] which in particular gives good sunflower bounds [18]; in probabilistic +terms the application of the trick for the “spread lemma” is described in [12]. Kahn, +Narayanan and Park [8] and further D´ıaz and Person [3] used the “planted trick” to +prove their results on threshold probabilities as well. In essence, the key idea is to +“plant” a graph F from the family Fn and to combine it with the noise produced +by G(n, p). Then, it turns out that whp there exists a graph F ′ ∈ Fn which is +entirely inside the perturbed planted hyperedge F ∪ G(n, p) such that the size of +F ′ \ G(n, p) is quite small. This allows to replace Fn with the set of fragments of +F ∈ Fn equal to F ′ \ G(n, p), to draw independently edges of another G(n, p) and +to apply the same argument once again. If the number of steps in this procedure +is bounded by a constant, then we get that the threshold probability has the same +order of magnitude as p∗. +In the proof of Theorem 2 we show that it is sufficient to apply this trick only +once. Actually the usual second moment method (but for the uniform model instead +of the binomial) works as well. However, we give the proof of Theorem 2 in terms +of the planted hyperedge for the sake of convenience and coherence. In particular, +we want to explicitly show the borders between the following three phenomena: +1) it is sufficient to apply the “planted trick” once, 2) it is sufficient to apply the +“planted trick” constantly many times, 3) the number of applications of the trick is +unbounded. We claim that our analysis is optimal, and the method in its current +form cannot be used to weaken the bound on edge boundaries in Theorem 2 for +d ≥ 5. Our main achievement is that we make a step beyond the usage of the notions +of spreadness and superspreadness. We obtain optimal bounds on the number of +hyperedges containing a given set of edges I (commonly denoted by |Fn ∩ ⟨I⟩|) +and on the number of subgraphs of Fn with a fixed number of vertices, edges and +components (see Section 5 and Claim 6). The main ingredient of the proof of Claim 6 +is a very nice property of d-regular graphs satisfying the requirements of Theorem 1: +for every v, there are not too many subgraphs on v vertices with the maximum +possible number of edges +dv +2 − +� d+1 +2 +� +(see Section 2). +We describe the “planted +trick” in Section 3. Then we prove both theorems in Section 4. Sections 6 and 7 are +devoted to the proof of Claim 6 and the key lemma (Lemma 3 from Section 3) that +validates the application of the planted trick respectively. +2 +Linearly many closed subgraphs +Let us call a graph Fn with the second property from the requirement (on the edge +boundary) in Theorem 1 locally sparse. Note that this (local sparsity) property is +that the edge boundary of every subgraph ˜F with 3 ≤ |V ( ˜F)| ≤ n − 3 is at least +d + 1. Clearly d + 1 can be replaced with d + 2 for even d since in this case the edge +boundary δ( ˜F) cannot be odd. Let ∆ = d+1 for odd d and ∆ = d+2 for even d. It +5 + +is easy to see that the condition |δ( ˜F)| ≥ ∆ holds for all ˜F with 2 ≤ |V ( ˜F)| ≤ d − 1 +just due to the d-regularity of Fn. Let us call a subgraph ˜F with the edge boundary +exactly ∆ closed (note that a closed subgraph is always connected). For j < d, let +us call a vertex w of a connected subgraph ˜F ⊂ Fn j-free, if its degree in ˜F equals +j; w is simply free, if it is j-free for some j < d. +Let F be a locally sparse d-regular graph on [n]. +Claim 1. Every closed subgraph of F with at least 3 vertices has minimum degree +at least d/2. +Proof. Assume that ˜F is a closed subgraph of F with a vertex w having degree +d′ < d/2. If we remove the vertex w from ˜F, then we get the graph ˜F \ w with edge +boundary δ(F) + 2d′ −d < δ(F) = ∆. This contradicts the local sparsity of F when +|V ( ˜F)| ≥ 4. Otherwise it contradicts the fact that a subgraph on 2 vertices has the +edge boundary at least 2d − 2 ≥ ∆. +Claim 2. For any pair of adjacent vertices x, y in F and for every 3 ≤ v ≤ n − 3, +there are at most two closed subgraphs in F on v vertices containing x and not +containing y. +Proof. Fix adjacent vertices x, y and 3 ≤ v ≤ n − 3. +A closed graph ˜F ⊂ F sends exactly ∆ edges to F \ ˜F implying that F \ ˜F is +also closed. Assume that v ≥ n/2, and that there are at least 3 closed graphs on +v vertices that share x and do not contain y. Then their complements are closed +graphs on n − v ≤ n/2 vertices that share y and do not share x. Therefore, it is +sufficient to prove the claim for v ≤ n/2. +Let H1, H2 be different closed subgraphs of F on v vertices that contain x and do +not contain y. Note that H1, H2 should have at least one other common vertex since +otherwise the degree of x is bigger than d due to Claim 1. Then |V (H1) ∪ V (H2)| ≤ +n − 2. +Let H0 = H1 ∩ H2. Note that |E(H0)| ≤ d +2|V (H0)| − ∆ +2 implying that |E(Hj) \ +E(H0)| ≥ +d +2|V (Hj \ H0)| for both j = 1 and j = 2 since H1, H2 are closed. On +the other hand, if, say |E(H2) \ E(H0)| > +d +2|V (H2 \ H0)|, then |E(H1 ∪ H2)| > +d +2|V (H1∪H2)|− ∆ +2 which contradicts the local sparsity of F since |V (H1∪H2)| ≤ n−2. +Therefore, |E(Hj) \ E(H0)| = d +2|V (Hj \ H0)| for both j = 1 and j = 2, but then H0 +is closed. +Then, there are exactly ∆ edges between H0 and F \ H0, and one of them is the +edge between x and y. It means that Hj \ H0, j ∈ {1, 2}, send at most ∆ − 1 edges +(in total) to H0. This may happen only if |V (Hj \ H0)| = 1 for both j = 1 and +j = 2. Indeed, |V (H1 \ H0)| = |V (H2 \ H0)|. Moreover, the number of edges that +6 + +Hj \ H0 sends to H0 equals +|E(Hj) \ E(H0)| − |E(Hj \ H0)| ≥ d +2|V (Hj \ H0)| − +�d +2|V (Hj \ H0)| − ∆ +2 +� += ∆ +2 +whenever |V (Hj \ H0)| ≥ 2. +Assume that there exists a closed graph H3 ̸⊂ H1∪H2 on v vertices that contains +x and does not contain y. From the above it follows that H3 ∩ H1 = H3 ∩ H2 = H0. +Each vertex of Hj \ H0 sends at least d +2 edges to H0 due to Claim 1. But then the +vertices from Hj \ H0 send at least 3d +2 ≥ ∆ edges to H0 — a contradiction (since +there is one additional edge {x, y} in the edge boundary of H0). Therefore, any +other closed graph that contains x and does not contain y should be entirely inside +H1 ∪ H2. Assume that such a graph H3 exists. Let w1 ∈ H1 \ H0, w2 ∈ H2 \ H0. +Clearly, H3 contains w1, w2 and all but 1 vertex of H0. In the same way as above +we get that H1 ∩ H2 = H0, H1 ∩ H3 and H2 ∩ H3 are three closed graphs on v − 1 +vertices that contain x and do not contain y. These three closed graphs on v − 1 +vertices have the property that none of them is inside the union of the other two — +this is only possible when v − 1 = 2, i.e. v = 3. The only possible closed graph on +3 vertices is a triangle. Moreover, a triangle is closed only when d = 4. So, H1, H2 +are triangles sharing an edge, but then H3 adds another edge to the union H1 ∪ H2 +implying that H1 ∪ H2 ∪ H3 is a 4-clique. We get a contradiction with the local +sparsity since the edge boundary of a 4-clique is 4 < ∆ = 6. +From this, it immediately follows, that for every v, there are at most Cn closed +subgraphs on v vertices in F for a certain universal constant C. More precisely, the +following is true. +Claim 3. Let k ∈ N, and let F ′ be the induced subgraph of F on [k]. For every +3 ≤ v ≤ n − 3, the number of closed subgraphs of F ′ with v vertices is at most 2dk +3 . +Proof. Fix a vertex w in F ′ and let us bound the number (denoted by µ(w)) of +closed subgraphs of F ′ on v vertices containing w such that the vertex w is free in +these graphs. Due to Claim 2, µ(w) ≤ 2d. On the other hand, Claim 1 implies +that every closed subgraph contains at least 3 free vertices. Letting f to be the +number of closed subgraphs in F ′ on v vertices, by double counting, we get that +3f ≤ � +w∈V (F ′) µ(w) ≤ 2dk as needed. +For d = 3, 4, we need sharper bounds. Let us start from d = 3. +Claim 4. Let d = 3, k ∈ N. Let F ′ be the induced subgraph of F on [k]. Then for +every 3 ≤ v ≤ n − 3, there are at most 3 +4k closed subgraphs in F ′ on v vertices. +7 + +Proof. Fix a vertex w in F ′ and let us compute the number µ(w) of closed subgraphs +of F ′ on v vertices containing w such that the vertex w is free in these graphs. +Reviewing the proof of Claim 2, we may see that in the case d = 3, every vertex +x may be inside only a single closed subgraph on v vertices that does not contain +another vertex y — otherwise H1 \ H0 sends at least 2 edges to H0, and the same +for H2 \ H0 implying that H0 cannot be closed. Then, for every w, µ(w) ≤ 3. On +the other hand, Claim 1 implies that every closed subgraph contains at least 4 free +vertices. Letting f to be the number of closed subgraphs in F ′ on v vertices, by +double counting, we get that 4f ≤ � +w∈V (F ′) µ(w) ≤ 3k as needed. +For d = 4, we get the following. +Claim 5. Let d = 4, k ∈ N. Let F ′ be the induced subgraph of F on [k]. Then for +every 3 ≤ v ≤ n − 3, there are at most 4 +3k closed subgraphs in F ′ on v vertices. +Proof. Fix a vertex w in F ′ and let us compute the number of closed subgraphs of +F ′ on v vertices containing w such that the vertex w is free in these graphs. Let +µj(w) be the number of closed subgraphs on v vertices such that w is j-free in these +graphs. Due to Claim 1, µ1(w) = 0. Due to Claim 2, µ3(w) + 2µ2(w) ≤ 8. On the +other hand, letting f to be the number of closed subgraphs in F ′ on v vertices, since +every closed graph has the edge boundary equal to 6, we get that 6f is exactly the +number of pairs (a closed graph ˜F, an edge from the boundary of ˜F). Therefore, +6f = � +w(µ3(w) + 2µ2(w)) ≤ 8k implying that the number of closed graphs on v +vertices is at most 4 +3k as needed. +3 +Planted hyperedge +Let dn be even, Fn be a d-regular graph on [n] satisfying the requirements of +Theorem 1, and let F be a uniformly chosen random element of Fn. Let ε > 0, +w = (1 + ε + o(1))(e/n)2/d�n +2 +� +be a sequence of integers and W be a random graph +on [n] with w edges chosen uniformly at random. In this section, we review the +constructions and follow the terminology from [8]. For the sake of convenience, we +give the argument in full. Our achievement is Lemma 3 that we state in the end of +the section. +Fix a non-negative integer ℓ0. Let us call a pair (F ∈ Fn, W ⊂ +�[n] +2 +� +) ℓ0-bad, if +for every ℓ0-subset L ⊂ F (we hereinafter assume that F is the set of edges), we +have that L ⊔ [W \ F] does not contain a graph from Fn. +Fix t ∈ +� +0, 1, . . . , dn +2 +� +and let w′ = w − t. Note that, for F ∈ Fn, W ⊂ +�[n] +2 +� +, +such that |F ∩ W| = t, we have +|F ∪ W| = dn +2 + w − t = w′ + dn +2 . +8 + +Call Z ∈ +� ([n] +2 ) +w′+dn/2 +� +ℓ0-pathological if +|{F ⊂ Z : F ∈ Fn, (F, Z) is ℓ0-bad}| > 1 +n|Fn| +��n +2 +� +− dn/2 +w′ +� +/ +� +�n +2 +� +w′ + dn/2 +� +=: M. +Note that +P(|F ∩ W| = t) = +�dn/2 +t +���n +2 +� +− dn/2 +w′ +� +/ +��n +2 +� +w +� +. +We have therefore +P((F, W) is ℓ0-bad, F ∪ W is not ℓ0-pathological | |F ∩ W| = t) += P((F, W) is ℓ0-bad, F ∪ W is not ℓ0-pathological, |F ∩ W| = t) +P(|F ∩ W| = t) +≤ +� +(n +2) +w′+dn/2 +� +|{F ⊂ Z : F ∈ Fn, (F, Z) is ℓ0-bad}| +�dn/2 +t +� +|Fn| +�(n +2) +w +��dn/2 +t +��(n +2)−dn/2 +w′ +� +/ +�(n +2) +w +� +≤ 1 +n, +where Z is a not ℓ0-pathological hyperedge from +� ([n] +2 ) +w′+dn/2 +� +with maximum possible +value of |{F ⊂ Z : F ∈ Fn, (F, Z) is ℓ0-bad}|. +Fix F ∈ Fn and a set B ⊂ F of size t. Let W′ be chosen uniformly at random +from +�([n] +2 )\F +w′ +� +. Note that if F ∪ W′ is ℓ0-pathological, then there are at least M +subgraphs from Fn in F ∪ W′. On the other hand, if (F, W′) is ℓ0-bad, then, for +every F ′ ∈ Fn such that F ′ ⊂ F ∪ W′, +|F ′ ∩ F| = |F ′ \ W′| ≥ ℓ0 + 1. +Let X count the number of F ′ ∈ Fn such that F ′ ⊂ F ∪ W′ and |F ′ ∩ F| ≥ ℓ0 + 1. +We get that the event “(F, W′) is ℓ0-bad and F ∪ W′ is ℓ0-pathological” implies +X ≥ M. Then +P((F, W) is ℓ0-bad, F ∪ W is ℓ0-pathological | |F ∩ W| = t) = +P((F, W) is ℓ0-bad, F ∪ W is ℓ0-pathological | F = F, F ∩ W = B) = +P((F, W′) is ℓ0-bad, F ∪ W′ is ℓ0-pathological) ≤ P(X ≥ M) ≤ EX +M . +For ℓ ≥ ℓ0 + 1, let πℓ := P(|F ∩ F| = ℓ). Then +EX = +� +ℓ≥ℓ0+1 +|Fn|πℓ +� +w′ +dn/2 − ℓ +� +/ +��n +2 +� +− dn/2 +dn/2 − ℓ +� +. +(1) +9 + +We have +EX +M = +� +ℓ≥ℓ0+1 +|Fn| πℓ +M +� +w′ +dn/2 − ℓ +� +/ +��n +2 +� +− dn/2 +dn/2 − ℓ +� += n +� +ℓ≥ℓ0+1 +πℓ +� +w′ +dn/2−ℓ +� +/ +�(n +2)−dn/2 +dn/2−ℓ +� +�(n +2)−dn/2 +w′ +� +/ +� +(n +2) +w′+dn/2 +�. +(2) +Note that +� +w′ +dn/2−ℓ +� +/ +�(n +2)−dn/2 +dn/2−ℓ +� +�(n +2)−dn/2 +w′ +� +/ +� +(n +2) +w′+dn/2 +� ∼ +w′2w′( +�n +2 +� +− dn + ℓ)(n +2)−dn+ℓ�n +2 +�(n +2) +(w′ − dn/2 + ℓ)w′−dn/2+ℓ(w′ + dn/2)w′+dn/2( +�n +2 +� +− dn/2)2(n +2)−dn +< e− (dn/2−ℓ)2 +2w +− (dn/2)2 +2w ++O(1) +�� +n +(1 + ε)e +�2/d�ℓ +. +(3) +In Section 7, we prove the following. +Lemma 3. If one of the following two conditions hold +• ℓ0 = +� +d2 +2 ln(1+ε/2)n1−(∆−d)/d� +, or +• Fn is good and ℓ0 = 0, +then +EX +M ≤ n +� +ℓ≥ℓ0+1 +πℓe− (dn/2−ℓ)2 +2w +− (dn/2)2 +2w +�� +n +(1 + ε)e +�2/d�ℓ += o +�1 +n +� +. +(4) +4 +Proofs of Theorems 1, 2 +Lemma 3 implies Theorem 2 immediately. +It remains to prove Theorem 1 for d ≥ 5. Let p > (1 + ε)d +� e +n +�2/d. We use +the first assertion of Lemma 3 for that. Consider d independent copies G1, . . . , Gd +of G(n, p′), p′ = (1 + ε) +� e +n +�2/d. For every F ∈ Fn, consider a minimum possible +R = R(F) ⊂ F such that R ∪ G1 contains a graph from Fn. By Lemma 3 whp +|R| ≤ +d2 +2 ln(1+ε/2)n1−(∆−d)/d. +Let Σ be the set of all F ∈ Fn such that |R(F)| ≤ +d2 +2 ln(1+ε/2)n1−(∆−d)/d. We have that E|Σ| = (1 − o(1))|Fn|. By Markov’s inequality, +P +� +|Σ| ≤ |Fn| +2 +� += P +� +|Fn| − |Σ| ≥ |Fn| +2 +� +≤ 2(|Fn| − E|Σ|) +|Fn| +→ 0. +10 + +Let R = {R(F) : F ∈ Σ} be a multiset, i.e. |R| = |Σ|. We may assume that all sets +R ∈ R have equal cardinality exactly ℓ1 := +� +d2 +2 ln(1+ε/2)n1−(∆−d)/d� +. We then apply +the same proof (as in Section 3) but for R instead of Fn. +Let ℓ0 = 1 +2 +� +d2 +ln(1+ε/2) +�2 +n1−2(∆−d)/d. Let us call a pair (R ∈ R, W ∈ +�([n] +2 ) +w +� +) bad, if +for every ℓ0-subset L ⊂ R, we have that L ⊔ [W \ R] does not contain a graph from +R. For a fixed size of intersection t, a set Z ∈ +� ([n] +2 ) +w−t+ℓ1 +� +is pathological if +|{R ⊂ Z : R ∈ R, (R, Z) is bad}| > 1 +n|R| +��n +2 +� +− ℓ1 +w − t +� +/ +� +�n +2 +� +w − t + ℓ1 +� +=: M. +In order to show that for a fixed R ∈ R, whp in R∪G2 there exists a subset R′ ∈ R +such that |R′ \ G2| ≤ ℓ0, it is sufficient to prove an analogue of the first assertion of +Lemma 3: let +• W′ be chosen uniformly at random from +�([n] +2 )\R +w−t +� +; +• X be the number of R′ ∈ R such that R′ ⊂ R ∪ W′ and |R′ ∩ R| ≥ ℓ0 + 1, +then EX/M = o(1/n). Note that an analogue of the first inequality in (4) holds true +with dn/2 replaced by ℓ1. Note that R has at most 2ℓ1 vertices. Defining p(ℓ, x, c) +in the same way as in Section 5 and applying Claim 6, we get +p(ℓ, x, c) ≤ max +a +α2ℓ1(a, ℓ, x, c)β(a, ℓ, x, c) +|R| +≤ +�2ℓ1 +c +� +[(n − x + c)! + O(1)] +|R| +× +× max +a +�x − 2c +c − a +��c +a +��˜σ + c +c − a +��d +2 +�a �4d +3 +�c−a +22d(d+5)˜σ +max +o≤(d+4)˜σ +�x +o +�2 +. +Note that this bound differs from (8) only in the first binomial coefficient with n +replaced by 2ℓ1. Therefore, applying the same arguments as in Section 7.1, we get +that, for every ℓ > ℓ0, +πℓ = +� +x,c +p(x, ℓ, c) ≤ n2 � e +n +� 2ℓ +d (1 + δ)ℓe +d2 +2 +d2 +ln(1+ε/2) n1−2(∆−d)/d. +Therefore the analogues of (1), (2) and (3) imply that +EX +M ≤ +� +ℓ>ℓ0 +n3 +�1 + δ +1 + ε +�ℓ +e +d2 +2 +d2 +2 ln(1+ε/2) n1−2(∆−d)/d = o +�1 +n +� +. +Applying repeatedly the whole argument +d +∆−d − 1 ≤ d − 1 times, we arrive to +fragments of graphs from Fn of sizes at most ℓd−1 = +� +1 +2 +� +d2 +ln(1+ε/2) +�d−1 +n(∆−d)/d +� +. +11 + +Defining R (whp |R| ≥ |Fn|/2d−1) in a usual way as the multiset of fragments of +size exactly ℓd−1, letting +M := 1 +n|R| +��n +2 +� +− ℓd−1 +w − t +� +/ +� +�n +2 +� +w − t + ℓd−1 +� +, +considering a fixed fragment R, a uniformly chosen W′ ∈ +�([n] +2 )\R +w−t +� +, and defining X +as the number of R′ ∈ R such that R′ ⊂ R ∪ W′ and |R′ ∩ R| ≥ 1, it remains to +show that EX/M = o(1/n). We then consider p(ℓ, x, c) and apply Claim 6: +p(ℓ, x, c) ≤ max +a +α2ℓd−1(a, ℓ, x, c)β(a, ℓ, x, c) +|R| +≤ +�2ℓd−1 +c +� +[(n − x + c)! + O(1)] +|R| +× +× max +a +�x − 2c +c − a +��c +a +��˜σ + c +c − a +��d +2 +�a �4d +3 +�c−a +22d(d+5)˜σ +max +o≤(d+4)˜σ +�x +o +�2 +. +In the same way as in Section 7.1, the only non-trivial case is σ < ε′x, x < ε′n, +where 0 < ε′ ≪ δ is small enough (otherwise, p(ℓ, x, c) is even smaller). In this case, +for large enough constant C = C(d), +p(ℓ, x, c) ≤ 2d−1 +�2ℓd−1 +c +� +e2ℓ/d+(x−c)2/n+o(n2/d) +n2ℓ/d+(∆−d)c/d +(1 + δ/2)ℓ +�d2 +2 e2/d−5/6 +�c +≤ 2d−1 +�� +d2 +ln(1 + ε/2) +�d−1 e +c +�c +e2ℓ/d+(x−c)2/n+o(n2/d) +n2ℓ/d +(1 + δ/2)ℓ +�d2 +2 e2/d−5/6 +�c +≤ C e2ℓ/d+o(n2/d) +n2ℓ/d +(1 + δ)ℓ. +Finally, for every ℓ ≥ 1, +πℓ = +� +x,c +p(x, ℓ, c) ≤ n2C e2ℓ/d+o(n2/d) +n2ℓ/d +(1 + δ)ℓ. +Therefore the analogues of (1), (2) and (3) imply that +EX +M ≤ +� +ℓ>ℓ0 +n3 +�1 + δ +1 + ε +�ℓ +e−Θ(n2/d) = o +�1 +n +� +. +5 +Spread +We here follow the notations of Section 3: F ∈ Fn is fixed, F ∈ Fn is chosen +uniformly at random, and πℓ = P(|F ∩ F| = ℓ). +12 + +Fix c ∈ [ℓ], x ∈ +� 2ℓ +d + ∆ +d c, ℓ + c +� +. Denote +σ := d +2x − +� +ℓ + ∆ +2 c +� +. +Let p(ℓ, x, c) be the probability that the intersection of F with F is a graph on x +vertices with ℓ edges and c connected components (we think about graphs as about +sets of their edges, so there are no isolated vertices in |F∩F|). Let integers ℓ1, . . . , ℓc +and x1, . . . , xc be chosen in a way such that +• +2ℓi +d + ∆ +d ≤ xi ≤ ℓi + 1 for all i ∈ [c]; +• �c +i=1 ℓi = ℓ, �c +i=1 xi = x. +Let p(ℓ1, . . . , ℓc, x1, . . . , xc) be the probability that the intersection of F with F con- +sists of c connected components R1, . . . , Rc such that |V (Ri)| = xi, |E(Ri)| = ℓi. +Clearly, +p(ℓ, x, c) = +� +ℓi,xi +p(ℓi, xi, i ∈ [c]), +(5) +where the summation is over all unordered choices of ℓ1, . . . , ℓc, x1, . . . , xc. Note +that, in the case of the ordered choice, the number of ways to choose the values of +xi ≥ 2 is at most +�x−c +c +� +. The number of ways to choose the respective ℓi is at most +�σ+c +c +� +. +We will use the following claim. Let ˜F be a subgraph of F on k vertices. We +assume that either k = n and ˜F = F, or k ≪ n. +Claim 6. The number of ways to choose a subgraph R1 ⊔ . . . ⊔ Rc from ˜F without +isolated vertices with x vertices, ℓ edges and c components such that a is the number +of Ri that are either an edge or a full vertex-complement to an edge (that comprises +n − 2 vertices and dn/2 − (2d − 1) edges) is +αk(a, ℓ, x, c) ≤ +�k +c +��x − 2c +c − a +��c +a +��˜σ + c +c − a +� �d +2 +�a +γc−a +max +o≤(d+4)˜σ +�x +o +� +2d(d+5)˜σ, +(6) +where +γ = 2d +3 I(d ≥ 5) + 4 +3I(d = 4) + 3 +4I(d = 3), +˜σ = σ − a(d − 1 − ∆/2). +Given disjoint non-trivial connected R1, . . . , Rc ⊂ ˜F such that their union has x +vertices and ℓ edges, and there are exactly a graphs Ri that are either an edge or a +full vertex-complement to an edge, the number of ways to extend R1 ⊔ . . . ⊔ Rc to a +graph from Fn is +β(a, ℓ, x, c) ≤ (n − x + c)!(d − 1)a2c−a +max +o≤(d+4)˜σ +�x +o +� +2d(d+4)˜σ + O(1). +(7) +13 + +6 +Proof of Claim 6 +Fix ℓ1, . . . , ℓc and x1, . . . , xc. Let us compute the number of ways to choose connected +vertex-disjoint subgraphs R1, . . . , Rc from ˜F with the respective numbers of edges +and vertices. Let us call Ri dense, if one of the following holds: 1) xi = 2 and ℓi = 1, +2) Ri is closed, 3) xi = n − 2 and ℓi = d +2n − (2d − 1), 4) xi = n − 1 and ℓi = d +2n − d, +5) xi = n and ℓi = d +2n. +For i ∈ [c], set σi = +d +2xi − ℓi − ∆ +2 . Note that a is the number of i such that +xi = 2 and ℓi = 1, or xi = n − 2 and ℓi = d +2n − (2d − 1). Let us call the respective +Ri edge-components. +The number of ways to choose i ∈ [c] such that Ri is an +edge component equals +�c +a +� +, while the number of ways to choose the values of the +remaining xi is at most +�x−2c +c−a +� +. The number of ways to choose the respective ℓi is at +most +�˜σ+c +c−a +� +. +We first choose dense graphs. If c = 1, x1 = n − 1 and ℓ1 = d +2n − d, then there +are exactly n ways to choose R1. If c = 1, x1 = n and ℓ1 = 2n, there is only one way +to choose R1. Otherwise, we first choose edge-components Ri: for j = 1, 2, . . . , a, +the jth edge is chosen out of the set of kj remaining vertices in dkj +2 ways, and then +kj−1 ≤ kj −2. After that, we choose all the remaining dense graphs from R1, . . . , Rc. +Note that the remaining dense graphs from R1, . . . , Rc are closed. Assume that we +want to choose a closed Rj, and kj is the number of remaining vertices. Then by +Claims 3, 4, and 5, the number of ways to choose Rj is at most γkj. After that, +kj − |V (Rj)| ≤ kj − 3 vertices remain. Assuming that c − ˜c is the number of dense +Rj, we get that there are at most +� d +2 +�a γc−˜c−a +n! +(n−(c−˜c))! number of ways to choose +dense subgraphs. +Without loss of generality, we assume that it remains to choose R1, . . . , R˜c. +Note that the component Ri might have at most ∆ + 2σi free vertices. +For ev- +ery i = 1, 2, . . . , ˜c, we expose Ri in ˜F in the following way. +Assume that F ′ is +the current graph (obtained by removing all the chosen subgraphs R1, . . . , Ri−1 and +R˜c+1, . . . , Rc) on k′ vertices, +1. choose the number of free vertices oi ≤ d + 2 + 2σi; +2. choose the iterations of the below algorithm (out of the total number of iter- +ations xi) that produce a free vertex of Ri in +�xi +oi +� +ways; +3. choose a vertex w in F ′ which is minimum in Ri (here we mean the ordering +of the vertices of Ri induced by the ordering of the vertices of F ′) in at most +k′ ways, and activate it; +4. at every step, choose the minimum vertex (in the ordering of the vertices from +F ′) in the set of active vertices: +14 + +• if it should be free (in accordance to the above choice), then add to Ri +some of its neighbours (in at most 2d ways), deactivate it and activate all +its chosen neighbours, +• if it should not be free, then add to Ri all its neighbours, deactivate the +vertex and activate all its neighbours. +We get that the number of ways to choose Ri is at most k′(d+2+2σi) +max +oi≤d+2+2σi +�xi +oi +� +2doi. +Eventually we get that the number of ordered choices of components with pa- +rameters ℓi, xi, i ∈ [c], in ˜F is at most +c! +�k +c +� �d +2 +�a +γc−˜c−a +˜c� +i=1 +(d + 2 + 2σi) +max +oi≤d+2+2σi +�xi +oi +� +2doi +≤ c! +�k +c +� �d +2 +�a +γc−˜c−a +max +o≤(d+4)˜σ +�x +o +� +2d(d+5)˜σ. +Note that this bound does not depend on the order of the choice of all these com- +ponents R1, . . . , Rc, thus it implies (6). +Let us now fix connected subgraphs R1, . . . , Rc of F as above. Let us bound +the number of ways to extend R1 ⊔ . . . ⊔ Rc to an F ′ ∈ Fn. We construct such an +extension in the following way. +First, let us consider the two special cases: 1) c = 1, x1 ≥ n − 2 and R1 is dense; +2) c = 2, x1 = 2, x2 = n − 2 and R1, R2 are dense. In the first case, if x1 = n − 2, +then there are at most +�2(d−1) +d−1 +� +ways to construct F ′ (the remaining 2 vertices should +be adjacent — so we should only draw missing edges in constantly many ways); if +x1 ≥ n − 1, then there is a unique way to construct F ′. In the second case, there +are also at most +�2(d−1) +d−1 +� +ways to draw F ′. +We then forget the labels of the vertices from R1, . . . , Rc and assume (without +loss of generality) that the desired F ′ ∈ Fn is defined on [n] in a way such that +every i ∈ {2, . . . , n} has a neighbour in [i − 1], denote one such neighbour (chosen +randomly) by ν(i) (note that the graph F ′ is connected due to the restriction on +edge boundaries). Let H be obtained from F by deleting all the edges that do not +belong to R1 ⊔ . . . ⊔ Rc. Then let Z be the set of all components in H (together +with the isolated vertices), i.e. |Z| = n − x + c. We should compute the number of +ways to embed the elements of Z in F ′ disjointly. +Let z1, . . . , zn−x+c be an ordering of Z. At every step i = 1, . . . , n−x+c, consider +the minimum vertex κi of F ′ such that none of the embedded elements of Z in F ′ +contain this vertex. If zi /∈ {R1, . . . , Rc}, then we assign κi with zi and proceed +with the next step. Otherwise, we distinguish between the following cases. We let +zi = R1 without loss of generality. +15 + +First, we assume that R1 is dense. +If |V (R1)| = 2, then there are at most +d − 1 ways to choose the image of the edge R1, since the edge {κi, ν(κi)} is already +‘occupied’. If 2 < |V (R1)| < n − 2, then due to Claim 2, there are at most 2 ways +to choose a copy of R1 in F ′ containing κi and not containing ν(κi), as desired. +Second, let R1 be not dense. Choose the iterations of the below algorithm (out +of the total number of iterations xi) that produce a free vertex of Ri in +�xi +oi +� +ways. +Activate κi. +At every step, choose the minimum vertex (in the ordering of the +vertices from F ′) in the set of active vertices: +• if it should be free (in accordance to the above choice), then add to the image of +R1 under construction some of its neighbours (in at most 2d ways), deactivate +it and activate all its neighbours, +• if it should not free, then add all its neighbours, deactivate the vertex and +activate all its neighbours. +We get that the number of ways to construct the image of R1 is at most +�xi +oi +� +2doi. +Eventually we get that there are at most +(n − x + c)!(d − 1)a2c−˜c−a +˜c� +i=1 +max +oi≤d+2+2σi +�xi +oi +� +2doi + O(1) +≤ (n − x + c)!(d − 1)a2c−˜c−a +max +o≤(d+4)˜σ +�x +o +� +2d(d+4)˜σ + O(1). +ways to expose F ′ as needed. +7 +Proof of Lemma 3 +Summing up, from (5), (6) and (7), we get that +p(ℓ, x, c) ≤ max +a +αn(a, ℓ, x, c)β(a, ℓ, x, c) +|Fn| +≤ +�n +c +� +[(n − x + c)! + O(1)] +|Fn| +× +× max +a +�x − 2c +c − a +��c +a +��˜σ + c +c − a +��d +2 +�a +(2γ)c−a22d(d+5)˜σ +max +o≤(d+4)˜σ +�x +o +�2 +. +(8) +We let +Υ := max +a +�˜σ + c +c − a +� +22d(d+5)˜σ +max +o≤(d+4)˜σ +�x +o +�2 +. +Let us find the maximum value of ϕ(x) = +�x−2c +c−a +� +e∆c/d−x as a function of x. Since +ϕ(x + 1) +ϕ(x) += 1 +e +� +1 + +c − a +x + 1 − 3c + a +� +, +16 + +we get that the maximum value is achieved at x = 2c+(c−a) +e +e−1 +O(1). Therefore, +p(ℓ, x, c) ≤ +�n +c +� +e2σ/d+2ℓ/d+(x−c)2/n+o(n2/d) +n2ℓ/d+(∆−d)c/d+2σ/d +Υ× +× (2γ)c max +a +�c +a +� �d(d − 1) +4γ +�a �� +(c − a) +e +e−1 +� +c − a +� +e−(2−∆/d)c−(c−a) +e +e−1. +Since +�c +a +� �d(d − 1) +4γ +�a �� +(c − a) +e +e−1 +� +c − a +� +e−(c−a) +e +e−1 ≤ +�c +a +� �d(d − 1) +4γ +�a �2 +e +� +e +e−1 (c−a) +≤ +� +d(d − 1) +4γ ++ +�2 +e +�e/(e−1)�c +, +we eventually get +p(ℓ, x, c) ≤ +�n +c +� +e2σ/d+2ℓ/d+(x−c)2/n+o(n2/d) +n2ℓ/d+(∆−d)c/d+2σ/d +Υ +� +4d +3 e−(2−∆/d) +� +3(d − 1) +8 ++ +�2 +e +�e/(e−1)��c +≤ +�n +c +� +e2σ/d+2ℓ/d+(x−c)2/n+o(n2/d) +n2ℓ/d+(∆−d)c/d+2σ/d +Υ +�d2 +2 e2/d−5/6 +�c +for all d ≥ 5; +p(ℓ, x, c) ≤ +�n +c +� +eσ/2+ℓ/2+(x−c)2/n+o(√n) +nℓ/2+c/2+σ/2 +Υ +� +8 +3√e +� +9 +4 + +�2 +e +�e/(e−1)��c +≤ +�n +c +� +eσ/2+ℓ/2+(x−c)2/n+o(√n) +nℓ/2+c/2+σ/2 +Υ +�7.8 +√e +�c +for d = 4; +p(ℓ, x, c) ≤ +�n +c +� +e2σ/3+2ℓ/3+(x−c)2/n+o(n2/3) +n2ℓ/3+c/3+2σ/3 +Υ +� +3 +2e2/3 +� +2 + +�2 +e +�e/(e−1)��c +≤ +�n +c +� +e2σ/3+2ℓ/3+(x−c)2/n+o(n2/3) +n2ℓ/3+c/3+2σ/3 +Υ +� 4 +e2/3 +�c +for d = 3. +From now on, we separately proof three assertions of Lemma 3. +7.1 +d ≥ 5: existence of a fragment +First we assume that d ≥ 5, ℓ0 = +d2 +2 ln(1+ε/2)n1−(∆−d)/d, ℓ > ℓ0. Let δ > 0 be small +enough. Choose ε′ = ε′(δ) > 0 small enough in a way such that σ < ε′x implies +17 + +Υ ≤ (1 + δ/3)ℓ and c < ε′x implies +�x−2c +c−a +��c +a +��d +2 +�a(2γ)c−a ≤ (1 + δ/3)ℓ for all a as +well. +Assume that σ < ε′x. In this case, +p(ℓ, x, c) ≤ +�n +c +� +e2ℓ/d+(x−c)2/n+o(n2/d) +n2ℓ/d+(∆−d)c/d +(1 + δ)ℓ +�d2 +2 e2/d−5/6 +�c +. +If x > ε′n and c > ε′x, then p(ℓ, x, c) = +� e +n +� 2ℓ +d exp(−Θ(n ln n)). +If x > ε′n and c ≤ ε′x, then +p(ℓ, x, c) ≤ +�n +c +� +n(∆−d)c/d +� e +n +� 2ℓ +d (1 + δ)ℓ ≤ en1−(∆−d)c/d � e +n +� 2ℓ +d (1 + δ)ℓ. +Finally, let x ≤ ε′n. +The maximum value of +�n +c +� +(d2e2/d−5/6/2)cn−(∆−d)c/d is +achieved when c = d2 +2 e2/d−5/6n1−(∆−d)/d + O(1) and is at most +�en +c +�c +(d2e2/d−5/6/2)cn−(∆−d)c/d ≤ exp +�d2 +2 e2/d−5/6n1−(∆−d)/d + O(1) +� +implying that +p(ℓ, x, c) ≤ +� e +n +� 2ℓ +d (1 + δ)ℓe +d2 +2 n1−(∆−d)/d. +Let σ ≥ ε′x. Then, for some large enough C > 0, +p(ℓ, x, c) ≤ +�n +c +� +[(n − x + c)! + O(1)] +|Fn| +23x+σ+2d(d+5)σ +�d +2 +�a +(2γ)c−a +≤ n−2ℓ/dCx22d(d−5)σ +n2σ/d += o(n−2ℓ/d). +Summing up, for every ℓ > ℓ0, +πℓ = +� +x,c +p(x, ℓ, c) ≤ n2 � e +n +� 2ℓ +d (1 + δ)ℓe +d2 +2 n1−(∆−d)/d. +Therefore (1), (2) and (3) imply that +EX +M ≤ +� +ℓ>ℓ0 +n3 +�1 + δ +1 + ε +�ℓ +e +d2 +2 n1−(∆−d)/d = o +�1 +n +� +. +18 + +7.2 +d ≥ 5: sharp threshold for a good sequence +Now, we assume that d ≥ 5, Fn is good, ℓ0 = 0 and ℓ ≥ 1. In this case ˜σ ≥ +(c − a)(d − ∆/2). Therefore, +(∆ − d)c +d ++ 2σ +d ≥ c +� +1 − 2 +d +� ++ +˜σ +d − ∆/2. +In the same way as above, if x > ε′n and c > ε′x, then p(ℓ, x, c) = +� e +n +� 2ℓ +d exp(−Θ(n ln n)). +If x > ε′n and c ≤ ε′x, then p(ℓ, x, c) ≤ exp +� +n(∆−d)c/d� � e +n +� 2ℓ +d (1 + δ)ℓ. +Finally, let x ≤ ε′n. Assume first that c − a ≤ ε′x. Then +p(ℓ, x, c) ≤ +�n +c +� +[(n − x + c)! + O(1)] +|Fn| +�˜σ + c +c − a +��d +2 +�c +(1 + δ/2)ℓ22d(d+5)˜σ +max +o≤(d+4)˜σ +�x +o +�2 +≤ +�n +c +� +e(x−c)2/n+o(n2/d) +n2ℓ/d+(∆−d)c/d+2σ/d +�˜σ + c +c − a +��d +2 +�c +(1 + δ/2)ℓ22d(d+5)˜σ +max +o≤(d+4)˜σ +�x +o +�2 +≤ +� e +n +�2ℓ/d +e(d +2)(n/e)2/d e(x−c)2/n+o(n2/d) +n˜σ/(d−∆/2) +�˜σ + c +c − a +� +(1 + δ/2)ℓ22d(d+5)˜σ +max +o≤(d+4)˜σ +�x +o +�2 +≤ +� e +n +�2ℓ/d +e(d +2)(n/e)2/d+o(n2/d)(1 + δ)ℓ. +Let c − a > ε′x. Then ˜σ > ε′x(d − ∆/2). Therefore, +p(ℓ, x, c) ≤ +�n +c +� +[(n − x + c)! + O(1)] +|Fn| +2x+˜σ +�d +2 +�c +(2γ)c−a22d(d+5)˜σ +max +o≤(d+4)˜σ +�x +o +�2 +≤ +�n +c +� +e(x−c)2/n+o(n2/d) +n2ℓ/d+c(1−2/d)+˜σ/(d−∆/2) 23x+˜σ +�d +2 +�c +(2γ)c−a22d(d+5)˜σ +≤ +�1 +n +�2ℓ/d +�n +c +� +nc(1−2/d) (1 + δ)ℓ ≤ +�1 +n +�2ℓ/d +en2/d(1 + δ)ℓ. +Summing up, for every ℓ > 0, +πℓ = +� +x,c +p(x, ℓ, c) ≤ n2 � e +n +�2ℓ/d +e(d +2)(n/e)2/d+o(n2/d)(1 + δ)ℓ. +Therefore (1), (2) and (3) imply that (assuming that ε < 1/d) +EX +M ≤ +� +ℓ>0 +n3 +�1 + δ +1 + ε +�ℓ +e− d(1−dε) +2 +(n/e)2/d = o +�1 +n +� +. +(9) +19 + +7.3 +d = 4 +We now consider d = 4. The maximum value of +�n +c +� +(7.8/√e)cn−c/2 is achieved when +c = 7.8 +√e +√n + O(1) and is at most +�en +c +�c +(7.8/√e)cn−c/2 ≤ +�7.8√e√n +c +�c +≤ e +7.8 +√e +√n+O(1) +implying that +p(ℓ, x, c) ≤ e(x−c)2/n+o(√n) +nℓ/2+σ/2 +Υeℓ/2+σ/2e +7.8 +√e +√n. +Let us assume that σ < ε′ℓ. If 1 ≤ ℓ ≤ ε′n, then +p(ℓ, c, x) ≤ (1 + δ)ℓe +� +7.8 +√e +o(1) +�√n(e/n)ℓ/2. +If ℓ > ε′n and c > ε′x, then p(ℓ, c, x) ≤ n−ℓ/2 exp(−Ω(n ln n)). If c ≤ ε′x, then +�x−2c +c−a2 +� +≤ (1 + δ/3)ℓ. Therefore, +p(ℓ, x, c) ≤ +�n +c +� +[(n − x + c)! + O(1)] +|Fn| +(1+2δ/3)ℓ32c ≤ +�n +c +� +e(x−c)2/n+o(√n) +nℓ/2+c/2+σ/2 +(1+2δ/3)ℓ32c. +Since +�n +c +� +n−c/232c ≤ e32√n+O(1), we get +p(ℓ, x, c) ≤ e32√n+o(√n)e(x−c)2/n(1 + 2δ/3)ℓn−ℓ/2 ≤ (1 + δ)ℓ(e/n)ℓ/2. +Finally, let σ ≥ ε′ℓ. Note that this is possible only when ℓ is large enough. Since +�c +a +��x − 2c +c − a +��σ + c − a +c − a +� +6a +�8 +3 +�c−a �x +o +�2 +≤ 23x+σ3a +�8 +3 +�c−a +≤ 23x+σ3c, +we get that +p(ℓ, x, c) ≤ +�n +c +� +[(n − x + c)! + O(1)] +|Fn| +23x+73σ3c ≤ +�n +c +� +eo(√n) +nℓ/2+c/2+σ/2 e(x−c)2/n23x+73σ3c +≤ e +√n+o(√n)n−ℓ/2−σ/2ex−c23x+73σ3c ≤ e +√n+o(√n)n−ℓ/2. +Therefore, +πℓ = +� +x,c +p(x, ℓ, c) ≤ n2(1 + δ)ℓe +� +7.8 +√e +o(1) +�√n(e/n)ℓ/2. +Then (1), (2) and (3) imply (9) as needed. +20 + +7.4 +d = 3 +It remains to consider d = 3. We only need to consider the case σ < ε′ℓ, 1 ≤ ℓ ≤ ε′n. +For all the other values of the parameters, the proof is absolutely the same as for +d = 4. The maximum value of +�n +c +� +(4/e2/3)cn−c/3 is achieved when c = +4 +e2/3n2/3+O(1) +and is at most +�en +c +�c +(4/e2/3)cn−c/3 ≤ +�4e1/3n2/3 +c +�c +≤ e4(n/e)2/3+O(1) +implying that +p(ℓ, x, c) ≤ e(x−c)2/n+o(√n) +n2ℓ/3+2σ/3 +Υe2ℓ/3+2σ/3e4(n/e)2/3 ≤ (1 + δ)ℓe(4/e2/3+o(1))n2/3(e/n)2ℓ/3. +Therefore, +πℓ = +� +x,c +p(x, ℓ, c) ≤ n2(1 + δ)ℓe(4/e2/3+o(1))n2/3(e/n)2ℓ/3. +Then (1), (2) and (3) imply (9) as needed. +Acknowledgements +This work was originated when the author was a visitor at Tel Aviv University. +The author is grateful to Wojciech Samotij for his kind hospitality during the visit +and for helpful discussions. The author would like to thank Michael Krivelevich for +helpful remarks and valuable comments on the paper. +References +[1] R. Alweiss, S. Lovett, K. Wu, J. Zhang, Improved bounds for the sunflower +lemma, Ann. of Math. (2), 194:3 (2021) 795–815. +[2] B. 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B 31:2 (1981) 168–182. +22 + diff --git a/4NE2T4oBgHgl3EQf6Qhs/content/tmp_files/load_file.txt b/4NE2T4oBgHgl3EQf6Qhs/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cbb26a0e65754fb49b3a563c370c41a1b29cdf2f --- /dev/null +++ b/4NE2T4oBgHgl3EQf6Qhs/content/tmp_files/load_file.txt @@ -0,0 +1,589 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf,len=588 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='04198v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='CO] 10 Jan 2023 Sharp thresholds for spanning regular graphs Maksim Zhukovskii∗ Abstract Let d ≥ 3 be a constant and let F be a d-regular graph on [n] with not too many symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' The expectation threshold for the existence of a spanning subgraph in G(n, p) isomorphic to F is p∗(n) = (1 + o(1))(e/n)2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We give a tight bound on the edge expansion of F guaranteeing that the probability threshold for the appearance of a copy of F has the same order of magnitude as p∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We also prove that, within a slight strengthening of this bound, the probability threshold is asymptotically equal to p∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In particular, it proves the conjecture of Kahn, Narayanan and Park on a sharp threshold for the containment of a square of a Hamilton cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' It also implies that, for d ≥ 4 and (asymptotically) almost all d-regular graphs F on [n], p(n) = (e/n)2/d is a sharp threshold for F-containment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' 1 Introduction Let d ≥ 3 be a fixed constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Given a d-regular graph Fn on the vertex set [n] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , n}, what is the threshold probability to contain its isomorphic copy by the binomial random graph G(n, p) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' the unique p = p(n) such that the probability that G(n, p) contains an isomorphic copy of Fn equals 1/2)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Note that the threshold probability exists since the considered property is monotone [7, Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' If Fn has a small enough automorphism group, then, by the union bound, the threshold probability is at least (1 + o(1))(e/n)2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Indeed, let Fn be the set of all isomorphic copies of Fn on [n], and let the number of automorphisms of Fn be eo(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Clearly |Fn| = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' eo(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let X be the number of graphs from Fn that are subgraphs of G(n, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We get EX = |Fn|pdn/2 = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' eo(n)pdn/2 → 0 as n → ∞ if p < (1 − ε) � e n �2/d, implying that with high probability (whp for brevity) G(n, p) does not contain any graph from Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let us denote by p∗(n) = (1+o(1))(e/n)2/d the ∗The University of Sheffield;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' zhukmax@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='com 1 expectation threshold for the existence of a spanning subgraph in G(n, p) isomorphic to Fn (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' p∗(n) is the unique solution of the equation EX = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' On the other hand, from the recently resolved “expectation–threshold” conjec- ture of Kahn and Kalai [16] it follows that the threshold does not exceed Cp∗(n) log n for some constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' For some specific Fn it is known that the logarithmic fac- tor can be removed, and the threshold probability equals Θ(n−2/d): it is true for example for powers of a Hamilton cycle [15] and for the square tori T√n×√n [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' On the other hand, if Fn has many small subgraphs with a small edge boundary, this is no longer true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' More precisely, assume that, for some constant v, every vertex of Fn belongs to a subgraph on v vertices with the edge boundary at most d (the edge boundary of a subgraph ˜F is the number of edges between ˜F and its vertex complement) or, equivalently, with at least dv 2 − d 2 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Then, a polylogarithmic factor arises since in order to contain a copy of Fn, the random graph should have every vertex inside a graph with v vertices and at least dv 2 − d 2 edges — see [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We prove that, when the number of automorphisms of Fn is small enough, this condition on the edge boundary is the only obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let d ≥ 3 and let Fn be a sequence of d-regular graphs on [n], n ∈ N, such that for every ε > 0 and all large enough n the number of automorphisms of Fn is less then eεn2/d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' for every ˜F ⊂ Fn with 3 ≤ |V ( ˜F)| ≤ n − 3, the edge boundary of ˜F is at least d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' If p > (1 + ε)dp∗, then whp (assuming that dn is even) G(n, p) contains a copy of Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' It immediately implies that the threshold probability for containing a copy of Fn equals p(n) = Θ(n−2/d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' As we mentioned above, the restriction on edge boundaries is tight — if we allow subgraphs with edge boundary d instead of d + 1, then the assertion becomes false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Note that a bound on the number of symmetries can not be omitted — as soon as the number of automorphisms of Fn becomes larger, the expectation threshold p∗ becomes larger as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In particular, p(n) = (d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' log n) 2 d(d+1) n−2/(d+1) is a sharp threshold for the existence of a Kd+1-factor [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In [15] Riordan proved a general result that for d-regular graphs can be stated as follows: p(n) = Θ(n−2/d) is the threshold probability for containing a copy of Fn if the d-regular graph Fn (the automorphism group should be at most exponential 2 in n) satisfies a stronger condition on the edge boundary: for every ˜F ⊂ Fn with 3 ≤ |V ( ˜F)| ≤ n−3, the edge boundary of ˜F is at least 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' For powers of a Hamilton cycle, this result implies the following: for every k ≥ 3, the threshold probability for containing the kth power of a Hamilton cycle equals Θ(n−1/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' However, the proof of Riordan does not work for k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In [10], K¨uhn and Osthus proved that n−1/2+o(1) is the threshold probability for containing the second power of a Hamilton cycle and conjectured that the threshold is actually Θ(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In [13], Nenadov and ˇSkori´c proved the upper bound n−1/2(log n)4, which was improved to n−1/2(log n)3 by Fischer, ˇSkori´c, Steger and Truji´c in [4], and to n−1/2(log n)2 in an unpublished work of Montgomery (see [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Eventually, the conjecture was solved by Kahn, Narayanan and Park in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' However, they did non settle a right constant in front of n−1/2 and conjectured that the right constant is √e and that the threshold p(n) = � e/n(1 + o(1)) is sharp (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=', if p > (1 + ε) � e/n, then whp G(n, p) contains the second power of a Hamilton cycle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In this paper, we prove this conjecture and even more: for d ≤ 4 the requirement from Theorem 1 guarantees that p(n) = (1 + o(1)) � e/n is even sharp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' however, for d ≥ 5 we need to strengthen the bound on the edge boundary to 2d − 2 (note that this is still better than the condition of Riordan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let d ≥ 3 and let Fn be a sequence of d-regular graphs on [n], n ∈ N, such that for every ε > 0 and all large enough n the number of automorphisms of Fn is less then eεn2/d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' either d ∈ {3, 4} and, for every ˜F ⊂ Fn with 3 ≤ |V ( ˜F)| ≤ n − 3, the edge boundary of ˜F is at least d + 1, or d ≥ 5 and, for every ˜F ⊂ Fn with 3 ≤ |V ( ˜F)| ≤ n − 3, the edge boundary of ˜F is at least 2d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' If p > (1+ε) � e n �2/d, then whp (assuming that dn is even) G(n, p) contains a copy of Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Kahn, Narayanan and Park in [8] noted that the crucial fact that can be used to prove that the threshold for appearance of the second power of a Hamilton cycle equals Θ(n−1/2) is that the hypergraph of all copies of the second power of a cycle on [n] is (1 + o(1)) � e/n-spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Actually, they refined the notion of spreadness by incorporating the count of the number of components in a subhyperedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' This re- fined notion was distilled by D´ıaz and Person in [3], named superspreadness and used to generalise the result of Kahn, Narayanan and Park to a wider class of spanning subgraphs in G(n, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In particular, they answered a question of Frieze asked in [5] — they showed that the threshold for appearance of spanning 2-overlapping 4-cycles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' the copies of C4 are ordered cyclically, two consecutive C4 overlap in exactly 3 one edge, whereby each C4 overlaps with two copies of C4 in opposite edges) equals Θ(n−2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Clearly, Theorem 2 implies that p(n) = (e/n)2/3 is a sharp threshold for appearance of spanning 2-overlapping 4-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let us call a sequence of d-regular graphs on [n] satisfying the conditions of Theorem 2 good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Note that, for every d ≥ 4, almost all d-regular graphs are good (see [2, 9, 11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In particular, if d ≥ 5, then whp in a random d-regular graph on [n] there are no subgraphs with 3 ≤ v ≤ n − 3 vertices and the edge boundary at most 2d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' If d = 4, then whp there are no subgraphs with 3 ≤ v ≤ n − 3 vertices and the edge boundary at most d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' If d = 3, then whp there are no subgraphs with 3 ≤ v ≤ n − 3 vertices and the edge boundary at most d + 1 other than C3, C4 and their vertex-complements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Since the edge boundary of C4 is exactly d + 1 = 4, a random 3-regular graph is good whp under the condition that it does not contain triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' For every good sequence Fn, p(n) = � e n �2/d is a sharp threshold for containing a copy of Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In particular, for every ℓ ≥ 2, p(n) = (e/n)ℓ is a sharp threshold for containing the ℓth power of a Hamilton cycle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' p(n) = (e/n)2/3 is a sharp threshold for containing a spanning 2-overlapping 4-cycle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' for every 3 ≤ m ≤ √n, p(n) = � e/n is a sharp threshold for containing rectangular tori Tm×n/m (assuming that n is divisible by m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' for every d ≥ 4 and (asymptotically) almost all d-regular graphs Fn on [n], as- suming that dn is even, p(n) = (e/n)2/d is a sharp threshold for Fn-containment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' for (asymptotically) almost all triangle-free 3-regular graphs Fn on [n], assum- ing that n is odd, p(n) = (e/n)2/3 is a sharp threshold for Fn-containment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Actually we are able to establish the same sharp threshold for almost all 3-regular graphs — the condition of the absence of triangles is redundant, since the number of triangles converges in probability to a Poisson random variable [19], and so it is bounded in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In other words, we may allow Fn to have a bounded number of subgraphs with a smaller edge boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' However, we do not want to overload the proof with technical details, and so we formulate Theorem 1 and Theorem 2 as well as Corollary 1 in their current laconic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We prove Theorem 1 using the “planted trick” that in different forms appears in many applications — one of them is the well-known and very useful “spread 4 lemma” [1] which in particular gives good sunflower bounds [18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' in probabilistic terms the application of the trick for the “spread lemma” is described in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Kahn, Narayanan and Park [8] and further D´ıaz and Person [3] used the “planted trick” to prove their results on threshold probabilities as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In essence, the key idea is to “plant” a graph F from the family Fn and to combine it with the noise produced by G(n, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Then, it turns out that whp there exists a graph F ′ ∈ Fn which is entirely inside the perturbed planted hyperedge F ∪ G(n, p) such that the size of F ′ \\ G(n, p) is quite small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' This allows to replace Fn with the set of fragments of F ∈ Fn equal to F ′ \\ G(n, p), to draw independently edges of another G(n, p) and to apply the same argument once again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' If the number of steps in this procedure is bounded by a constant, then we get that the threshold probability has the same order of magnitude as p∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In the proof of Theorem 2 we show that it is sufficient to apply this trick only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Actually the usual second moment method (but for the uniform model instead of the binomial) works as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' However, we give the proof of Theorem 2 in terms of the planted hyperedge for the sake of convenience and coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In particular, we want to explicitly show the borders between the following three phenomena: 1) it is sufficient to apply the “planted trick” once, 2) it is sufficient to apply the “planted trick” constantly many times, 3) the number of applications of the trick is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We claim that our analysis is optimal, and the method in its current form cannot be used to weaken the bound on edge boundaries in Theorem 2 for d ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Our main achievement is that we make a step beyond the usage of the notions of spreadness and superspreadness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We obtain optimal bounds on the number of hyperedges containing a given set of edges I (commonly denoted by |Fn ∩ ⟨I⟩|) and on the number of subgraphs of Fn with a fixed number of vertices, edges and components (see Section 5 and Claim 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' The main ingredient of the proof of Claim 6 is a very nice property of d-regular graphs satisfying the requirements of Theorem 1: for every v, there are not too many subgraphs on v vertices with the maximum possible number of edges dv 2 − � d+1 2 � (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We describe the “planted trick” in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Then we prove both theorems in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Sections 6 and 7 are devoted to the proof of Claim 6 and the key lemma (Lemma 3 from Section 3) that validates the application of the planted trick respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' 2 Linearly many closed subgraphs Let us call a graph Fn with the second property from the requirement (on the edge boundary) in Theorem 1 locally sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Note that this (local sparsity) property is that the edge boundary of every subgraph ˜F with 3 ≤ |V ( ˜F)| ≤ n − 3 is at least d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Clearly d + 1 can be replaced with d + 2 for even d since in this case the edge boundary δ( ˜F) cannot be odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let ∆ = d+1 for odd d and ∆ = d+2 for even d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' It 5 is easy to see that the condition |δ( ˜F)| ≥ ∆ holds for all ˜F with 2 ≤ |V ( ˜F)| ≤ d − 1 just due to the d-regularity of Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let us call a subgraph ˜F with the edge boundary exactly ∆ closed (note that a closed subgraph is always connected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' For j < d, let us call a vertex w of a connected subgraph ˜F ⊂ Fn j-free, if its degree in ˜F equals j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' w is simply free, if it is j-free for some j < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let F be a locally sparse d-regular graph on [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Every closed subgraph of F with at least 3 vertices has minimum degree at least d/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Assume that ˜F is a closed subgraph of F with a vertex w having degree d′ < d/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' If we remove the vertex w from ˜F, then we get the graph ˜F \\ w with edge boundary δ(F) + 2d′ −d < δ(F) = ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' This contradicts the local sparsity of F when |V ( ˜F)| ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Otherwise it contradicts the fact that a subgraph on 2 vertices has the edge boundary at least 2d − 2 ≥ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' For any pair of adjacent vertices x, y in F and for every 3 ≤ v ≤ n − 3, there are at most two closed subgraphs in F on v vertices containing x and not containing y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Fix adjacent vertices x, y and 3 ≤ v ≤ n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' A closed graph ˜F ⊂ F sends exactly ∆ edges to F \\ ˜F implying that F \\ ˜F is also closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Assume that v ≥ n/2, and that there are at least 3 closed graphs on v vertices that share x and do not contain y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Then their complements are closed graphs on n − v ≤ n/2 vertices that share y and do not share x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Therefore, it is sufficient to prove the claim for v ≤ n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let H1, H2 be different closed subgraphs of F on v vertices that contain x and do not contain y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Note that H1, H2 should have at least one other common vertex since otherwise the degree of x is bigger than d due to Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Then |V (H1) ∪ V (H2)| ≤ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let H0 = H1 ∩ H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Note that |E(H0)| ≤ d 2|V (H0)| − ∆ 2 implying that |E(Hj) \\ E(H0)| ≥ d 2|V (Hj \\ H0)| for both j = 1 and j = 2 since H1, H2 are closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' On the other hand, if, say |E(H2) \\ E(H0)| > d 2|V (H2 \\ H0)|, then |E(H1 ∪ H2)| > d 2|V (H1∪H2)|− ∆ 2 which contradicts the local sparsity of F since |V (H1∪H2)| ≤ n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Therefore, |E(Hj) \\ E(H0)| = d 2|V (Hj \\ H0)| for both j = 1 and j = 2, but then H0 is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Then, there are exactly ∆ edges between H0 and F \\ H0, and one of them is the edge between x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' It means that Hj \\ H0, j ∈ {1, 2}, send at most ∆ − 1 edges (in total) to H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' This may happen only if |V (Hj \\ H0)| = 1 for both j = 1 and j = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Indeed, |V (H1 \\ H0)| = |V (H2 \\ H0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Moreover, the number of edges that 6 Hj \\ H0 sends to H0 equals |E(Hj) \\ E(H0)| − |E(Hj \\ H0)| ≥ d 2|V (Hj \\ H0)| − �d 2|V (Hj \\ H0)| − ∆ 2 � = ∆ 2 whenever |V (Hj \\ H0)| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Assume that there exists a closed graph H3 ̸⊂ H1∪H2 on v vertices that contains x and does not contain y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' From the above it follows that H3 ∩ H1 = H3 ∩ H2 = H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Each vertex of Hj \\ H0 sends at least d 2 edges to H0 due to Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' But then the vertices from Hj \\ H0 send at least 3d 2 ≥ ∆ edges to H0 — a contradiction (since there is one additional edge {x, y} in the edge boundary of H0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Therefore, any other closed graph that contains x and does not contain y should be entirely inside H1 ∪ H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Assume that such a graph H3 exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let w1 ∈ H1 \\ H0, w2 ∈ H2 \\ H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Clearly, H3 contains w1, w2 and all but 1 vertex of H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In the same way as above we get that H1 ∩ H2 = H0, H1 ∩ H3 and H2 ∩ H3 are three closed graphs on v − 1 vertices that contain x and do not contain y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' These three closed graphs on v − 1 vertices have the property that none of them is inside the union of the other two — this is only possible when v − 1 = 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' v = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' The only possible closed graph on 3 vertices is a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Moreover, a triangle is closed only when d = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' So, H1, H2 are triangles sharing an edge, but then H3 adds another edge to the union H1 ∪ H2 implying that H1 ∪ H2 ∪ H3 is a 4-clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We get a contradiction with the local sparsity since the edge boundary of a 4-clique is 4 < ∆ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' From this, it immediately follows, that for every v, there are at most Cn closed subgraphs on v vertices in F for a certain universal constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' More precisely, the following is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let k ∈ N, and let F ′ be the induced subgraph of F on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' For every 3 ≤ v ≤ n − 3, the number of closed subgraphs of F ′ with v vertices is at most 2dk 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Fix a vertex w in F ′ and let us bound the number (denoted by µ(w)) of closed subgraphs of F ′ on v vertices containing w such that the vertex w is free in these graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Due to Claim 2, µ(w) ≤ 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' On the other hand, Claim 1 implies that every closed subgraph contains at least 3 free vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Letting f to be the number of closed subgraphs in F ′ on v vertices, by double counting, we get that 3f ≤ � w∈V (F ′) µ(w) ≤ 2dk as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' For d = 3, 4, we need sharper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let us start from d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let d = 3, k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let F ′ be the induced subgraph of F on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Then for every 3 ≤ v ≤ n − 3, there are at most 3 4k closed subgraphs in F ′ on v vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Fix a vertex w in F ′ and let us compute the number µ(w) of closed subgraphs of F ′ on v vertices containing w such that the vertex w is free in these graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Reviewing the proof of Claim 2, we may see that in the case d = 3, every vertex x may be inside only a single closed subgraph on v vertices that does not contain another vertex y — otherwise H1 \\ H0 sends at least 2 edges to H0, and the same for H2 \\ H0 implying that H0 cannot be closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Then, for every w, µ(w) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' On the other hand, Claim 1 implies that every closed subgraph contains at least 4 free vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Letting f to be the number of closed subgraphs in F ′ on v vertices, by double counting, we get that 4f ≤ � w∈V (F ′) µ(w) ≤ 3k as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' For d = 4, we get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let d = 4, k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let F ′ be the induced subgraph of F on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Then for every 3 ≤ v ≤ n − 3, there are at most 4 3k closed subgraphs in F ′ on v vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Fix a vertex w in F ′ and let us compute the number of closed subgraphs of F ′ on v vertices containing w such that the vertex w is free in these graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let µj(w) be the number of closed subgraphs on v vertices such that w is j-free in these graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Due to Claim 1, µ1(w) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Due to Claim 2, µ3(w) + 2µ2(w) ≤ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' On the other hand, letting f to be the number of closed subgraphs in F ′ on v vertices, since every closed graph has the edge boundary equal to 6, we get that 6f is exactly the number of pairs (a closed graph ˜F, an edge from the boundary of ˜F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Therefore, 6f = � w(µ3(w) + 2µ2(w)) ≤ 8k implying that the number of closed graphs on v vertices is at most 4 3k as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' 3 Planted hyperedge Let dn be even, Fn be a d-regular graph on [n] satisfying the requirements of Theorem 1, and let F be a uniformly chosen random element of Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let ε > 0, w = (1 + ε + o(1))(e/n)2/d�n 2 � be a sequence of integers and W be a random graph on [n] with w edges chosen uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In this section, we review the constructions and follow the terminology from [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' For the sake of convenience, we give the argument in full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Our achievement is Lemma 3 that we state in the end of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Fix a non-negative integer ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let us call a pair (F ∈ Fn, W ⊂ �[n] 2 � ) ℓ0-bad, if for every ℓ0-subset L ⊂ F (we hereinafter assume that F is the set of edges), we have that L ⊔ [W \\ F] does not contain a graph from Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Fix t ∈ � 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , dn 2 � and let w′ = w − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Note that, for F ∈ Fn, W ⊂ �[n] 2 � , such that |F ∩ W| = t, we have |F ∪ W| = dn 2 + w − t = w′ + dn 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' 8 Call Z ∈ � ([n] 2 ) w′+dn/2 � ℓ0-pathological if |{F ⊂ Z : F ∈ Fn, (F, Z) is ℓ0-bad}| > 1 n|Fn| ��n 2 � − dn/2 w′ � / � �n 2 � w′ + dn/2 � =: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Note that P(|F ∩ W| = t) = �dn/2 t ���n 2 � − dn/2 w′ � / ��n 2 � w � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We have therefore P((F, W) is ℓ0-bad, F ∪ W is not ℓ0-pathological | |F ∩ W| = t) = P((F, W) is ℓ0-bad, F ∪ W is not ℓ0-pathological, |F ∩ W| = t) P(|F ∩ W| = t) ≤ � (n 2) w′+dn/2 � |{F ⊂ Z : F ∈ Fn, (F, Z) is ℓ0-bad}| �dn/2 t � |Fn| �(n 2) w ��dn/2 t ��(n 2)−dn/2 w′ � / �(n 2) w � ≤ 1 n, where Z is a not ℓ0-pathological hyperedge from � ([n] 2 ) w′+dn/2 � with maximum possible value of |{F ⊂ Z : F ∈ Fn, (F, Z) is ℓ0-bad}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Fix F ∈ Fn and a set B ⊂ F of size t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let W′ be chosen uniformly at random from �([n] 2 )\\F w′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Note that if F ∪ W′ is ℓ0-pathological, then there are at least M subgraphs from Fn in F ∪ W′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' On the other hand, if (F, W′) is ℓ0-bad, then, for every F ′ ∈ Fn such that F ′ ⊂ F ∪ W′, |F ′ ∩ F| = |F ′ \\ W′| ≥ ℓ0 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let X count the number of F ′ ∈ Fn such that F ′ ⊂ F ∪ W′ and |F ′ ∩ F| ≥ ℓ0 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We get that the event “(F, W′) is ℓ0-bad and F ∪ W′ is ℓ0-pathological” implies X ≥ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Then P((F, W) is ℓ0-bad, F ∪ W is ℓ0-pathological | |F ∩ W| = t) = P((F, W) is ℓ0-bad, F ∪ W is ℓ0-pathological | F = F, F ∩ W = B) = P((F, W′) is ℓ0-bad, F ∪ W′ is ℓ0-pathological) ≤ P(X ≥ M) ≤ EX M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' For ℓ ≥ ℓ0 + 1, let πℓ := P(|F ∩ F| = ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Then EX = � ℓ≥ℓ0+1 |Fn|πℓ � w′ dn/2 − ℓ � / ��n 2 � − dn/2 dn/2 − ℓ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' (1) 9 We have EX M = � ℓ≥ℓ0+1 |Fn| πℓ M � w′ dn/2 − ℓ � / ��n 2 � − dn/2 dn/2 − ℓ � = n � ℓ≥ℓ0+1 πℓ � w′ dn/2−ℓ � / �(n 2)−dn/2 dn/2−ℓ � �(n 2)−dn/2 w′ � / � (n 2) w′+dn/2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' (2) Note that � w′ dn/2−ℓ � / �(n 2)−dn/2 dn/2−ℓ � �(n 2)−dn/2 w′ � / � (n 2) w′+dn/2 � ∼ w′2w′( �n 2 � − dn + ℓ)(n 2)−dn+ℓ�n 2 �(n 2) (w′ − dn/2 + ℓ)w′−dn/2+ℓ(w′ + dn/2)w′+dn/2( �n 2 � − dn/2)2(n 2)−dn < e− (dn/2−ℓ)2 2w − (dn/2)2 2w +O(1) �� n (1 + ε)e �2/d�ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' (3) In Section 7, we prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' If one of the following two conditions hold ℓ0 = � d2 2 ln(1+ε/2)n1−(∆−d)/d� , or Fn is good and ℓ0 = 0, then EX M ≤ n � ℓ≥ℓ0+1 πℓe− (dn/2−ℓ)2 2w − (dn/2)2 2w �� n (1 + ε)e �2/d�ℓ = o �1 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' (4) 4 Proofs of Theorems 1, 2 Lemma 3 implies Theorem 2 immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' It remains to prove Theorem 1 for d ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let p > (1 + ε)d � e n �2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We use the first assertion of Lemma 3 for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Consider d independent copies G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , Gd of G(n, p′), p′ = (1 + ε) � e n �2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' For every F ∈ Fn, consider a minimum possible R = R(F) ⊂ F such that R ∪ G1 contains a graph from Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' By Lemma 3 whp |R| ≤ d2 2 ln(1+ε/2)n1−(∆−d)/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let Σ be the set of all F ∈ Fn such that |R(F)| ≤ d2 2 ln(1+ε/2)n1−(∆−d)/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We have that E|Σ| = (1 − o(1))|Fn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' By Markov’s inequality, P � |Σ| ≤ |Fn| 2 � = P � |Fn| − |Σ| ≥ |Fn| 2 � ≤ 2(|Fn| − E|Σ|) |Fn| → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' 10 Let R = {R(F) : F ∈ Σ} be a multiset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' |R| = |Σ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We may assume that all sets R ∈ R have equal cardinality exactly ℓ1 := � d2 2 ln(1+ε/2)n1−(∆−d)/d� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We then apply the same proof (as in Section 3) but for R instead of Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let ℓ0 = 1 2 � d2 ln(1+ε/2) �2 n1−2(∆−d)/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let us call a pair (R ∈ R, W ∈ �([n] 2 ) w � ) bad, if for every ℓ0-subset L ⊂ R, we have that L ⊔ [W \\ R] does not contain a graph from R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' For a fixed size of intersection t, a set Z ∈ � ([n] 2 ) w−t+ℓ1 � is pathological if |{R ⊂ Z : R ∈ R, (R, Z) is bad}| > 1 n|R| ��n 2 � − ℓ1 w − t � / � �n 2 � w − t + ℓ1 � =: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In order to show that for a fixed R ∈ R, whp in R∪G2 there exists a subset R′ ∈ R such that |R′ \\ G2| ≤ ℓ0, it is sufficient to prove an analogue of the first assertion of Lemma 3: let W′ be chosen uniformly at random from �([n] 2 )\\R w−t � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' X be the number of R′ ∈ R such that R′ ⊂ R ∪ W′ and |R′ ∩ R| ≥ ℓ0 + 1, then EX/M = o(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Note that an analogue of the first inequality in (4) holds true with dn/2 replaced by ℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Note that R has at most 2ℓ1 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Defining p(ℓ, x, c) in the same way as in Section 5 and applying Claim 6, we get p(ℓ, x, c) ≤ max a α2ℓ1(a, ℓ, x, c)β(a, ℓ, x, c) |R| ≤ �2ℓ1 c � [(n − x + c)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' + O(1)] |R| × × max a �x − 2c c − a ��c a ��˜σ + c c − a ��d 2 �a �4d 3 �c−a 22d(d+5)˜σ max o≤(d+4)˜σ �x o �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Note that this bound differs from (8) only in the first binomial coefficient with n replaced by 2ℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Therefore, applying the same arguments as in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='1, we get that, for every ℓ > ℓ0, πℓ = � x,c p(x, ℓ, c) ≤ n2 � e n � 2ℓ d (1 + δ)ℓe d2 2 d2 ln(1+ε/2) n1−2(∆−d)/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Therefore the analogues of (1), (2) and (3) imply that EX M ≤ � ℓ>ℓ0 n3 �1 + δ 1 + ε �ℓ e d2 2 d2 2 ln(1+ε/2) n1−2(∆−d)/d = o �1 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Applying repeatedly the whole argument d ∆−d − 1 ≤ d − 1 times, we arrive to fragments of graphs from Fn of sizes at most ℓd−1 = � 1 2 � d2 ln(1+ε/2) �d−1 n(∆−d)/d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' 11 Defining R (whp |R| ≥ |Fn|/2d−1) in a usual way as the multiset of fragments of size exactly ℓd−1, letting M := 1 n|R| ��n 2 � − ℓd−1 w − t � / � �n 2 � w − t + ℓd−1 � , considering a fixed fragment R, a uniformly chosen W′ ∈ �([n] 2 )\\R w−t � , and defining X as the number of R′ ∈ R such that R′ ⊂ R ∪ W′ and |R′ ∩ R| ≥ 1, it remains to show that EX/M = o(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We then consider p(ℓ, x, c) and apply Claim 6: p(ℓ, x, c) ≤ max a α2ℓd−1(a, ℓ, x, c)β(a, ℓ, x, c) |R| ≤ �2ℓd−1 c � [(n − x + c)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' + O(1)] |R| × × max a �x − 2c c − a ��c a ��˜σ + c c − a ��d 2 �a �4d 3 �c−a 22d(d+5)˜σ max o≤(d+4)˜σ �x o �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In the same way as in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='1, the only non-trivial case is σ < ε′x, x < ε′n, where 0 < ε′ ≪ δ is small enough (otherwise, p(ℓ, x, c) is even smaller).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In this case, for large enough constant C = C(d), p(ℓ, x, c) ≤ 2d−1 �2ℓd−1 c � e2ℓ/d+(x−c)2/n+o(n2/d) n2ℓ/d+(∆−d)c/d (1 + δ/2)ℓ �d2 2 e2/d−5/6 �c ≤ 2d−1 �� d2 ln(1 + ε/2) �d−1 e c �c e2ℓ/d+(x−c)2/n+o(n2/d) n2ℓ/d (1 + δ/2)ℓ �d2 2 e2/d−5/6 �c ≤ C e2ℓ/d+o(n2/d) n2ℓ/d (1 + δ)ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Finally, for every ℓ ≥ 1, πℓ = � x,c p(x, ℓ, c) ≤ n2C e2ℓ/d+o(n2/d) n2ℓ/d (1 + δ)ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Therefore the analogues of (1), (2) and (3) imply that EX M ≤ � ℓ>ℓ0 n3 �1 + δ 1 + ε �ℓ e−Θ(n2/d) = o �1 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' 5 Spread We here follow the notations of Section 3: F ∈ Fn is fixed, F ∈ Fn is chosen uniformly at random, and πℓ = P(|F ∩ F| = ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' 12 Fix c ∈ [ℓ], x ∈ � 2ℓ d + ∆ d c, ℓ + c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Denote σ := d 2x − � ℓ + ∆ 2 c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let p(ℓ, x, c) be the probability that the intersection of F with F is a graph on x vertices with ℓ edges and c connected components (we think about graphs as about sets of their edges, so there are no isolated vertices in |F∩F|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let integers ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , ℓc and x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , xc be chosen in a way such that 2ℓi d + ∆ d ≤ xi ≤ ℓi + 1 for all i ∈ [c];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' �c i=1 ℓi = ℓ, �c i=1 xi = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let p(ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , ℓc, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , xc) be the probability that the intersection of F with F con- sists of c connected components R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , Rc such that |V (Ri)| = xi, |E(Ri)| = ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Clearly, p(ℓ, x, c) = � ℓi,xi p(ℓi, xi, i ∈ [c]), (5) where the summation is over all unordered choices of ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , ℓc, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Note that, in the case of the ordered choice, the number of ways to choose the values of xi ≥ 2 is at most �x−c c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' The number of ways to choose the respective ℓi is at most �σ+c c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We will use the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let ˜F be a subgraph of F on k vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We assume that either k = n and ˜F = F, or k ≪ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' The number of ways to choose a subgraph R1 ⊔ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' ⊔ Rc from ˜F without isolated vertices with x vertices, ℓ edges and c components such that a is the number of Ri that are either an edge or a full vertex-complement to an edge (that comprises n − 2 vertices and dn/2 − (2d − 1) edges) is αk(a, ℓ, x, c) ≤ �k c ��x − 2c c − a ��c a ��˜σ + c c − a � �d 2 �a γc−a max o≤(d+4)˜σ �x o � 2d(d+5)˜σ, (6) where γ = 2d 3 I(d ≥ 5) + 4 3I(d = 4) + 3 4I(d = 3), ˜σ = σ − a(d − 1 − ∆/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Given disjoint non-trivial connected R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , Rc ⊂ ˜F such that their union has x vertices and ℓ edges, and there are exactly a graphs Ri that are either an edge or a full vertex-complement to an edge, the number of ways to extend R1 ⊔ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' ⊔ Rc to a graph from Fn is β(a, ℓ, x, c) ≤ (n − x + c)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' (d − 1)a2c−a max o≤(d+4)˜σ �x o � 2d(d+4)˜σ + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' (7) 13 6 Proof of Claim 6 Fix ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , ℓc and x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let us compute the number of ways to choose connected vertex-disjoint subgraphs R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , Rc from ˜F with the respective numbers of edges and vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let us call Ri dense, if one of the following holds: 1) xi = 2 and ℓi = 1, 2) Ri is closed, 3) xi = n − 2 and ℓi = d 2n − (2d − 1), 4) xi = n − 1 and ℓi = d 2n − d, 5) xi = n and ℓi = d 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' For i ∈ [c], set σi = d 2xi − ℓi − ∆ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Note that a is the number of i such that xi = 2 and ℓi = 1, or xi = n − 2 and ℓi = d 2n − (2d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let us call the respective Ri edge-components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' The number of ways to choose i ∈ [c] such that Ri is an edge component equals �c a � , while the number of ways to choose the values of the remaining xi is at most �x−2c c−a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' The number of ways to choose the respective ℓi is at most �˜σ+c c−a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We first choose dense graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' If c = 1, x1 = n − 1 and ℓ1 = d 2n − d, then there are exactly n ways to choose R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' If c = 1, x1 = n and ℓ1 = 2n, there is only one way to choose R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Otherwise, we first choose edge-components Ri: for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , a, the jth edge is chosen out of the set of kj remaining vertices in dkj 2 ways, and then kj−1 ≤ kj −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' After that, we choose all the remaining dense graphs from R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Note that the remaining dense graphs from R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , Rc are closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Assume that we want to choose a closed Rj, and kj is the number of remaining vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Then by Claims 3, 4, and 5, the number of ways to choose Rj is at most γkj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' After that, kj − |V (Rj)| ≤ kj − 3 vertices remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Assuming that c − ˜c is the number of dense Rj, we get that there are at most � d 2 �a γc−˜c−a n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' (n−(c−˜c))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' number of ways to choose dense subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Without loss of generality, we assume that it remains to choose R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , R˜c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Note that the component Ri might have at most ∆ + 2σi free vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' For ev- ery i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , ˜c, we expose Ri in ˜F in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Assume that F ′ is the current graph (obtained by removing all the chosen subgraphs R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , Ri−1 and R˜c+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , Rc) on k′ vertices, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' choose the number of free vertices oi ≤ d + 2 + 2σi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' choose the iterations of the below algorithm (out of the total number of iter- ations xi) that produce a free vertex of Ri in �xi oi � ways;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' choose a vertex w in F ′ which is minimum in Ri (here we mean the ordering of the vertices of Ri induced by the ordering of the vertices of F ′) in at most k′ ways, and activate it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' at every step, choose the minimum vertex (in the ordering of the vertices from F ′) in the set of active vertices: 14 if it should be free (in accordance to the above choice), then add to Ri some of its neighbours (in at most 2d ways), deactivate it and activate all its chosen neighbours, if it should not be free, then add to Ri all its neighbours, deactivate the vertex and activate all its neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We get that the number of ways to choose Ri is at most k′(d+2+2σi) max oi≤d+2+2σi �xi oi � 2doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Eventually we get that the number of ordered choices of components with pa- rameters ℓi, xi, i ∈ [c], in ˜F is at most c!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' �k c � �d 2 �a γc−˜c−a ˜c� i=1 (d + 2 + 2σi) max oi≤d+2+2σi �xi oi � 2doi ≤ c!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' �k c � �d 2 �a γc−˜c−a max o≤(d+4)˜σ �x o � 2d(d+5)˜σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Note that this bound does not depend on the order of the choice of all these com- ponents R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , Rc, thus it implies (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let us now fix connected subgraphs R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , Rc of F as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let us bound the number of ways to extend R1 ⊔ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' ⊔ Rc to an F ′ ∈ Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We construct such an extension in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' First, let us consider the two special cases: 1) c = 1, x1 ≥ n − 2 and R1 is dense;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' 2) c = 2, x1 = 2, x2 = n − 2 and R1, R2 are dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In the first case, if x1 = n − 2, then there are at most �2(d−1) d−1 � ways to construct F ′ (the remaining 2 vertices should be adjacent — so we should only draw missing edges in constantly many ways);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' if x1 ≥ n − 1, then there is a unique way to construct F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In the second case, there are also at most �2(d−1) d−1 � ways to draw F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We then forget the labels of the vertices from R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , Rc and assume (without loss of generality) that the desired F ′ ∈ Fn is defined on [n] in a way such that every i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , n} has a neighbour in [i − 1], denote one such neighbour (chosen randomly) by ν(i) (note that the graph F ′ is connected due to the restriction on edge boundaries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let H be obtained from F by deleting all the edges that do not belong to R1 ⊔ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' ⊔ Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Then let Z be the set of all components in H (together with the isolated vertices), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' |Z| = n − x + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We should compute the number of ways to embed the elements of Z in F ′ disjointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , zn−x+c be an ordering of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' At every step i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , n−x+c, consider the minimum vertex κi of F ′ such that none of the embedded elements of Z in F ′ contain this vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' If zi /∈ {R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' , Rc}, then we assign κi with zi and proceed with the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Otherwise, we distinguish between the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We let zi = R1 without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' 15 First, we assume that R1 is dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' If |V (R1)| = 2, then there are at most d − 1 ways to choose the image of the edge R1, since the edge {κi, ν(κi)} is already ‘occupied’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' If 2 < |V (R1)| < n − 2, then due to Claim 2, there are at most 2 ways to choose a copy of R1 in F ′ containing κi and not containing ν(κi), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Second, let R1 be not dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Choose the iterations of the below algorithm (out of the total number of iterations xi) that produce a free vertex of Ri in �xi oi � ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Activate κi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' At every step, choose the minimum vertex (in the ordering of the vertices from F ′) in the set of active vertices: if it should be free (in accordance to the above choice), then add to the image of R1 under construction some of its neighbours (in at most 2d ways), deactivate it and activate all its neighbours, if it should not free, then add all its neighbours, deactivate the vertex and activate all its neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We get that the number of ways to construct the image of R1 is at most �xi oi � 2doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Eventually we get that there are at most (n − x + c)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' (d − 1)a2c−˜c−a ˜c� i=1 max oi≤d+2+2σi �xi oi � 2doi + O(1) ≤ (n − x + c)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' (d − 1)a2c−˜c−a max o≤(d+4)˜σ �x o � 2d(d+4)˜σ + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' ways to expose F ′ as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' 7 Proof of Lemma 3 Summing up, from (5), (6) and (7), we get that p(ℓ, x, c) ≤ max a αn(a, ℓ, x, c)β(a, ℓ, x, c) |Fn| ≤ �n c � [(n − x + c)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' + O(1)] |Fn| × × max a �x − 2c c − a ��c a ��˜σ + c c − a ��d 2 �a (2γ)c−a22d(d+5)˜σ max o≤(d+4)˜σ �x o �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' (8) We let Υ := max a �˜σ + c c − a � 22d(d+5)˜σ max o≤(d+4)˜σ �x o �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let us find the maximum value of ϕ(x) = �x−2c c−a � e∆c/d−x as a function of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Since ϕ(x + 1) ϕ(x) = 1 e � 1 + c − a x + 1 − 3c + a � , 16 we get that the maximum value is achieved at x = 2c+(c−a) e e−1 +O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Therefore, p(ℓ, x, c) ≤ �n c � e2σ/d+2ℓ/d+(x−c)2/n+o(n2/d) n2ℓ/d+(∆−d)c/d+2σ/d Υ× × (2γ)c max a �c a � �d(d − 1) 4γ �a �� (c − a) e e−1 � c − a � e−(2−∆/d)c−(c−a) e e−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Since �c a � �d(d − 1) 4γ �a �� (c − a) e e−1 � c − a � e−(c−a) e e−1 ≤ �c a � �d(d − 1) 4γ �a �2 e � e e−1 (c−a) ≤ � d(d − 1) 4γ + �2 e �e/(e−1)�c , we eventually get p(ℓ, x, c) ≤ �n c � e2σ/d+2ℓ/d+(x−c)2/n+o(n2/d) n2ℓ/d+(∆−d)c/d+2σ/d Υ � 4d 3 e−(2−∆/d) � 3(d − 1) 8 + �2 e �e/(e−1)��c ≤ �n c � e2σ/d+2ℓ/d+(x−c)2/n+o(n2/d) n2ℓ/d+(∆−d)c/d+2σ/d Υ �d2 2 e2/d−5/6 �c for all d ≥ 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' p(ℓ, x, c) ≤ �n c � eσ/2+ℓ/2+(x−c)2/n+o(√n) nℓ/2+c/2+σ/2 Υ � 8 3√e � 9 4 + �2 e �e/(e−1)��c ≤ �n c � eσ/2+ℓ/2+(x−c)2/n+o(√n) nℓ/2+c/2+σ/2 Υ �7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='8 √e �c for d = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' p(ℓ, x, c) ≤ �n c � e2σ/3+2ℓ/3+(x−c)2/n+o(n2/3) n2ℓ/3+c/3+2σ/3 Υ � 3 2e2/3 � 2 + �2 e �e/(e−1)��c ≤ �n c � e2σ/3+2ℓ/3+(x−c)2/n+o(n2/3) n2ℓ/3+c/3+2σ/3 Υ � 4 e2/3 �c for d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' From now on, we separately proof three assertions of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='1 d ≥ 5: existence of a fragment First we assume that d ≥ 5, ℓ0 = d2 2 ln(1+ε/2)n1−(∆−d)/d, ℓ > ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let δ > 0 be small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Choose ε′ = ε′(δ) > 0 small enough in a way such that σ < ε′x implies 17 Υ ≤ (1 + δ/3)ℓ and c < ε′x implies �x−2c c−a ��c a ��d 2 �a(2γ)c−a ≤ (1 + δ/3)ℓ for all a as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Assume that σ < ε′x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In this case, p(ℓ, x, c) ≤ �n c � e2ℓ/d+(x−c)2/n+o(n2/d) n2ℓ/d+(∆−d)c/d (1 + δ)ℓ �d2 2 e2/d−5/6 �c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' If x > ε′n and c > ε′x, then p(ℓ, x, c) = � e n � 2ℓ d exp(−Θ(n ln n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' If x > ε′n and c ≤ ε′x, then p(ℓ, x, c) ≤ �n c � n(∆−d)c/d � e n � 2ℓ d (1 + δ)ℓ ≤ en1−(∆−d)c/d � e n � 2ℓ d (1 + δ)ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Finally, let x ≤ ε′n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' The maximum value of �n c � (d2e2/d−5/6/2)cn−(∆−d)c/d is achieved when c = d2 2 e2/d−5/6n1−(∆−d)/d + O(1) and is at most �en c �c (d2e2/d−5/6/2)cn−(∆−d)c/d ≤ exp �d2 2 e2/d−5/6n1−(∆−d)/d + O(1) � implying that p(ℓ, x, c) ≤ � e n � 2ℓ d (1 + δ)ℓe d2 2 n1−(∆−d)/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let σ ≥ ε′x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Then, for some large enough C > 0, p(ℓ, x, c) ≤ �n c � [(n − x + c)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' + O(1)] |Fn| 23x+σ+2d(d+5)σ �d 2 �a (2γ)c−a ≤ n−2ℓ/dCx22d(d−5)σ n2σ/d = o(n−2ℓ/d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Summing up, for every ℓ > ℓ0, πℓ = � x,c p(x, ℓ, c) ≤ n2 � e n � 2ℓ d (1 + δ)ℓe d2 2 n1−(∆−d)/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Therefore (1), (2) and (3) imply that EX M ≤ � ℓ>ℓ0 n3 �1 + δ 1 + ε �ℓ e d2 2 n1−(∆−d)/d = o �1 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' 18 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='2 d ≥ 5: sharp threshold for a good sequence Now, we assume that d ≥ 5, Fn is good, ℓ0 = 0 and ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In this case ˜σ ≥ (c − a)(d − ∆/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Therefore, (∆ − d)c d + 2σ d ≥ c � 1 − 2 d � + ˜σ d − ∆/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' In the same way as above, if x > ε′n and c > ε′x, then p(ℓ, x, c) = � e n � 2ℓ d exp(−Θ(n ln n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' If x > ε′n and c ≤ ε′x, then p(ℓ, x, c) ≤ exp � n(∆−d)c/d� � e n � 2ℓ d (1 + δ)ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Finally, let x ≤ ε′n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Assume first that c − a ≤ ε′x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Then p(ℓ, x, c) ≤ �n c � [(n − x + c)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' + O(1)] |Fn| �˜σ + c c − a ��d 2 �c (1 + δ/2)ℓ22d(d+5)˜σ max o≤(d+4)˜σ �x o �2 ≤ �n c � e(x−c)2/n+o(n2/d) n2ℓ/d+(∆−d)c/d+2σ/d �˜σ + c c − a ��d 2 �c (1 + δ/2)ℓ22d(d+5)˜σ max o≤(d+4)˜σ �x o �2 ≤ � e n �2ℓ/d e(d 2)(n/e)2/d e(x−c)2/n+o(n2/d) n˜σ/(d−∆/2) �˜σ + c c − a � (1 + δ/2)ℓ22d(d+5)˜σ max o≤(d+4)˜σ �x o �2 ≤ � e n �2ℓ/d e(d 2)(n/e)2/d+o(n2/d)(1 + δ)ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let c − a > ε′x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Then ˜σ > ε′x(d − ∆/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Therefore, p(ℓ, x, c) ≤ �n c � [(n − x + c)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' + O(1)] |Fn| 2x+˜σ �d 2 �c (2γ)c−a22d(d+5)˜σ max o≤(d+4)˜σ �x o �2 ≤ �n c � e(x−c)2/n+o(n2/d) n2ℓ/d+c(1−2/d)+˜σ/(d−∆/2) 23x+˜σ �d 2 �c (2γ)c−a22d(d+5)˜σ ≤ �1 n �2ℓ/d �n c � nc(1−2/d) (1 + δ)ℓ ≤ �1 n �2ℓ/d en2/d(1 + δ)ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Summing up, for every ℓ > 0, πℓ = � x,c p(x, ℓ, c) ≤ n2 � e n �2ℓ/d e(d 2)(n/e)2/d+o(n2/d)(1 + δ)ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Therefore (1), (2) and (3) imply that (assuming that ε < 1/d) EX M ≤ � ℓ>0 n3 �1 + δ 1 + ε �ℓ e− d(1−dε) 2 (n/e)2/d = o �1 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' (9) 19 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='3 d = 4 We now consider d = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' The maximum value of �n c � (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='8/√e)cn−c/2 is achieved when c = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='8 √e √n + O(1) and is at most �en c �c (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='8/√e)cn−c/2 ≤ �7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='8√e√n c �c ≤ e 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='8 √e √n+O(1) implying that p(ℓ, x, c) ≤ e(x−c)2/n+o(√n) nℓ/2+σ/2 Υeℓ/2+σ/2e 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='8 √e √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Let us assume that σ < ε′ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' If 1 ≤ ℓ ≤ ε′n, then p(ℓ, c, x) ≤ (1 + δ)ℓe � 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='8 √e +o(1) �√n(e/n)ℓ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' If ℓ > ε′n and c > ε′x, then p(ℓ, c, x) ≤ n−ℓ/2 exp(−Ω(n ln n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' If c ≤ ε′x, then �x−2c c−a2 � ≤ (1 + δ/3)ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Therefore, p(ℓ, x, c) ≤ �n c � [(n − x + c)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' + O(1)] |Fn| (1+2δ/3)ℓ32c ≤ �n c � e(x−c)2/n+o(√n) nℓ/2+c/2+σ/2 (1+2δ/3)ℓ32c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Since �n c � n−c/232c ≤ e32√n+O(1), we get p(ℓ, x, c) ≤ e32√n+o(√n)e(x−c)2/n(1 + 2δ/3)ℓn−ℓ/2 ≤ (1 + δ)ℓ(e/n)ℓ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Finally, let σ ≥ ε′ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Note that this is possible only when ℓ is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Since �c a ��x − 2c c − a ��σ + c − a c − a � 6a �8 3 �c−a �x o �2 ≤ 23x+σ3a �8 3 �c−a ≤ 23x+σ3c, we get that p(ℓ, x, c) ≤ �n c � [(n − x + c)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' + O(1)] |Fn| 23x+73σ3c ≤ �n c � eo(√n) nℓ/2+c/2+σ/2 e(x−c)2/n23x+73σ3c ≤ e √n+o(√n)n−ℓ/2−σ/2ex−c23x+73σ3c ≤ e √n+o(√n)n−ℓ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Therefore, πℓ = � x,c p(x, ℓ, c) ≤ n2(1 + δ)ℓe � 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='8 √e +o(1) �√n(e/n)ℓ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Then (1), (2) and (3) imply (9) as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' 20 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content='4 d = 3 It remains to consider d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' We only need to consider the case σ < ε′ℓ, 1 ≤ ℓ ≤ ε′n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' For all the other values of the parameters, the proof is absolutely the same as for d = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' The maximum value of �n c � (4/e2/3)cn−c/3 is achieved when c = 4 e2/3n2/3+O(1) and is at most �en c �c (4/e2/3)cn−c/3 ≤ �4e1/3n2/3 c �c ≤ e4(n/e)2/3+O(1) implying that p(ℓ, x, c) ≤ e(x−c)2/n+o(√n) n2ℓ/3+2σ/3 Υe2ℓ/3+2σ/3e4(n/e)2/3 ≤ (1 + δ)ℓe(4/e2/3+o(1))n2/3(e/n)2ℓ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Therefore, πℓ = � x,c p(x, ℓ, c) ≤ n2(1 + δ)ℓe(4/e2/3+o(1))n2/3(e/n)2ℓ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Then (1), (2) and (3) imply (9) as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' Acknowledgements This work was originated when the author was a visitor at Tel Aviv University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' The author is grateful to Wojciech Samotij for his kind hospitality during the visit and for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' The author would like to thank Michael Krivelevich for helpful remarks and valuable comments on the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQf6Qhs/content/2301.04198v1.pdf'} +page_content=' References [1] R.' metadata={'source': 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a/4tAzT4oBgHgl3EQf9v5K/content/tmp_files/2301.01923v1.pdf.txt b/4tAzT4oBgHgl3EQf9v5K/content/tmp_files/2301.01923v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4e4e0f7075aace881fb5723df5fe9d87a34ab590 --- /dev/null +++ b/4tAzT4oBgHgl3EQf9v5K/content/tmp_files/2301.01923v1.pdf.txt @@ -0,0 +1,696 @@ +Two-dimensional Heisenberg models with materials-dependent superexchange +interactions +Jia-Wen Li,1 Zhen Zhang,2 Jing-Yang You,3 Bo Gu,1, 4, 5, ∗ and Gang Su1, 4, 5, 6, † +1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijng 100049, China +2Key Laboratory of Multifunctional Nanomaterials and Smart Systems, +Division of Advanced Materials, Suzhou Institute of Nano-Tech and Nano-Bionics, +Chinese Academy of Sciences, Suzhou, 215123 China +3Department of Physics, National University of Singapore, Science Drive, Singapore 117551 +4CAS Center for Excellence in Topological Quantum Computation, +University of Chinese Academy of Sciences, Beijng 100190, China +5Physical Science Laboratory, Huairou National Comprehensive Science Center, Beijing 101400, China +6School of Physical Sciences, University of Chinese Academy of Sciences, Beijng 100049, China +The two-dimensional (2D) van der Waals ferromagnetic semiconductors, such as CrI3 and +Cr2Ge2Te6, and the 2D ferromagnetic metals, such as Fe3GeTe2 and MnSe2, have been obtained in +recent experiments and attracted a lot of attentions. The superexchange interaction has been sug- +gested to dominate the magnetic interactions in these 2D magnetic systems. In the usual theoretical +studies, the expression of the 2D Heisenberg models were fixed by hand due to experiences. Here, we +propose a method to determine the expression of the 2D Heisenberg models by counting the possible +superexchange paths with the density functional theory (DFT) and Wannier function calculations. +With this method, we obtain a 2D Heisenberg model with six different nearest-neighbor exchange +coupling constants for the 2D ferromagnetic metal Cr3Te6, which is very different for the crystal +structure of Cr atoms in Cr3Te6. The calculated Curie temperature Tc = 328 K is close to the +Tc = 344 K of 2D Cr3Te6 reported in recent experiment. In addition, we predict two stable 2D +ferromagnetic semiconductors Cr3O6 and Mn3O6 sharing the same crystal structure of Cr3Te6. The +similar Heisenberg models are obtained for 2D Cr3O6 and Mn3O6, where the calculated Tc is 218 +K and 208 K, respectively. Our method offers a general approach to determine the expression of +Heisenberg models for these 2D magnetic semiconductors and metals, and builds up a solid basis +for further studies. +I. +INTRODUCTION +Recently, the successful synthesis of two-dimensional +(2D) van der Waals ferromagnetic semiconductors in ex- +periments, such as CrI3 [1] and Cr2Ge2Te6 [2] has at- +tracted extensive attentions to 2D ferromagnetic materi- +als. According to Mermin-Wagner theorem [3], the mag- +netic anisotropy is essential to produce the long-range +magnetic order in 2D systems. For the 2D magnetic semi- +conductors obtained in experiments, the Curie tempera- +ture Tc is still much lower than room temperature. For +example, Tc = 45 K in CrI3 [1], 30 K in Cr2Ge2Te6 [2], +34 K in CrBr3 [4], 17 K in CrCl3 [5], 75 K in Cr2S3 [6, 7], +etc. For applications, the ferromagnetic semiconductors +with Tc higher than room temperature are highly re- +quired [8–11]. On the other hand, the 2D van der Waals +ferromagnetic metals with high Tc have been obtained in +recent experiments. For example, Tc = 140 K in CrTe +[12], 300 K in CrTe2 [13, 14], 344 K in Cr3Te6 [15], 160 +K in Cr3Te4 [16], 280 K in CrSe [17], 300 K in Fe3GeTe2 +[18, 19], 270 K in Fe4GeTe2 [20], 229 K in Fe5GeTe2 +[21, 22], 300 K in MnSe2 [23], etc. +In these 2D van der Waals ferromagnetic materials, the +superexchange interaction has been suggested to domi- +∗ gubo@ucas.ac.cn +† gsu@ucas.ac.cn +nate the magnetic interactions. The superexchange in- +teraction describes the indirect magnetic interaction be- +tween two magnetic cations mediated by the neighboring +non-magnetic anions [24–26]. The superexchange inter- +action has been discussed in the 2D magnetic semicon- +ductors. +Based on the superexchange interaction, the +strain-enhanced Tc in 2D ferromagnetic semiconductor +Cr2Ge2Se6 can be understood by the decreased energy +difference between the d electrons of cation Cr atoms +and the p electrons of anion Se atoms [27]. The similar +superexchange picture was obtained in several 2D ferro- +magnetic semiconductors, including the great enhance- +ment of Tc in bilayer heterostructures Cr2Ge2Te6/PtSe2 +[28], the high Tc in technetium-based semiconductors +TcSiTe3, TcGeSe3 and TcGeTe3 [29], and the electric +field enhanced Tc in the monolayer MnBi2Te4 [30]. The +superexchange interaction has also been discussed in +the semiconductor heterostructure CrI3/MoTe2 [31], and +2D semiconductor Cr2Ge2Te6 with molecular adsorption +[32]. +In addition, the superexchange interaction has also +been obtained in the 2D van der Waals ferromagnetic +metals. By adding vacancies, the angles of the superex- +change interaction paths of 2D metals VSe2 and MnSe2 +will change, thereby tuning the superexchange coupling +strength [33]. It is found that biaxial strain changes the +angle of superexchange paths in 2D metal Fe3GeTe2, and +affects Tc [34]. Under tensile strain, the ferromagnetism +of the 2D magnetic metal CoB6 is enhanced, due to the +arXiv:2301.01923v1 [cond-mat.mtrl-sci] 5 Jan 2023 + +2 +competition between superexchange and direct exchange +interactions [35]. +It is important to determine the spin Hamiltonian for +the magnetic materials, in order to theoretically study +the magnetic properties, such as Tc. In the usual theoret- +ical studies, the expression of the spin Hamiltonian needs +to be fixed by hand according to the experiences. By the +four-state method and density functional theory (DFT) +calculations [36–38], the exchange coupling parameters of +the spin Hamiltonian, such as the nearest neighbor, the +next nearest neighbor, inter-layer, etc, can be obtained. +Then the Tc can be estimated through Monte Carlo sim- +ulations [38]. With different spin Hamiltonians chosen by +hand, sometimes different results are obtained in calcu- +lations. Is it possible to determine the spin Hamiltonian +by the help of calculations rather than by the experiences +? +In this paper, we propose a method to establish the 2D +Heisenberg models for the 2D van der Waals magnetic +materials, when the superexchange interactions domi- +nate. +Through the DFT and Wannier function calcu- +lations, we can calculate the exchange coupling between +any two magnetic cations, by counting the possible su- +perexchange paths. +By this method, we obtain a 2D +Heisenberg model with six different nearest-neighbor ex- +change coupling constants for the 2D van der Waals fer- +romagnetic metal Cr3Te6 [15], where the calculated Tc += 328 K is close to the Tc = 344 K reported in the ex- +periment. In addition, based on the crystal structure of +2D Cr3Te6, we predict two 2D magnetic semiconductors +Cr3O6 and Mn3O6 with Tc of 218 K and 208 K, and +energy gap of 0.99 eV and 0.75 eV, respectively. +II. +COMPUTATIONAL METHODS +Our calculations were based on the DFT as im- +plemented in the Vienna ab initio simulation package +(VASP) [39]. The exchange-correlation potential is de- +scribed with the Perdew-Burke-Ernzerhof (PBE) form +of the generalized gradient approximation (GGA) [40]. +The electron-ion potential is described by the projector- +augmented wave (PAW) method [41]. +We carried out +the calculation of GGA + U with U = 3.2 eV, a rea- +sonable U value for the 3d electrons of Cr in Cr3Te6 +[15]. The band structures for 2D Cr3O6 and Mn3O6 were +calculated in HSE06 hybrid functional [42]. The plane- +wave cutoff energy is set to be 500 eV. Spin polariza- +tion is taken into account in structure optimization. To +prevent interlayer interaction in the supercell of 2D sys- +tems, the vacuum layer of 16 ˚A is included. The 5×9×1, +5×9×1 and 7×11×1 Monkhorst Pack k-point meshed +were used for the Brillouin zone (BZ) sampling for 2D +Cr3O6, Cr3Te6 and Mn3O6, respectively [43]. The struc- +tures of 2D Cr3O6 and Mn3O6 were fully relaxed, where +the convergence precision of energy and force were 10−6 +and 10−3 eV/˚A, respectively. The phonon spectra were +obtained in a 3×3×1 supercell with the PHONOPY pack- +age [44]. The Wannier90 code was used to construct a +tight-binding Hamiltonian [45, 46] to calculate the mag- +netic coupling constant. +In the calculation of molecu- +lar dynamics, a 3×4×1 supercell (108 atoms) was built, +and we took the NVT ensemble (constant-temperature, +constant-volume ensemble) and maintained a tempera- +ture of 250 K with a step size of 3 fs and a total duration +of 6 ps. +III. +Method to determine the 2D Heisenberg +model: an example of 2D Cr3Te6 +A. +Calculate exchange coupling J from +superexchange paths +The crystal structure of 2D Cr3Te6 is shown in Fig. 1, +where the space goup is Pm (No.6). In experiment, it +is a ferromagnetic metal with high Tc = 344 K [15]. To +theoretically study its magnetic properties, we considered +seven different magnetic configurations, including a ferro- +magnetic (FM) , a ferrimagnetic (FIM), and five antifer- +romagnetic (AFM) configurations, as discussed in Sup- +plemental Materials [47]. The calculation results show +that the magnetic ground state is ferromagnetic, consis- +tent with the experimental results. Since the superex- +change interaction has been suggested to dominate the +magnetic interactions in these 2D van der Waals ferro- +magnetic semiconductors and metals, we study the su- +perexchange interactions in 2D Cr3Te6. +The superexchange interaction can be reasonably de- +scried by a simple Cr-Te-Cr model [48], as shown in Fig. +2. +There are two Cr atoms at sites i and j, and one +Te atom at site k between the two Cr atoms. By the +perturbation calculation, the superexchange coupling Jij +between the two Cr atoms can be obtained as [48], +Jij =( 1 +E2 +↑↓ +− +1 +E2 +↑↑ +) +� +k,p,d +|Vik|2Jpd +kj += 1 +A +� +k,p,d +|Vik|2Jpd +kj . +(1) +The indirect exchange coupling Jij is consisting of two +processes. One is the direct exchange process between the +d electron of Cr at site j and the p electrons of Te at site +k, presented by Jpd +kj. The other is the electron hopping +process between p electrons of Te atom at site k and d +electrons of Cr atom at site i, presented by —Vik|2/A. +Vik is the hopping parameter between d electrons of Cr +atom at site i and p electrons of Te atom at site k. Here, A += 1/(1/E2 +↑↓-1/E2 +↑↑), and is taken as a pending parameter. +E↑↑ and E↑↓ are energies of two d electrons at Cr atom +at site i with parallel and antiparallel spins, respectively. +The direct exchange coupling Jpd +kj can be expressed as +[27–30]: + +3 +FIG. 1. Crystal structure of Cr3Te6 . (a) Top view (b) Side view. +Jpd +kj = +2|Vkj|2 +|Ep +k − Ed +j |. +(2) +Vkj is the hopping parameter between p electrons of +Te atom at site k and d electrons of Cr atom at site j. +Ep +k is the energy of p electrons of Te atom at site k, and +Ed +j is the energy of d electrons of Cr atom at site j. +FIG. 2. Schematic picture of superexchange interaction by +a Cr-Te-Cr model. There are two process, one is direct ex- +change process between Crj and Tek, noted as Jpd +kj, and the +other is electron hopping between Tek and Cri, noted as +|Vik|2/A. See text for details. +By the DFT and Wannier function calculations, the +parameters Vik, Vkj, Ep +k, and Ed +j in Eqs. (1) and (2) can +be calculated. The JijA can be obtained by counting all +the possible k sites of Te atoms, p orbitals of Te atoms, +and d orbitals of Cr atoms. +From the calculated results in Table I, it is suggested +that there are six different nearest-neighbor couplings, +denoted as J11, J22, J33, J12, J13, and J23, as shown in +Fig. 3(b). Accordingly, there are three kinds of Cr atoms, +noted as Cr1, Cr2, and Cr3. Based on the results in Table +I, the effective spin Hamiltonian can be written as +H =J11 +� +n +⃗S1n · ⃗S1n + J22 +� +n +⃗S2n · ⃗S2n + J33 +� +n +⃗S3n · ⃗S3n ++J12 +� +n +⃗S1n · ⃗S2n + J13 +� +n +⃗S1n · ⃗S3n + J23 +� +n +⃗S2n · ⃗S3n ++D +� +n +(S2 +1nz + S2 +2nz + S2 +3nz), +(3) +where Jij means magnetic coupling between Cri and Crj, +as indicated in Fig. +3(b). +D represents the magnetic +anisotropy energy (MAE) of Cr3Te6. +B. +Determine the parameters D and A +The single-ion magnetic anisotropy parameter DS2 can +be obtained by: DS2=(E⊥-E∥)/6, where E⊥ and E∥ are +energies of Cr3Te6 with out-of-plane and in-plane polar- +izations in FM state, respectively. It has DS2 = -0.14 +meV/Cr for 2D Cr3Te6, which is in agreement with the +value of -0.13 meV/Cr reported in the previous study of +Cr3Te6 [15]. +The parameter A can be calculated in the following +way. Considering a FM and an AFM configurations, the +total energy of Eq. (3) without MAE term can be re- +spectively expressed as [47]: +EF M = 2J11S2 +1 + 2J22S2 +2 + 2J33S2 +3 + 8J12S1S2 ++2J23S2S3 + 8J13S1S3 + E0 += 11838/A + E0, +EAF M1 = 2J11S2 +1 + 2J22S2 +2 − 2J33S2 +3 − 8J12S1S2 + E0 += −2502/A + E0. +(4) +The results in Table I are used to obtain the final ex- +pressions in Eq. (4). Since two parameters A and E0 are +kept, two spin configurations FM and AFM1 are consid- +ered here. Discussion on the choice of spin configurations + +(b) +(a) +y +Te +X +X1 +2 +Tek +Tpd +A +kj +d1 +P1 +P2 +d24 +FIG. 3. (a) The crystal structure of Cr atoms in 2D Cr3Te6. (b) The magnetic structure of Cr atoms in 2D Cr3Te6, calculated +by Eqs. (1) and (2). +TABLE I. For 2D Cr3Te6, the calculated exchange coupling parameters JijA in Eqs.(1) and (2), by the density functional +theory and Wannier functional calculations. A is a pending parameter. The unit of JijA is meV3. +J11A +J22A +J33A +J12A +J13A +J23A +40 +26 +53 +29 +44 +83 +is given in Supplemental Materials [47]. For the FM spin +configuration, the ground state of Cr3Te6, the total en- +ergy is taken as EF M = 0 for the energy reference. The +total energy of AFM1, EAF M1 = 535 meV is obtained by +the DFT calculation. The parameters A and E0 are ob- +tained by solving Eq. (4), and the six exchange coupling +parameters Jij can be obtained by Table I. The results +are given in Table II. +C. +Estimate Tc by Monte Carlo simulation +To calculate the Curie temperature, we used the Monte +Carlo program for the Heisenberg-type Hamiltonian in +Eq. (3) with parameters in Table II. The Monte Carlo +simulation was performed on a 30 +√ +3 ×30 +√ +3 lattice with +more than 1×106 steps for each temperature. The first +two-third steps were discarded, and the last one-thirds +steps were used to calculate the temperature-dependent +physical quantities. As shown in Table II and Fig. 4 (d), +the calculated Tc = 328 K for 2D Cr3Te6, close to the Tc += 344 K of 2D Cr3Te6 in the experiment [15]. Discussion +on the choice of spin configurations and the estimation of +exchange couplings Jij and Tc is given in Supplemental +Materials [47]. +IV. +Prediction of Two High Curie Temperature +Magnetic Semiconductors Cr3O6 and Mn3O6 +Inspired by the high Tc in the 2D magnetic metal +Cr3Te6, we explore the possible high Tc magnetic semi- +conductors with the same crystal structure of Cr3Te6 by +FIG. 4. (a) Band structures of Cr3O6 with a bandgap of 0.99 +eV. (b) Band structures of Mn3O6 with a bandgap of 0.75 eV. +(c) Energy gap of Cr3O6 and Mn3O6 under external electric +field out-plane. (d) The magnetic moment of Cr3Te6, Cr3O6, +and Mn3O6 varies with temperature. +the DFT calculations. We obtain two stable ferromag- +netic semiconductors Cr3O6 and Mn3O6. +In order to +study the stability of the 2D Cr3O6 and Mn3O6, we cal- +culate the phonon spectrum. As shown in Supplemental +Materials [47], there is no imaginary frequency, indicat- +ing the dynamical stability. In addition, we performed +molecular dynamics simulations of Cr3O6 and Mn3O6 +at 250 K, taking the NVT ensemble (constant temper- +ature and volume) and run for 6 ps. The results show +that 2D Cr3O6 and Mn3O6 are thermodynamically sta- + +(a) +(b) +22 +J11 +J23 +J33 +J13 +2(a) +(b) +-1.0 +-1.0 +(eV) +(eV) +-0.5 +-0.5 +- Spin up +2DCr306 +2D/Mn.06 +- Spin up +E +0.0 +0.0 + Spin down + Spin down +E +-0.5 +-0.5 +-1.0 +-1.0 +-1.5 +-1.5 +X +S +X +S +(c) +(d) +1.0 +1.4 +- 2D C +1.2 +- 2D Mn3O6 +0.8 +Exp + (eV) +1.0 +ref. 15) +0.6 +0.8 +Gap +0.6 +Mas +Cr,Te6" +0.4 +Cr,O6 +0.2 +0.2 +Mn,O6 +0.0 +-0.3 -0.2 -0.1 +0.3 +400 +0 +0.1 0.2 +100 +200 +300 +500 +Electric field (V/A) +Temepture (K)5 +TABLE II. For 2D magnetic metal Cr3Te6 and semiconductors Cr3O6 and Mn3O6, the parameter A (in unit of meV−2) in Eq. +(1), the exchange couping parameters JijS2 and the magnetic anisotropy parameter DS2 (in unit of meV) in the Hamiltonian +in Eq. (3), and the estimated Curie temperature Tc. See text for details. +Materials +A +J11S2 +J22S2 +J33S2 +J12S2 +J13S2 +J23S2 +DS2 +Tc (K) +Cr3Te6 +-27 +-17.1 +-11.5 +-24.4 +-12.6 +-19.6 +-37.4 +-0.14 +328 +Cr3O6 +-36 +-18.9 +-14.6 +-10.1 +-18.7 +-1.8 +-3.1 +0.04 +218 +Mn3O6 +-465 +-11.9 +-7.6 +-50.4 +-15.9 +-5.2 +-10.7 +-0.09 +208 +ble [47]. These calculation results suggest that 2D Cr3O6 +and Mn3O6 may be feasible in experiment. +The band structure of 2D Cr3O6 and Mn3O6 is shown +in Figs. 4(a) and 4(b), respectively, where the band gap +is 0.99 eV for Cr3O6 and 0.75 eV for Mn3O6. As shown +in Figs. 4(a) and (b), the band gap for 2D Cr3O6 and +Mn3O6 is 0.99 eV and 0.75 eV, respectively. When ap- +plying an out-of-plane electric field with a range of ± 0.3 +V/˚A, which is possible in experiment [49], the band gap +of Cr3O6 (Mn3O6) increases (decreases) with increasing +electric field, as shown in Fig. 4(c). By the same calcula- +tion method above, the parameter A, the similar Heisen- +berg models in Eq. 3 with six nearest-neighbor exchange +coupling Jij are obtained for the 2D Cr3O6 and Mn3O6. +The parameters A, Jij and D are calculated and shown +in Table II. The spin polarization of Cr3O6 and Mn3O6 +is in-plane (DS2 = 0.04 meV) and out-of-plane (DS2 = +-0.09 meV), respectively. Fig. 4(d) shows the magnetiza- +tion as a function of temperature for 2D Cr3Te6, Cr3O6 +and Mn3O6. The calculated Curie temperature is Tc = +218 K for 2D Cr3O6 and Tc = 208 K for 2D Mn3O6, +respectively. +V. +CONCLUSION +Based on the DFT and Wannier function calculations, +we propose a method for constructing the 2D Heisen- +berg model with the superexchange interactions. By this +method, we obtain a 2D Heisenberg model with six differ- +ent nearest-neighbor exchange couplings for the 2D fer- +romagnetic metal Cr3Te6. The calculated Curie temper- +ature Tc = 328 K is close to the Tc = 344 K of Cr3Te6 in +the experiment. In addition, we predicted two 2D mag- +netic semiconductors: Cr3O6 with band gap of 0.99 eV +and Tc = 218 K, and Mn3O6 with band gap of 0.75 eV +and Tc = 208 K, where the similar 2D Heisenberg models +are obtained. The complex Heisenberg model developed +from the simple crystal structure shows the power of our +method to study the magnetic properties in these 2D +magnetic metals and semiconductors. +ACKNOWLEDGEMENTS +This work is supported in part by the National Natu- +ral Science Foundation of China (Grants No. 12074378 +and No. 11834014), the Beijing Natural Science Foun- +dation (Grant No. +Z190011), the National Key R&D +Program of China (Grant No. +2018YFA0305800), the +Beijing Municipal Science and Technology Commission +(Grant No. Z191100007219013), the Chinese Academy of +Sciences (Grants No. YSBR-030 and No. Y929013EA2), +and the Strategic Priority Research Program of Chinese +Academy of Sciences (Grants No. XDB28000000 and No. +XDB33000000). +[1] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, +R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. +McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo- +Herrero, and X. Xu, Nature 546, 270 (2017). +[2] C. Gong, L. Li, Z. Li, H. 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Nan- +otechnol. 17, 1078 (2022). + diff --git a/4tAzT4oBgHgl3EQf9v5K/content/tmp_files/load_file.txt b/4tAzT4oBgHgl3EQf9v5K/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a95b1df4f53219cbeeb2d099e7bd9d1bdefd42bd --- /dev/null +++ b/4tAzT4oBgHgl3EQf9v5K/content/tmp_files/load_file.txt @@ -0,0 +1,880 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf,len=879 +page_content='Two-dimensional Heisenberg models with materials-dependent superexchange interactions Jia-Wen Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='1 Zhen Zhang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='2 Jing-Yang You,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='3 Bo Gu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' ∗ and Gang Su1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' † 1Kavli Institute for Theoretical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Beijng 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' China 2Key Laboratory of Multifunctional Nanomaterials and Smart Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Division of Advanced Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Suzhou Institute of Nano-Tech and Nano-Bionics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Suzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 215123 China 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' National University of Singapore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Science Drive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Singapore 117551 4CAS Center for Excellence in Topological Quantum Computation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Beijng 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' China 5Physical Science Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Huairou National Comprehensive Science Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Beijing 101400,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' China 6School of Physical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Beijng 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' China The two-dimensional (2D) van der Waals ferromagnetic semiconductors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' such as CrI3 and Cr2Ge2Te6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' and the 2D ferromagnetic metals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' such as Fe3GeTe2 and MnSe2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' have been obtained in recent experiments and attracted a lot of attentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The superexchange interaction has been sug- gested to dominate the magnetic interactions in these 2D magnetic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' In the usual theoretical studies, the expression of the 2D Heisenberg models were fixed by hand due to experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Here, we propose a method to determine the expression of the 2D Heisenberg models by counting the possible superexchange paths with the density functional theory (DFT) and Wannier function calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' With this method, we obtain a 2D Heisenberg model with six different nearest-neighbor exchange coupling constants for the 2D ferromagnetic metal Cr3Te6, which is very different for the crystal structure of Cr atoms in Cr3Te6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The calculated Curie temperature Tc = 328 K is close to the Tc = 344 K of 2D Cr3Te6 reported in recent experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' In addition, we predict two stable 2D ferromagnetic semiconductors Cr3O6 and Mn3O6 sharing the same crystal structure of Cr3Te6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The similar Heisenberg models are obtained for 2D Cr3O6 and Mn3O6, where the calculated Tc is 218 K and 208 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Our method offers a general approach to determine the expression of Heisenberg models for these 2D magnetic semiconductors and metals, and builds up a solid basis for further studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' INTRODUCTION Recently, the successful synthesis of two-dimensional (2D) van der Waals ferromagnetic semiconductors in ex- periments, such as CrI3 [1] and Cr2Ge2Te6 [2] has at- tracted extensive attentions to 2D ferromagnetic materi- als.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' According to Mermin-Wagner theorem [3], the mag- netic anisotropy is essential to produce the long-range magnetic order in 2D systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' For the 2D magnetic semi- conductors obtained in experiments, the Curie tempera- ture Tc is still much lower than room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' For example, Tc = 45 K in CrI3 [1], 30 K in Cr2Ge2Te6 [2], 34 K in CrBr3 [4], 17 K in CrCl3 [5], 75 K in Cr2S3 [6, 7], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' For applications, the ferromagnetic semiconductors with Tc higher than room temperature are highly re- quired [8–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' On the other hand, the 2D van der Waals ferromagnetic metals with high Tc have been obtained in recent experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' For example, Tc = 140 K in CrTe [12], 300 K in CrTe2 [13, 14], 344 K in Cr3Te6 [15], 160 K in Cr3Te4 [16], 280 K in CrSe [17], 300 K in Fe3GeTe2 [18, 19], 270 K in Fe4GeTe2 [20], 229 K in Fe5GeTe2 [21, 22], 300 K in MnSe2 [23], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' In these 2D van der Waals ferromagnetic materials, the superexchange interaction has been suggested to domi- ∗ gubo@ucas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='cn † gsu@ucas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='cn nate the magnetic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The superexchange in- teraction describes the indirect magnetic interaction be- tween two magnetic cations mediated by the neighboring non-magnetic anions [24–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The superexchange inter- action has been discussed in the 2D magnetic semicon- ductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Based on the superexchange interaction, the strain-enhanced Tc in 2D ferromagnetic semiconductor Cr2Ge2Se6 can be understood by the decreased energy difference between the d electrons of cation Cr atoms and the p electrons of anion Se atoms [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The similar superexchange picture was obtained in several 2D ferro- magnetic semiconductors, including the great enhance- ment of Tc in bilayer heterostructures Cr2Ge2Te6/PtSe2 [28], the high Tc in technetium-based semiconductors TcSiTe3, TcGeSe3 and TcGeTe3 [29], and the electric field enhanced Tc in the monolayer MnBi2Te4 [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The superexchange interaction has also been discussed in the semiconductor heterostructure CrI3/MoTe2 [31], and 2D semiconductor Cr2Ge2Te6 with molecular adsorption [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' In addition, the superexchange interaction has also been obtained in the 2D van der Waals ferromagnetic metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' By adding vacancies, the angles of the superex- change interaction paths of 2D metals VSe2 and MnSe2 will change, thereby tuning the superexchange coupling strength [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' It is found that biaxial strain changes the angle of superexchange paths in 2D metal Fe3GeTe2, and affects Tc [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Under tensile strain, the ferromagnetism of the 2D magnetic metal CoB6 is enhanced, due to the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='01923v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='mtrl-sci] 5 Jan 2023 2 competition between superexchange and direct exchange interactions [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' It is important to determine the spin Hamiltonian for the magnetic materials, in order to theoretically study the magnetic properties, such as Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' In the usual theoret- ical studies, the expression of the spin Hamiltonian needs to be fixed by hand according to the experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' By the four-state method and density functional theory (DFT) calculations [36–38], the exchange coupling parameters of the spin Hamiltonian, such as the nearest neighbor, the next nearest neighbor, inter-layer, etc, can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Then the Tc can be estimated through Monte Carlo sim- ulations [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' With different spin Hamiltonians chosen by hand, sometimes different results are obtained in calcu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Is it possible to determine the spin Hamiltonian by the help of calculations rather than by the experiences ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' In this paper, we propose a method to establish the 2D Heisenberg models for the 2D van der Waals magnetic materials, when the superexchange interactions domi- nate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Through the DFT and Wannier function calcu- lations, we can calculate the exchange coupling between any two magnetic cations, by counting the possible su- perexchange paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' By this method, we obtain a 2D Heisenberg model with six different nearest-neighbor ex- change coupling constants for the 2D van der Waals fer- romagnetic metal Cr3Te6 [15], where the calculated Tc = 328 K is close to the Tc = 344 K reported in the ex- periment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' In addition, based on the crystal structure of 2D Cr3Te6, we predict two 2D magnetic semiconductors Cr3O6 and Mn3O6 with Tc of 218 K and 208 K, and energy gap of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='99 eV and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='75 eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' COMPUTATIONAL METHODS Our calculations were based on the DFT as im- plemented in the Vienna ab initio simulation package (VASP) [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The exchange-correlation potential is de- scribed with the Perdew-Burke-Ernzerhof (PBE) form of the generalized gradient approximation (GGA) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The electron-ion potential is described by the projector- augmented wave (PAW) method [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' We carried out the calculation of GGA + U with U = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='2 eV, a rea- sonable U value for the 3d electrons of Cr in Cr3Te6 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The band structures for 2D Cr3O6 and Mn3O6 were calculated in HSE06 hybrid functional [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The plane- wave cutoff energy is set to be 500 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Spin polariza- tion is taken into account in structure optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' To prevent interlayer interaction in the supercell of 2D sys- tems, the vacuum layer of 16 ˚A is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The 5×9×1, 5×9×1 and 7×11×1 Monkhorst Pack k-point meshed were used for the Brillouin zone (BZ) sampling for 2D Cr3O6, Cr3Te6 and Mn3O6, respectively [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The struc- tures of 2D Cr3O6 and Mn3O6 were fully relaxed, where the convergence precision of energy and force were 10−6 and 10−3 eV/˚A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The phonon spectra were obtained in a 3×3×1 supercell with the PHONOPY pack- age [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The Wannier90 code was used to construct a tight-binding Hamiltonian [45, 46] to calculate the mag- netic coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' In the calculation of molecu- lar dynamics, a 3×4×1 supercell (108 atoms) was built, and we took the NVT ensemble (constant-temperature, constant-volume ensemble) and maintained a tempera- ture of 250 K with a step size of 3 fs and a total duration of 6 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Method to determine the 2D Heisenberg model: an example of 2D Cr3Te6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Calculate exchange coupling J from superexchange paths The crystal structure of 2D Cr3Te6 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 1, where the space goup is Pm (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' In experiment, it is a ferromagnetic metal with high Tc = 344 K [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' To theoretically study its magnetic properties, we considered seven different magnetic configurations, including a ferro- magnetic (FM) , a ferrimagnetic (FIM), and five antifer- romagnetic (AFM) configurations, as discussed in Sup- plemental Materials [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The calculation results show that the magnetic ground state is ferromagnetic, consis- tent with the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Since the superex- change interaction has been suggested to dominate the magnetic interactions in these 2D van der Waals ferro- magnetic semiconductors and metals, we study the su- perexchange interactions in 2D Cr3Te6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The superexchange interaction can be reasonably de- scried by a simple Cr-Te-Cr model [48], as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' There are two Cr atoms at sites i and j, and one Te atom at site k between the two Cr atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' By the perturbation calculation, the superexchange coupling Jij between the two Cr atoms can be obtained as [48], Jij =( 1 E2 ↑↓ − 1 E2 ↑↑ ) � k,p,d |Vik|2Jpd kj = 1 A � k,p,d |Vik|2Jpd kj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' (1) The indirect exchange coupling Jij is consisting of two processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' One is the direct exchange process between the d electron of Cr at site j and the p electrons of Te at site k, presented by Jpd kj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The other is the electron hopping process between p electrons of Te atom at site k and d electrons of Cr atom at site i, presented by —Vik|2/A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Vik is the hopping parameter between d electrons of Cr atom at site i and p electrons of Te atom at site k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Here, A = 1/(1/E2 ↑↓-1/E2 ↑↑), and is taken as a pending parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' E↑↑ and E↑↓ are energies of two d electrons at Cr atom at site i with parallel and antiparallel spins, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The direct exchange coupling Jpd kj can be expressed as [27–30]: 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Crystal structure of Cr3Te6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' (a) Top view (b) Side view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Jpd kj = 2|Vkj|2 |Ep k − Ed j |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' (2) Vkj is the hopping parameter between p electrons of Te atom at site k and d electrons of Cr atom at site j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Ep k is the energy of p electrons of Te atom at site k, and Ed j is the energy of d electrons of Cr atom at site j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Schematic picture of superexchange interaction by a Cr-Te-Cr model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' There are two process, one is direct ex- change process between Crj and Tek, noted as Jpd kj, and the other is electron hopping between Tek and Cri, noted as |Vik|2/A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' By the DFT and Wannier function calculations, the parameters Vik, Vkj, Ep k, and Ed j in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' (1) and (2) can be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The JijA can be obtained by counting all the possible k sites of Te atoms, p orbitals of Te atoms, and d orbitals of Cr atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' From the calculated results in Table I, it is suggested that there are six different nearest-neighbor couplings, denoted as J11, J22, J33, J12, J13, and J23, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Accordingly, there are three kinds of Cr atoms, noted as Cr1, Cr2, and Cr3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Based on the results in Table I, the effective spin Hamiltonian can be written as H =J11 � n ⃗S1n · ⃗S1n + J22 � n ⃗S2n · ⃗S2n + J33 � n ⃗S3n · ⃗S3n +J12 � n ⃗S1n · ⃗S2n + J13 � n ⃗S1n · ⃗S3n + J23 � n ⃗S2n · ⃗S3n +D � n (S2 1nz + S2 2nz + S2 3nz), (3) where Jij means magnetic coupling between Cri and Crj, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' D represents the magnetic anisotropy energy (MAE) of Cr3Te6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Determine the parameters D and A The single-ion magnetic anisotropy parameter DS2 can be obtained by: DS2=(E⊥-E∥)/6, where E⊥ and E∥ are energies of Cr3Te6 with out-of-plane and in-plane polar- izations in FM state, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' It has DS2 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='14 meV/Cr for 2D Cr3Te6, which is in agreement with the value of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='13 meV/Cr reported in the previous study of Cr3Te6 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The parameter A can be calculated in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Considering a FM and an AFM configurations, the total energy of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' (3) without MAE term can be re- spectively expressed as [47]: EF M = 2J11S2 1 + 2J22S2 2 + 2J33S2 3 + 8J12S1S2 +2J23S2S3 + 8J13S1S3 + E0 = 11838/A + E0, EAF M1 = 2J11S2 1 + 2J22S2 2 − 2J33S2 3 − 8J12S1S2 + E0 = −2502/A + E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' (4) The results in Table I are used to obtain the final ex- pressions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Since two parameters A and E0 are kept, two spin configurations FM and AFM1 are consid- ered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Discussion on the choice of spin configurations (b) (a) y Te X X1 2 Tek Tpd A kj d1 P1 P2 d24 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' (a) The crystal structure of Cr atoms in 2D Cr3Te6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' (b) The magnetic structure of Cr atoms in 2D Cr3Te6, calculated by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' For 2D Cr3Te6, the calculated exchange coupling parameters JijA in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' (1) and (2), by the density functional theory and Wannier functional calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' A is a pending parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The unit of JijA is meV3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' J11A J22A J33A J12A J13A J23A 40 26 53 29 44 83 is given in Supplemental Materials [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' For the FM spin configuration, the ground state of Cr3Te6, the total en- ergy is taken as EF M = 0 for the energy reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The total energy of AFM1, EAF M1 = 535 meV is obtained by the DFT calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The parameters A and E0 are ob- tained by solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' (4), and the six exchange coupling parameters Jij can be obtained by Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The results are given in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Estimate Tc by Monte Carlo simulation To calculate the Curie temperature, we used the Monte Carlo program for the Heisenberg-type Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' (3) with parameters in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The Monte Carlo simulation was performed on a 30 √ 3 ×30 √ 3 lattice with more than 1×106 steps for each temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The first two-third steps were discarded, and the last one-thirds steps were used to calculate the temperature-dependent physical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' As shown in Table II and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 4 (d), the calculated Tc = 328 K for 2D Cr3Te6, close to the Tc = 344 K of 2D Cr3Te6 in the experiment [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Discussion on the choice of spin configurations and the estimation of exchange couplings Jij and Tc is given in Supplemental Materials [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Prediction of Two High Curie Temperature Magnetic Semiconductors Cr3O6 and Mn3O6 Inspired by the high Tc in the 2D magnetic metal Cr3Te6, we explore the possible high Tc magnetic semi- conductors with the same crystal structure of Cr3Te6 by FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' (a) Band structures of Cr3O6 with a bandgap of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='99 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' (b) Band structures of Mn3O6 with a bandgap of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='75 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' (c) Energy gap of Cr3O6 and Mn3O6 under external electric field out-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' (d) The magnetic moment of Cr3Te6, Cr3O6, and Mn3O6 varies with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' the DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' We obtain two stable ferromag- netic semiconductors Cr3O6 and Mn3O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' In order to study the stability of the 2D Cr3O6 and Mn3O6, we cal- culate the phonon spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' As shown in Supplemental Materials [47], there is no imaginary frequency, indicat- ing the dynamical stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' In addition, we performed molecular dynamics simulations of Cr3O6 and Mn3O6 at 250 K, taking the NVT ensemble (constant temper- ature and volume) and run for 6 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The results show that 2D Cr3O6 and Mn3O6 are thermodynamically sta- (a) (b) 22 J11 J23 J33 J13 2(a) (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='0 (eV) (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='5 Spin up 2DCr306 2D/Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='06 Spin up E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='0 Spin down Spin down E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='5 X S X S (c) (d) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='4 2D C 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='2 2D Mn3O6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='8 Exp (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='0 ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 15) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='8 Gap 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='6 Mas Cr,Te6" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='4 Cr,O6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='2 Mn,O6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='3 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='2 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='3 400 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='2 100 200 300 500 Electric field (V/A) Temepture (K)5 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' For 2D magnetic metal Cr3Te6 and semiconductors Cr3O6 and Mn3O6, the parameter A (in unit of meV−2) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' (1), the exchange couping parameters JijS2 and the magnetic anisotropy parameter DS2 (in unit of meV) in the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' (3), and the estimated Curie temperature Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Materials A J11S2 J22S2 J33S2 J12S2 J13S2 J23S2 DS2 Tc (K) Cr3Te6 27 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='5 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='6 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='6 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='14 328 Cr3O6 36 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='9 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='1 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='04 218 Mn3O6 465 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='6 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='4 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='09 208 ble [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' These calculation results suggest that 2D Cr3O6 and Mn3O6 may be feasible in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The band structure of 2D Cr3O6 and Mn3O6 is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 4(a) and 4(b), respectively, where the band gap is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='99 eV for Cr3O6 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='75 eV for Mn3O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' As shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 4(a) and (b), the band gap for 2D Cr3O6 and Mn3O6 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='99 eV and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='75 eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' When ap- plying an out-of-plane electric field with a range of ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='3 V/˚A, which is possible in experiment [49], the band gap of Cr3O6 (Mn3O6) increases (decreases) with increasing electric field, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' By the same calcula- tion method above, the parameter A, the similar Heisen- berg models in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 3 with six nearest-neighbor exchange coupling Jij are obtained for the 2D Cr3O6 and Mn3O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The parameters A, Jij and D are calculated and shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The spin polarization of Cr3O6 and Mn3O6 is in-plane (DS2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='04 meV) and out-of-plane (DS2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='09 meV), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 4(d) shows the magnetiza- tion as a function of temperature for 2D Cr3Te6, Cr3O6 and Mn3O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The calculated Curie temperature is Tc = 218 K for 2D Cr3O6 and Tc = 208 K for 2D Mn3O6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' CONCLUSION Based on the DFT and Wannier function calculations, we propose a method for constructing the 2D Heisen- berg model with the superexchange interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' By this method, we obtain a 2D Heisenberg model with six differ- ent nearest-neighbor exchange couplings for the 2D fer- romagnetic metal Cr3Te6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The calculated Curie temper- ature Tc = 328 K is close to the Tc = 344 K of Cr3Te6 in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' In addition, we predicted two 2D mag- netic semiconductors: Cr3O6 with band gap of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='99 eV and Tc = 218 K, and Mn3O6 with band gap of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content='75 eV and Tc = 208 K, where the similar 2D Heisenberg models are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' The complex Heisenberg model developed from the simple crystal structure shows the power of our method to study the magnetic properties in these 2D magnetic metals and semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This work is supported in part by the National Natu- ral Science Foundation of China (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 12074378 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 11834014), the Beijing Natural Science Foun- dation (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Z190011), the National Key R&D Program of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 2018YFA0305800), the Beijing Municipal Science and Technology Commission (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Z191100007219013), the Chinese Academy of Sciences (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' YSBR-030 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Y929013EA2), and the Strategic Priority Research Program of Chinese Academy of Sciences (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' XDB28000000 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' XDB33000000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Huang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Clark, E.' metadata={'source': 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+page_content=' Yu, Nano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 19, 3138 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' [5] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Cai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Song, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Wilson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Clark, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Zhang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Taniguchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Watanabe, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Yao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Xiao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' McGuire, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Cobden, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Xu, Nano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 19, 3993 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' [6] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Cui, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Zhao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Tang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Shang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Shi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Huan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Liao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Hou, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Pen- nycook, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Zhang, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 32, 1905896 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Chu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Wen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Qiao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Wu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' He, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Yin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Cheng, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Xiong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Li, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' He, Nano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 19, 2154 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' [8] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Zhao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Dong, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Fu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Gu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Zhang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Yang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Xie, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Tang, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Ning, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Semicond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 43, 112501 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Li, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Xiong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Semicond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 42, 010302 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' [10] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Huang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Sun, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Cai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Wei, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Meng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Wu, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Zeng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Semicond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 41, 122501 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Kang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Su, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Dai, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Cheng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Han, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Zhai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Liu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Han, Nanoscale 12, 16427 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' [13] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Meng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Zhou, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Xu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Yang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 17, 778 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' [19] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Deng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Yu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Song, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Zhang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Sun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Yi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Chen, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Zhang, Nature 563, 94 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} 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Hwang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Kim, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Jang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Kim, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Eom, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Seo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Stania, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Muntwiler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Lee, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Taniguchi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Jo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Lee, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Min, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Ramesh, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Birgeneau, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 128, 1 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' May, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Ovchinnikov, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Zheng, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Hermann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Calder, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Huang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Fei, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' O’Hara, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Zhu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Trout, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Ahmed, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' K.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Goodenough and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Loeb, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 98, 391 (1955).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' [26] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Jena, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Kan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' C 123, 17987 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' [32] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Interfaces 12, 26367 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' [35] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Tang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Sun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} +page_content=' Gu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQf9v5K/content/2301.01923v1.pdf'} 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[math.CO] 12 Jan 2023 +Strengthening the Directed Brooks’ Theorem for oriented graphs +and consequences on digraph redicolouring * +Lucas Picasarri-Arrieta +Universit´e Cˆote d’Azur, CNRS, I3S, INRIA, Sophia Antipolis, France +lucas.picasarri-arrieta@inria.fr +Abstract +Let D = (V, A) be a digraph. We define ∆max(D) as the maximum of {max(d+(v), d−(v)) | v ∈ V } and +∆min(D) as the maximum of {min(d+(v), d−(v)) | v ∈ V }. It is known that the dichromatic number of D +is at most ∆min(D) + 1. In this work, we prove that every digraph D which has dichromatic number exactly +∆min(D) + 1 must contain the directed join of ←→ +Kr and ←→ +Ks for some r, s such that r + s = ∆min(D) + 1. In +particular, every oriented graph ⃗G with ∆min( ⃗G) ≥ 2 has dichromatic number at most ∆min( ⃗G). +Let ⃗G be an oriented graph of order n such that ∆min( ⃗G) ≤ 1. Given two 2-dicolourings of ⃗G, we show that +we can transform one into the other in at most n steps, by recolouring one vertex at each step while maintaining +a dicolouring at any step. Furthermore, we prove that, for every oriented graph ⃗G on n vertices, the distance +between two k-dicolourings is at most 2∆min( ⃗G)n when k ≥ ∆min( ⃗G) + 1. +We then extend a theorem of Feghali to digraphs. We prove that, for every digraph D with ∆max(D) = +∆ ≥ 3 and every k ≥ ∆ + 1, the k-dicolouring graph of D consists of isolated vertices and at most one further +component that has diameter at most c∆n2, where c∆ = O(∆2) is a constant depending only on ∆. +1 +Introduction +1.1 +Graph (re)colouring +Given a graph G = (V, E), a k-colouring of G is a function c : V −→ {1, . . ., k} such that, for every edge xy ∈ E, +we have c(x) ̸= c(y). So for every i ∈ {1, . . ., k}, c−1(i) induces an independent set on G. The chromatic number +of G, denoted by χ(G), is the smallest k such that G admits a k-colouring. The maximum degree of G, denoted +by ∆(G), is the degree of the vertex with the greatest number of edges incident to it. A simple greedy procedure +shows that, for any graph G, χ(G) ≤ ∆(G) + 1. The celebrated theorem of Brooks [6] characterizes the graphs +for which equality holds. +Theorem 1 (Brooks, [6]). A connected graph G satisfies χ(G) = ∆(G) + 1 if and only if G is an odd cycle or a +complete graph. +For any k ≥ χ(G), the k-colouring graph of G, denoted by Ck(G), is the graph whose vertices are the k- +colourings of G and in which two k-colourings are adjacent if they differ by the colour of exactly one vertex. A +path between two given colourings in Ck(G) corresponds to a recolouring sequence, that is a sequence of pairs +composed of a vertex of G, which is going to receive a new colour, and a new colour for this vertex. If Ck(G) +is connected, we say that G is k-mixing. A k-colouring of G is k-frozen if it is an isolated vertex in Ck(G). The +graph G is k-freezable if it admits a k-frozen colouring. In the last fifteen years, since the papers of Cereceda, +van den Heuvel and Johnson [8, 7], graph recolouring has been studied by many researchers in graph theory. We +refer the reader to the PhD thesis of Bartier [2] for a complete overview on graph recolouring and to the surveys +of van Heuvel [12] and Nishimura [15] for reconfiguration problems in general. Feghali [9] proved the following +analogue of Brooks’ Theorem for graphs recolouring. +*Research supported by research grant DIGRAPHS ANR-19-CE48-0013 and by the French government, through the EUR DS4H Invest- +ments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-17-EURE-0004. +1 + +Theorem 2 (Feghali, [9]). Let G = (V, E) be a connected graph with ∆(G) = ∆ ≥ 3, k ≥ ∆ + 1, and α, β two +k-colourings of G. Then at least one of the following holds: +• α is k-frozen, or +• β is k-frozen, or +• there is a recolouring sequence of length at most c∆|V |2 between α and β, where c∆ = O(∆) is a constant +depending on ∆. +1.2 +Digraph (re)dicolouring. +In this paper, we are looking for extensions of the previous results on graphs colouring and recolouring to digraphs. +Let D be a digraph. A digon is a pair of arcs in opposite directions between the same vertices. A simple arc +is an arc which is not in a digon. For any two vertices x, y ∈ V (D), the digon {xy, yx} is denoted by [x, y]. The +digon graph of D is the undirected graph with vertex set V (D) in which uv is an edge if and only if [u, v] is a +digon of D. An oriented graph is a digraph with no digon. The bidirected graph associated to a graph G, denoted +by ←→ +G , is the digraph obtained from G, by replacing every edge by a digon. The underlying graph of D, denoted +by UG(D), is the undirected graph G with vertex set V (D) in which uv is an edge if and only if uv or vu is an +arc of D. +Let v be a vertex of a digraph D. The out-degree (resp. in-degree) of a vertex v, denoted by d+(v) (resp. +d−(v)), is the number of arcs leaving (resp. entering) v. We define the maximum degree of v as dmax(v) = +max{d+(v), d−(v)}, and the minimum degree of v as dmin(v) = min{d+(v), d−(v)}. We can then define the cor- +responding maximum degrees of D: ∆max(D) = maxv∈V (D)(dmax(v)) and ∆min(D) = maxv∈V (D)(dmin(v)). +A digraph D is ∆-diregular if, for every vertex v ∈ V (D), d−(v) = d+(v) = ∆. +In 1982, Neumann-Lara [14] introduced the notions of dicolouring and dichromatic number, which generalize +the ones of colouring and chromatic number. A k-dicolouring of D is a function c : V (D) → {1, . . ., k} such +that c−1(i) induces an acyclic subdigraph in D for each i ∈ {1, . . . , k}. The dichromatic number of D, denoted +by ⃗χ(D), is the smallest k such that D admits a k-dicolouring. There is a one-to-one correspondence between +the k-colourings of a graph G and the k-dicolourings of the associated bidirected graph ←→ +G , and in particular +χ(G) = ⃗χ(←→ +G ). Hence every result on graph colourings can be seen as a result on dicolourings of bidirected +graphs, and it is natural to study whether the result can be extended to all digraphs. +The directed version of Brooks’ Theorem has been first proved by Mohar in [13], but people discovered a flaw +in the proof. Harutyunyan and Mohar then gave a stronger result in [10]. Finally, Aboulker and Aubian gave four +new proofs of the following theorem in [1]. +Theorem 3 (DIRECTED BROOKS’ THEOREM). Let D be a connected digraph. Then ⃗χ(D) ≤ ∆max(D) + 1 and +equality holds if and only if one of the following occurs: +• D is a directed cycle, or +• D is a bidirected odd cycle, or +• D is a bidirected complete graph (of order at least 4). +It is easy to prove, by induction on |V (D)|, that every digraph D can be dicoloured with ∆min(D) + 1 +colours. Hence, one can wonder if Brooks’ Theorem can be extended to digraphs using ∆min(D) instead of +∆max(D). Unfortunately, Aboulker and Aubian [1] proved that, given a digraph D, deciding whether D is +∆min(D)-dicolourable is NP-complete. Thus, unless P=NP, we cannot expect an easy characterization of digraphs +satisfying ⃗χ(D) = ∆min(D) + 1. +Let the maximum geometric mean of a digraph D be ˜∆(D) = max{ +� +d+(v)d−(v)|v ∈ V (D)}. By definition +we have ∆min(D) ≤ ˜∆(D) ≤ ∆max(D). Restricted to oriented graphs, Harutyunyan and Mohar [11] have +strengthened Theorem 3 by proving the following. +2 + +Theorem 4 (Harutyunyan and Mohar [11]). There is an absolute constant ∆1 such that every oriented graph ⃗G +with ˜∆(⃗G) ≥ ∆1 has ⃗χ(⃗G) ≤ (1 − e−13) ˜∆(⃗G). +In Section 2, we give another strengthening of Theorem 3 on a large class of digraphs which contains oriented +graphs. The directed join of H1 and H2, denoted by H1 ⇒ H2, is the digraph obtained from disjoint copies of H1 +and H2 by adding all arcs from the copy of H1 to the copy of H2 (H1 or H2 may be empty). +Theorem 5. Let D be a digraph. If D is not ∆min(D)-dicolourable, then one of the following holds: +• ∆min(D) ≤ 1, or +• ∆min(D) = 2 and D contains ←→ +K2, or +• ∆min(D) ≥ 3 and D contains ←→ +Kr ⇒ ←→ +Ks, for some r, s such that r + s = ∆min(D) + 1. +In particular, the following is a direct consequence of Theorem 5. +Corollary 6. Let D be a digraph. If ⃗χ(D) = ∆min(D) + 1, then D contains the complete bidirected graph on +� ∆min+1 +2 +� +vertices as a subdigraph. +Moreover, since an oriented graph does not contain any digon, Corollary 6 implies the following: +Corollary 7. Let ⃗G be an oriented graph. If ∆min(⃗G) ≥ 2, then ⃗χ(⃗G) ≤ ∆min(⃗G). +Corollary 6 is best possible: if we restrict D to not contain the complete bidirected graph on +� ∆min+1 +2 +� ++ 1, +then we show that deciding whether ⃗χ(D) ≤ ∆min(D) remains NP-complete (Theorem 11). +For any k ≥ ⃗χ(D), the k-dicolouring graph of D, denoted by Dk(D), is the graph whose vertices are the +k-dicolourings of D and in which two k-dicolourings are adjacent if they differ by the colour of exactly one vertex. +Observe that Ck(G) = Dk(←→ +G ) for any bidirected graph ←→ +G . A redicolouring sequence between two dicolourings is +a path between these dicolourings in Dk(D). The digraph D is k-mixing if Dk(D) is connected. A k-dicolouring of +D is k-frozen if it is an isolated vertex in Dk(D). The digraph D is k-freezable if it admits a k-frozen dicolouring. +A vertex v is blocked to its colour in a dicolouring α if, for every colour c ̸= α(v), recolouring v to c in α creates +a monochromatic directed cycle. +Digraph redicolouring was first introduced in [5], where the authors generalized different results on graph re- +colouring to digraphs, and proved some specific results on oriented graphs redicolouring. In particular, they studied +the k-dicolouring graph of digraphs with bounded degeneracy or bounded maximum average degree, and they show +that finding a redicolouring sequence between two given k-dicolouring of a digraph is PSPACE-complete. Dealing +with the maximum degree of a digraph, they proved that, given an orientation of a subcubic graph ⃗G on n ver- +tices, its 2-dicolouring graph D2(⃗G) is connected and has diameter at most 2n and they asked if this bound can be +improved. We answer this question in Section 3 by proving the following theorem. +Theorem 8. Let ⃗G be an oriented graph of order n such that ∆min(⃗G) ≤ 1. Then D2(⃗G) is connected and has +diameter exactly n. +In particular, if ⃗G is an orientation of a subcubic graph, then ∆min(⃗G) ≤ 1 (because d+(v) + d−(v) ≤ 3 for +every vertex v), and so D2(⃗G) has diameter exactly n. Furthermore, we prove the following as a consequence of +Corollary 7 and Theorem 8. +Corollary 9. Let ⃗G be oriented graph of order n with ∆min(⃗G) = ∆ ≥ 1, and let k ≥ ∆ + 1. Then Dk(⃗G) is +connected and has diameter at most 2∆n. +Corollary 9 does not hold for digraphs in general: indeed, ←→ +Pn, the bidirected path on n vertices, satisfies +∆min(←→ +Pn) = 2 and D3(←→ +Pn) = C3(Pn) has diameter Ω(n2), as proved in [4]. +In Section 4, we extend Theorem 2 to digraphs. +Theorem 10. Let D = (V, A) be a connected digraph with ∆max(D) = ∆ ≥ 3, k ≥ ∆ + 1, and α, β two +k-dicolourings of D. Then at least one of the following holds: +3 + +• α is k-frozen, or +• β is k-frozen, or +• there is a redicolouring sequence of length at most c∆|V |2 between α and β, where c∆ = O(∆2) is a +constant depending only on ∆. +Furthermore, we prove that a digraph D is k-freezable only if D is bidirected and its underlying graph is +k-freezable. Thus, an obstruction in Theorem 10 is exactly the bidirected graph of an obstruction in Theorem 2. +2 +Strengthening of Directed Brooks’ Theorem for oriented graphs +A digraph D is k-dicritical if ⃗χ(D) = k and for every vertex v ∈ V (D), ⃗χ(D − v) < k. Observe that every +digraph with dichromatic number at least k contains a k-dicritical subdigraph. +Let F2 be {←→ +K2}, and for each ∆ ≥ 3, we define F∆ = {←→ +Kr ⇒ ←→ +Ks | r, s ≥ 0 and r + s = ∆ + 1}. A digraph +D is F∆-free if it does not contain F as a subdigraph, for any F ∈ F∆. Theorem 5 can then be reformulated as +follows: +Theorem 5. Let D be a digraph with ∆min(D) = ∆ ≥ 2. If D is F∆-free, then ⃗χ(D) ≤ ∆. +Proof. Let D be a digraph such that ∆min(D) = ∆ ≥ 2 and ⃗χ(D) = ∆ + 1. We will show that D contains some +F ∈ F∆ as a subdigraph. +Let (X, Y ) be a partition of V (D) such that for each x ∈ X, d+(x) ≤ ∆, and for each y ∈ Y , d−(y) ≤ ∆. +We define the digraph ˜D as follows: +• V ( ˜D) = V (D), +• A( ˜D) = A(D⟨X⟩) ∪ A(D⟨Y ⟩) ∪ {xy, yx | xy ∈ A(D), x ∈ X, y ∈ Y }. +Claim 5.1: ⃗χ( ˜D) ≥ ∆ + 1. +Proof of claim. Assume for a contradiction that there exists a ∆-dicolouring c of ˜D. Then D, coloured with c, +must contain a monochromatic directed cycle C. Now C is not contained in X nor Y , for otherwise C would be a +monochromatic directed cycle of D⟨X⟩ or D⟨Y ⟩ and so a monochromatic directed cycle of ˜D. Thus C contains +an arc xy from X to Y . But then, [x, y] is a monochromatic digon in ˜D, a contradiction. +♦ +Since ⃗χ( ˜D) ≥ ∆ + 1, there is a (∆ + 1)-dicritical subdigraph H of ˜D. By dicriticality of H, for every vertex +v ∈ V (H), d+ +H(v) ≥ ∆ and d− +H(v) ≥ ∆, for otherwise a ∆-dicolouring of H − v could be extended to H by +choosing for v a colour which is not appearing in its out-neighbourhood or in its in-neighbourhood. We define XH +as X ∩ V (H) and YH as Y ∩ V (H). Note that both H⟨XH⟩ and H⟨YH⟩ are subdigraphs of D. +Claim 5.2: H is ∆-diregular. +Proof of claim. Let ℓ be the number of digons between XH and YH in H. Observe that, by definition of X and +H, for each vertex x ∈ XH, d+ +H(x) = ∆. Note also that, in H, ℓ is exactly the number of arcs leaving XH and +exactly the number of arcs entering XH. We get: +∆|XH| = +� +x∈XH +d+ +H(x) += ℓ + |A(H⟨XH⟩)| += +� +x∈XH +d− +H(x) +which implies, since H is dicritical, d+ +H(x) = d− +H(x) = ∆ for every vertex x ∈ XH. Using a symmetric argument, +we prove that ∆|YH| = � +y∈YH d+ +H(y), implying d+ +H(y) = d− +H(y) = ∆ for every vertex y ∈ YH. +♦ +4 + +Since H is ∆-diregular, then in particular ∆max(H) = ∆. Hence, because ⃗χ(H) = ∆ + 1, by Theorem 3, +either ∆ = 2 and H is a bidirected odd cycle, or ∆ ≥ 3 and H is the bidirected complete graph on ∆ + 1 vertices. +• If ∆ = 2 and H is a bidirected odd cycle, then at least one digon of H belongs to H⟨XH⟩ or H⟨YH⟩, for oth- +erwise H would be bipartite (with bipartition (XH, YH)). Since both H⟨XH⟩ and H⟨YH⟩ are subdigraphs +of D, this shows, as desired, that D contains a copy of ←→ +K2. +• If k ≥ 3 and H is the bidirected complete graph on ∆ + 1 vertices, let AH be all the arcs from YH to XH. +Then D⟨V (H)⟩ \ AH is a subdigraph of D which belongs to F∆. +Now we will justify that Corollary 6 is best possible. To do so, we prove that given a digraph D which does +not contain the bidirected complete graph on +� +∆min(D)+1 +2 +� ++ 1 vertices, deciding if it is ∆min(D)-dicolourable is +NP-complete. We shall use a reduction from k-DICOLOURABILITY which is defined as follows: +k-DICOLOURABILITY +Input: A digraph D +Question: Is D k-dicolourable ? +k-DICOLOURABILITY is NP-complete for every fixed k ≥ 2 [3]. It remains NP-complete when we restrict to +digraphs D with ∆min(D) = k [1]. +Theorem 11. For all k ≥ 2, k-DICOLOURABILITY remains NP-complete when restricted to digraphs D satisfying +∆min(D) = k and not containing the bidirected complete graph on +� k+1 +2 +� ++ 1 vertices. +Proof. Let D = (V, A) be an instance of k-DICOLOURABILITY for some fixed k ≥ 2. Then we build D′ = +(V ′, A′) as follows: +• For each vertex x ∈ V , we associate a copy of S− +x ⇒ S+ +x where S− +x is the bidirected complete graph on +� k+1 +2 +� +vertices, and S+ +x is the bidirected complete graph on +� k+1 +2 +� +vertices. +• For each arc xy ∈ A, we associate all possible arcs x+y− in A′, such that x+ ∈ S+ +x and y− ∈ S− +y . +First observe that ∆min(D′) = k. Let v be a vertex of D′, if v belongs to some S+ +x , then d−(v) = k, otherwise +it belongs to some S− +x and then d+(v) = k. Then observe that D′ does not contain the bidirected complete graph +on +� k+1 +2 +� ++ 1 vertices since every digon in D′ is contained in some S+ +x or S− +x . Thus we only have to prove that +⃗χ(D) ≤ k if and only if ⃗χ(D′) ≤ k to get the result. +• Let us first prove that ⃗χ(D) ≤ k implies ⃗χ(D′) ≤ k. +Assume that ⃗χ(D) ≤ k. Let φ : V −→ {1, . . . , k} be a k-dicolouring of D. Let φ′ be the k-dicolouring of +D′ defined as follows: for each vertex x ∈ V , choose arbitrarily x− ∈ S− +x , x+ ∈ S+ +x , and set φ′(x−) = +φ′(x+) = φ(x). Then choose a distinct colour for every other vertex v in S− +x ∪ S+ +x , and set φ′(v) to +this colour. We get that φ′ must be a k-dicolouring of D′: for each x ∈ V , every vertex but x− in S− +x +must be a sink in its colour class, and every vertex but x+ in S+ +x must be a source in its colour class. +Thus if D′, coloured with φ′, contains a monochromatic directed cycle C′, then C′ must be of the form +x− +1 x+ +1 x− +2 x+ +2 · · · x− +ℓ x+ +ℓ x− +1 . But then C = x1x2 · · · xℓx1 is a monochromatic directed cycle in D coloured +with φ: a contradiction. +• Reciprocally, let us prove that ⃗χ(D′) ≤ k implies ⃗χ(D) ≤ k. +Assume that ⃗χ(D′) ≤ k. Let φ′ : V ′ −→ {1, . . ., k} be a k-dicolouring of D′. Let φ be the k-dicolouring +of D defined as follows. For each vertex x ∈ V , we know that |S+ +x ∪ S− +x | = k + 1, thus there must be +two vertices x+ and x− in S+ +x ∪ S− +x such that φ′(x+) = φ′(x−). Moreover, since both S+ +x and S− +x are +bidirected, one of these two vertices belongs to S+ +x and the other one belongs to S− +x . We assume without +loss of generality x+ ∈ S+ +x and x− ∈ S− +x . Then we set φ(x) = φ′(x+). We get that φ must be a k- +dicolouring of D. If D, coloured with φ, contains a monochromatic directed cycle C = x1x2 · · · xℓx1, then +C′ = x− +1 x+ +1 x− +2 x+ +2 · · · x− +ℓ x+ +ℓ x− +1 is a monochromatic directed cycle in D′ coloured with φ′, a contradiction. +5 + +3 +Redicolouring oriented graphs +In this section, we restrict to oriented graphs. We first prove Theorem 8, let us restate it. +Theorem 8. Let ⃗G be an oriented graph of order n such that ∆min(⃗G) ≤ 1. Then D2(⃗G) is connected and has +diameter exactly n. +Observe that, if D2(⃗G) is connected, then its diameter must be at least n: for any 2-dicolouring α, we can +define its mirror ¯α where, for every vertex v ∈ V (⃗G), α(v) ̸= ¯α(v); then every redicolouring sequence between α +and ¯α has length at least n. +Lemma 12. Let C be a directed cycle of length at least 3. Then D2(C) is connected and has diameter exactly n. +Proof. Let α and β be any two 2-dicolourings of C. Let x = diff(α, β) = |{v ∈ V (C) | α(v) ̸= β(v)}|. By +induction on x ≥ 0, let us show that there exists a path of length at most x from α to β in D2(C). This clearly holds +for x = 0 (i.e., α = β). Assume x > 0 and the result holds for x − 1. Let v ∈ V (C) be such that α(v) ̸= β(v). +If v can be recoloured in β(v), then we recolour it and reach a new 2-dicolouring α′ such that diff(α′, β) = x−1 +and the result holds by induction. Else if v cannot be recoloured, then recolouring v must create a monochromatic +directed cycle, which must be C. Then there must be a vertex v′, different from v, such that β(v) = α(v′) ̸= β(v′), +and v′ can be recoloured. We recolour it and reach a new 2-dicolouring α′ such that diff(α′, β) = x − 1 and the +result holds by induction. +We are now ready to prove Theorem 8. +Proof of Theorem 8. Let α and β be any two 2-dicolourings of ⃗G. We will show that there exists a redicolouring +sequence of length at most n between α and β. We may assume that ⃗G is strongly connected, otherwise we +consider each strongly connected component independently. This implies in particular that ⃗G does not contain any +sink nor source. Let (X, Y ) be a partition of V (⃗G) such that, for every x ∈ X, d+(x) = 1, and for every y ∈ Y , +d−(y) = 1. +Assume first that ⃗G⟨X⟩ contains a directed cycle C. Since every vertex in X has exactly one out-neighbour, +there is no arc leaving C. Thus, since ⃗G is strongly connected, ⃗G must be exactly C, and the result holds by +Lemma 12. Using a symmetric argument, we get the result when ⃗G⟨Y ⟩ contains a directed cycle. +Assume now that both ⃗G⟨X⟩ and ⃗G⟨Y ⟩ are acyclic. Thus, since every vertex in X has exactly one out- +neighbour, ⃗G⟨X⟩ is the union of disjoint and independent in-trees, that are oriented trees in which all arcs are +directed towards the root. We denote by Xr the set of roots of these in-trees. Symmetrically, ⃗G⟨Y ⟩ is the union of +disjoint and independent out-trees (oriented trees in which all arcs are directed away from the root), and we denote +by Yr the set of roots of these out-trees. Set Xℓ = X \ Xr and Yℓ = Y \ Yr. Observe that the arcs from X to Y +form a perfect matching directed from Xr to Yr. We denote by Mr this perfect matching. Observe also that there +can be any arc from Y to X. Now we define X1 +r and Y 1 +r two subsets of Xr and Yr respectively, depending on the +two 2-dicolourings α and β, as follows: +X1 +r = {x | xy ∈ Mr, α(x) = β(y) ̸= α(y) = β(x)} +Y 1 +r = {y | xy ∈ Mr, α(x) = β(y) ̸= α(y) = β(x)} +Set X2 +r = Xr \ X1 +r and Y 2 +r = Yr \ Y 1 +r . We denote by M 1 +r (respectively M 2 +r ) the perfect matching from X1 +r to Y 1 +r +(respectively from X2 +r to Y 2 +r ). Figure 1 shows a partitioning of V (⃗G) into X1 +r , X2 +r, Xℓ, Y 1 +r , Y 2 +r , Yℓ. +Claim 8.1: There exists a redicolouring sequence of length sα from α to some 2-dicolouring α′ and a redicolouring +sequence of length sβ from β to some 2-dicolouring β′ such that each of the following holds: +(i) For any arc xy ∈ Mr, α′(x) ̸= α′(y) and β′(x) ̸= β′(y), +(ii) For any arc xy ∈ M 2 +r , α′(x) = β′(x) (and so α′(y) = β′(y) by (i)), and +(iii) sα + sβ ≤ |X2 +r| + |Y 2 +r |. +6 + +X1 +r +X2 +r +Xℓ +Y 1 +r +Y 2 +r +Yℓ +⃗G dicoloured with α +X1 +r +X2 +r +Xℓ +Y 1 +r +Y 2 +r +Yℓ +⃗G dicoloured with β +Figure 1: The partitioning of V (⃗G) into X1 +r, X2 +r , Xℓ, Y 1 +r , Y 2 +r , Yℓ. +Proof of claim. We consider the arcs xy of M 2 +r one after another and do the following recolourings depending on +the colours of x and y in both α and β to get α′ and β′. +• If α(x) = α(y) = β(x) = β(y), then we recolour x in both α and β; +• Else if α(x) = α(y) ̸= β(x) = β(y), then we recolour x in α and we recolour y in β; +• Else if α(x) = β(x) ̸= α(y) = β(y), then we do nothing; +• Else if α(x) ̸= α(y) = β(x) = β(y), then we recolour x in β; +• Finally if α(y) ̸= α(x) = β(x) = β(y), then we recolour y in β. +Each of these recolourings is valid because, when a vertex in X2 +r (respectively Y 2 +r ) is recoloured, it gets a colour +different from its only out-neighbour (respectively in-neighbour). Let α′ and β′ be the the two resulting 2- +dicolourings. By construction, α′ and β′ agree on X2 +r ∪ Y 2 +r . For each arc xy ∈ M 2 +r , either α(x) = α′(x) or +α(y) = α′(y), and the same holds for β and β′. This implies that sα + sβ ≤ 2|M 2 +r | = |X2 +r | + |Y 2 +r |. +♦ +Claim 8.2: There exists a redicolouring sequence from α′ to some 2-dicolouring ˜α of length s′ +α and a redicolouring +sequence from β′ to some 2-dicolouring ˜β of length s′ +β such that each of the following holds: +(i) ˜α and ˜β agree on V (⃗G) \ (X1 +r ∪ Y 1 +r ), +(ii) α′ and ˜α agree on Xr ∪ Yr, +(iii) β′ and ˜β agree on Xr ∪ Yr, +(iv) Xℓ ∪ Yℓ is monochromatic in ˜α (and in ˜β by (i)), and +(v) s′ +α + s′ +β ≤ |Xℓ| + |Yℓ|. +Proof of claim. Observe that in both 2-dicolourings α′ and β′, we are free to recolour any vertex of Xℓ ∪ Yℓ since +there is no monochromatic arc from X to Y and both ⃗G⟨X⟩ and ⃗G⟨Y ⟩ are acyclic. Let n1 (respectively n2) be the +number of vertices in Xℓ ∪ Yℓ that are coloured 1 (respectively 2) in both α′ and β′. Without loss of generality, +assume that n1 ≤ n2. Then we set each vertex of Xℓ ∪ Yℓ to colour 2 in both α′ and β′. Let ˜α and ˜β the resulting +2-dicolouring. Then s′ +α + s′ +β is exactly |Xℓ| + |Yℓ| + n1 − n2 ≤ |Xℓ| + |Yℓ|. +♦ +7 + +Claim 8.3: There is a redicolouring sequence between ˜α and ˜β of length |X1 +r| + |Y 1 +r |. +Proof of claim. By construction of ˜α and ˜β, we only have to exchange the colours of x and y for each arc xy ∈ M 1 +r . +Without loss of generality, we may assume that the colour of all vertices in Xℓ ∪ Yℓ by ˜α and ˜β is 1. +We first prove that, by construction, we can recolour any vertex of X1 +r ∪Y 1 +r from 1 to 2. Assume not, then there +is such a vertex x ∈ X1 +r ∪Y 1 +r such that recolouring x from 1 to 2 creates a monochromatic directed cycle C. Since +both ⃗G⟨X⟩ and ⃗G⟨Y ⟩ are acyclic, C must contain an arc of Mr. Since Mr does not contain any monochromatic +arc in ˜α, then this arc must be incident to x. Now observe that colour 2, in ˜α, induces an independent set on both +⃗G⟨X⟩ and ⃗G⟨Y ⟩. This implies that C must contain at least 2 arcs in Mr. This is a contradiction since recolouring +x creates exactly one monochromatic arc in Mr. +Then, for each arc xy ∈ M 1 +r , we can first recolour the vertex coloured 1 and then the vertex coloured 2. +Note that we maintain the invariant that colour 2 induces an independent set on both ⃗G⟨X⟩ and ⃗G⟨Y ⟩. We get a +redicolouring sequence from ˜α to ˜β in exactly 2|M 1 +r | = |X1 +r| + |Y 1 +r | steps. +♦ +Combining the three claims, we finally proved that there exists a redicolouring sequence between α and β of length +at most n. +In the following, when α is a dicolouring of a digraph D, and H is a subdigraph of D, we denote by α|H the +restriction of α to H. We will prove Corollary 9, let us restate it. +Corollary 9. Let ⃗G be an oriented graph of order n with ∆min(⃗G) = ∆ ≥ 1, and let k ≥ ∆ + 1. Then Dk(⃗G) is +connected and has diameter at most 2∆n. +Proof. We will show the result by induction on ∆. +Assume first that ∆ = 1, let k ≥ 2. Let α be any k-dicolouring of ⃗G and γ be any 2-dicolouring of ⃗G. To +ensure that Dk(⃗G) is connected and has diameter at most 2n, it is sufficient to prove that there is a redicolouring +sequence between α and γ of length at most n. Let H be the digraph induced by the set of vertices coloured 1 or +2 in α, and let J be V (⃗G) \ V (H). By Theorem 8, since ∆min(H) ≤ ∆min(⃗G) ≤ 1, we know that there exists +a redicolouring sequence, in H, from α|H to γ|H of length at most |V (H)|. This redicolouring sequence extends +in ⃗G because it only uses colours 1 and 2. Let α′ be the obtained dicolouring of ⃗G. Since α′(v) = γ(v) for every +v ∈ H, we can recolour each vertex in J to its colour in γ. This shows that there is a redicolouring sequence +between α and γ of length at most |V (H)| + |J| = |V (⃗G)|. This ends the case ∆ = 1. +Assume now that ∆ ≥ 2 and let k ≥ ∆ + 1. Let α and β be two k-dicolourings of ⃗G. By Corollary 7, we +know that ⃗χ(⃗G) ≤ ∆ ≤ k − 1. We first show that there is a redicolouring sequence of length at most 2n from +α to some (k − 1)-dicolouring γ of ⃗G. From α, whenever it is possible we recolour each vertex coloured 1, 2 or +k with a colour of {3, . . . , k − 1} (when k = 3 we do nothing). Let ˜α be the obtained dicolouring, and let M be +the set of vertices coloured in {3, . . . , k − 1} by ˜α (when k = 3, M is empty). We get that H = ⃗G − M satisfies +∆min(H) ≤ 2, since every vertex in H has at least one in-neighbour and one out-neighbour coloured c for every +c ∈ {3, . . ., k − 1}. By Corollary 7, there exists a 2-dicolouring γ|H of H. From ˜α|H, whenever it is possible, +we recolour a vertex coloured 1 or 2 to colour k. Let ˆα be the resulting dicolouring, and ˆH be the subdigraph of +H induced by the vertices coloured 1 or 2 in ˆα. We get that ∆min( ˆH) ≤ 1 since every vertex in ˆH has, in ⃗G, at +least one in-neighbour and one out-neighbour coloured c for every c ∈ {3, . . ., k}. In at most |V ( ˆH)| steps, using +Theorem 8, we can recolour the vertices of V ( ˆH) to their colour in γ|H (using only colours 1 and 2). Then we can +recolour each vertex coloured k to its colour in γ|H. This results in a redicolouring sequence of length at most 2n +from α to some (k − 1)-dicolouring γ of ⃗G , since colour k is not used in the resulting dicolouring (recall that M +is coloured with {3, . . ., k − 1}). +Now, from β, whenever it is possible we recolour each vertex to colour k. Let ˜β be the obtained k-dicolouring, +and let N be the set of vertices coloured k in ˜β. We get that J = ⃗G − N satisfies ∆min(J) ≤ ∆ − 1. Thus, +by induction, there exists a redicolouring sequence from ˜β|J to γ|J, in at most 2(∆ − 1)|V (J)| steps (using only +colours {1, . . . , k − 1}). Since N is coloured k in ˜β, this extends to a redicolouring sequence in ⃗G. Now, since γ +does not use colour k, we can recolour each vertex in N to its colour in γ. We finally get a redicolouring sequence +from β to γ of length at most 2(∆ − 1)n. Concatenating the redicolouring sequence from α to γ and the one from +γ to β, we get a redicolouring sequence from α to β in at most 2∆n steps. +8 + +4 +An analogue of Brook’s theorem for digraph redicolouring +Let us restate Theorem 10. +Theorem 10. Let D be a connected digraph with ∆max(D) = ∆ ≥ 3, k ≥ ∆ + 1, and α, β two k-dicolourings +of D. Then at least one of the following holds: +• α is k-frozen, or +• β is k-frozen, or +• there is a redicolouring sequence of length at most c∆|V |2 between α and β, where c∆ = O(∆2) is a +constant depending only on ∆. +An L-assignment of a digraph D is a function which associates to every vertex a list of colours. An L- +dicolouring of D is a dicolouring α where, for every vertex v of D, α(v) ∈ L(v). An L-redicolouring sequence is +a redicolouring sequence γ1, . . . , γr, such that for every i ∈ {1, . . . , r}, γi is an L-dicolouring of D. +Lemma 13. Let D = (V, A) be a digraph and L be a list-assignment of D such that, for every vertex v ∈ V , +|L(v)| ≥ dmax(v) + 1. Let α be an L-dicolouring of D. If u ∈ V is blocked in α, then for each colour c ∈ L(u) +different from α(u), u has exactly one out-neighbour u+ +c and one in-neighbour u− +c coloured c. Moreover, if +u+ +c ̸= u− +c , there must be a monochromatic directed path from u+ +c to u− +c . In particular, u is not incident to a +monochromatic arc. +Proof. Since u is blocked to its colour in α, for each colour c ∈ L(u) different from α(u), recolouring u to c must +create a monochromatic directed cycle C. Let v be the out-neighbour of u in C and w be the in-neighbour of u in +C. Then α(v) = α(w) = c, and there is a monochromatic directed path (in C) from v to w. +This implies that, for each colour c ∈ L(u) different from α(u), u has at least one out-neighbour and at least +one in-neighbour coloured c. Since |L(u)| ≥ dmax(u) + 1, then |L(u)| = dmax(u) + 1, and u must have exactly +one out-neighbour and exactly one in-neighbour coloured c. In particular, u cannot be incident to a monochromatic +arc. +Lemma 14. Let D = (V, A) be a digraph such that for every vertex v ∈ V , N +(v) \ N −(v) ̸= ∅ and N −(v) \ +N +(v) ̸= ∅. Let L be a list assignment of D, such that for every vertex v ∈ V , |L(v)| ≥ dmax(v) + 1. +Then for any pair of L-dicolourings α, β of D, there is an L-redicolouring sequence of length at most (|V | + +3)|V |. +Proof. Let x = diff(α, β) = |{v ∈ V | α(v) ̸= β(v)}|. We will show by induction on x that there is an L- +redicolouring sequence from α to β of length at most (|V | + 3)x. The result clearly holds for x = 0 (i.e. α = β). +Let v ∈ V be such that α(v) ̸= β(v). We denote α(v) by c and β(v) by c′. If v can be recoloured to c′, then we +recolour it and we get the result by induction. +Assume now that v cannot be recoloured to c′. Whenever v is contained in a directed cycle C of length at least +3, such that every vertex of C but v is coloured c′, we do the following: we choose w a vertex of C different from +v, such that β(w) ̸= c′. We know that such a w exists, for otherwise C would be a monochromatic directed cycle +in β. Now, since w is incident to a monochromatic arc in C, and because |L(w)| ≥ dmax(w) + 1, by Lemma 13, +we know that w can be recoloured to some colour different from c′. Thus we recolour w to this colour. Observe +that it does not increase x. +After repeating this process, maybe v cannot be recoloured to c′ because it is adjacent by a digon to some +vertices coloured c′. We know that these vertices are not coloured c′ in β. Thus, whenever such a vertex can be +recoloured, we recolour it. After this, let η be the obtained dicolouring. If v can be recoloured to c′ in η, we are +done. Otherwise, there must be some vertices, blocked to colour c′ in η, adjacent to v by a digon. Let S be the set +of such vertices. Observe that, by Lemma 13, for every vertex s ∈ S, c belongs to L(s), for otherwise s would not +be blocked in η. We distinguish two cases, depending on the size of S. +9 + +• If |S| ≥ 2, then by Lemma 13, v can be recoloured to a colour c′′, different from both c and c′, because v is +adjacent by a digon with two neighbours coloured c′. Hence we can successively recolour v to c′′, and every +vertex of S to c . This does not create any monochromatic directed cycle because for each s ∈ S, since s is +blocked in η, by Lemma 13 v must be the only neighbour of s coloured c in η. +We can finally recolour v to c′. +• If |S| = 1, let w be the only vertex in S. If v can be recoloured to any colour (different from c′ since w is +coloured c′), then we first recolour v, allowing us to recolour w to c, because v is the single neighbour of w +coloured c in η by Lemma 13. We finally can recolour v to c′. +Assume then that v is blocked to colour c in η. Let us fix w+ ∈ N +(w) \ N −(w). Since w is blocked to c′ +in η, by Lemma 13, there exists exactly one vertex w− ∈ N −(w) \ N +(w) such that η(w+) = η(w−) = c′′ +and there must be a monochromatic directed path from w+ to w−. +Since v is blocked to colour c in η, either vw− /∈ A or w+v /∈ A, otherwise, by Lemma 13, there must be +a monochromatic directed path from w− to w+, which is blocking v to its colour. But since there is also a +monochromatic directed path from w+ to w− (blocking w) there would be a monochromatic directed cycle, +a contradiction (see Figure 2). +w +v +w+ +w− +Figure 2: The vertices v, w, w+ and w−. +We distinguish the two possible cases: +– if vw− /∈ A, then we start by recolouring w− with a colour that does not appear in its in-neighbourhood. +This is possible because w− has a monochromatic entering arc, and because |L(w−)| ≥ dmax(w−)+1. +We first recolour w with c′′, since c′′ does not appear in its in-neighbourhood anymore (w− was the +only one by Lemma 13). Next we recolour v with c′: this is possible because v does not have any +out-neighbour coloured c′ since w was the only one by Lemma 13 and w− is not an out-neighbour of +v. We can finally recolour w to colour c and w− to c′′. After all these operations, we exchanged the +colours of v and w. +– if w+v /∈ A, then we use a symmetric argument. +Observe that we found an L-redicolouring sequence from α to a α′, in at most |V |+3 steps, such that diff(α′, β) < +diff(α, β). Thus by induction, we get an L-redicolouring sequence of length at most (|V | + 3)x between α and +β. +We are now able to prove Theorem 10. The idea of the proof is to divide the digraph D into two parts. One of +them is bidirected and we will use Theorem 2 as a black box on it. In the other part, we know that each vertex is +incident to at least two simple arcs, one leaving and one entering, and we will use Lemma 14 on it. +Proof of Theorem 10. Let D = (V, A) be a connected digraph with ∆max(D) = ∆, k ≥ ∆ + 1. Let α and β be +two k-dicolourings of D. Assume that neither α nor β is k-frozen. +We first make a simple observation. For any simple arc xy ∈ A, we may assume that N +(y) \ N −(y) ̸= ∅ +and N −(x) \ N +(x) ̸= ∅. If this is not the case, then every directed cycle containing xy must contain a digon, +implying that the k-dicolouring graph of D is also the k-dicolouring graph of D \ {xy}. Then we may look for a +redicolouring sequence in D \ {xy}. +10 + +Let X = {v ∈ V | N +(v) = N −(v)} and Y = V \ X. Observe that D⟨X⟩ is bidirected, and thus the +dicolourings of D⟨X⟩ are exactly the colourings of UG(D⟨X⟩). We first show that α|D⟨X⟩ and β|D⟨X⟩ are not +frozen k-colourings of D⟨X⟩. If Y is empty, then D⟨X⟩ = D and α|D⟨X⟩ and β|D⟨X⟩ are not k-frozen by +assumption. Otherwise, since D is connected, there exists x ∈ X such that, in D⟨X⟩, d+(x) = d−(x) ≤ ∆ − 1, +implying that x is not blocked in any dicolouring of D⟨X⟩. Thus, by Theorem 2, there is a redicolouring sequence +γ′ +1, . . . , γ′ +r in D⟨X⟩ from α|D⟨X⟩ to β|D⟨X⟩, where r ≤ c∆|X|2, and c∆ = O(∆) is a constant depending on ∆. +We will show that, for each i ∈ {1, . . . , r − 1}, if γi is a k-dicolouring of D which agrees with γ′ +i on X, then +there exist a k-dicolouring γi+1 of D that agrees with γ′ +i+1 on X and a redicolouring sequence from γi to γi+1 of +length at most ∆ + 2. +Observe that α agrees with γ′ +1 on X. Now assume that there is such a γi, which agrees with γ′ +i on X, and +let vi ∈ X be the vertex for which γ′ +i(vi) ̸= γ′ +i+1(vi). We denote by c (respectively c′) the colour of vi in γ′ +i +(respectively γ′ +i+1). If recolouring vi to c′ in γi is valid then we have the desired γi+1. Otherwise, we know that +vi is adjacent with a digon (since vi is only adjacent to digons) to some vertices (at most ∆) coloured c′ in Y . +Whenever such a vertex can be recoloured to a colour different from c′, we recolour it. Let ηi be the reached +k-dicolouring after these operations. If vi can be recoloured to c′ in ηi we are done. If not, then the neighbours of +vi coloured c′ in Y are blocked to colour c′ in ηi. We denote by S the set of these neighbours. We distinguish two +cases: +• If |S| ≥ 2, then by Lemma 13, vi can be recoloured to a colour c′′, different from both c and c′, because vi +has two neighbours with the same colour. Then we successively recolour vi to c′′, and every vertex of S to +c. This does not create any monochromatic directed cycle because, by Lemma 13, for each s ∈ S, vi is the +only neighbour of s coloured c in ηi. We can finally recolour vi to c′ to reach the desired γi+1. +• If |S| = 1, let y be the only vertex in S. Since y belongs to Y and is blocked to its colour in ηi, by Lemma 13, +we know that y has an out-neighbour y+ ∈ N +(y)\N −(y) and an in-neighbour y− ∈ N −(y)\N +(y) such +that there is a monochromatic directed path from y+ to y−. Observe that both y+ and y− are recolourable +in ηi by Lemma 13, because there are incident to a monochromatic arc. +– If vi is not adjacent to y+, then we recolour y+ to any possible colour, and we recolour y to ηi(y+). +We can finally recolour vi to c′ to reach the desired γi+1. +– If vi is not adjacent to y−, then we recolour y− to any possible colour, and we recolour y to ηi(y−). +We can finally recolour vi to c′ to reach the desired γi+1. +– Finally if vi is adjacent to both y+ and y−, since ηi(y+) = ηi(y−), then vi can be recoloured to a +colour c′′ different from c and c′. This allows us to recolour y to c, and we finally can recolour vi to c′ +to reach the desired γi+1. +We have shown that there is a redicolouring sequence of length at most (∆ + 2)c∆n2 from α to some α′ that +agrees with β on X. Now we define the list-assignment: for each y ∈ Y , +L(y) = {1, . . . , k} \ {β(x) | x ∈ N(y) ∩ X}. +Observe that, for every y ∈ Y , +|L(y)| ≥ k − |N +(y) ∩ X| ≥ ∆ + 1 − (∆ − d+ +Y (y)) ≥ d+ +Y (y) + 1. +Symmetrically, we get |L(y)| ≥ d− +Y (y) + 1. This implies, in D⟨Y ⟩, |L(y)| ≥ dmax(y) + 1. Note also that +both α′ +|D⟨Y ⟩ and β|D⟨Y ⟩ are L-dicolourings of D⟨Y ⟩. Note finally that, for each y ∈ Y , N +(y) \ N −(y) ̸= ∅ +and N +(y) \ N −(y) ̸= ∅ by choice of X and Y and by the initial observation. By Lemma 14, there is an L- +redicolouring sequence in D⟨Y ⟩ between α′ +|D⟨Y ⟩ and β|D⟨Y ⟩, with length at most (|Y | + 3)|Y |. By choice of L, +this extends directly to a redicolouring sequence from α′ to β on D of the same length. +The concatenation of the redicolouring sequence from α to α′ and the one from α′ to β leads to a redicolouring +sequence from α to β of length at most c′ +∆|V |2, where c′ +∆ = O(∆2) is a constant depending on ∆. +11 + +Remark 15. If α is a k-frozen dicolouring of a digraph D, with k ≥ ∆max(D) + 1, then D must be bidirected. +If D is not bidirected, then we choose v a vertex incident to a simple arc. If v cannot be recoloured in α, by +Lemma 13, since v is incident to a simple arc, there exists a colour c for which v has an out-neighbour w and an +in-neighbour u both coloured c, such that u ̸= w and there is a monochromatic directed path from w to u. But +then, every vertex on this path is incident to a monochromatic arc, and it can be recoloured by Lemma 13. Thus, α +is not k-frozen. This shows that an obstruction of Theorem 10 is exactly the bidirected graph of an obstruction of +Theorem 2. +5 +Further research +In this paper, we established some analogues of Brooks’ Theorem for the dichromatic number of oriented graphs +and for digraph redicolouring. Many open questions arise, we detail a few of them. +Restricted to oriented graphs, Mcdiarmid and Mohar (see [11]) conjectured that the Directed Brooks’ Theorem +can be improved to the following. +Conjecture 16 (Mcdiarmid and Mohar). Every oriented graph ⃗G has ⃗χ(⃗G) = O +� +∆max +log(∆max) +� +. +Concerning digraph redicolouring, we believe that Corollary 9 and Theorem 10 can be improved. We pose the +following two conjectures. +Conjecture 17. There is an absolute constant c such that for every integer k and every oriented graph ⃗G such that +k ≥ ∆min(⃗G) + 1, the diameter of Dk(⃗G) is bounded by cn. +Conjecture 18. There is an absolute constant d such that for every integer k and every digraph D with k ≥ +∆max(D) + 1, the diameter of Dk(D) is bounded by dn2. +Given an orientation ⃗G of a planar graph, a celebrated conjecture from Neumann-Lara [14] states that the +dichromatic number of ⃗G is at most 2. +It is known that it must be 4-mixing because planar graphs are 5- +degenerate [5]. It is also known that there exists 2-freezable orientations of planar graphs [5]. Thus the following +problem, stated in [5], remains open: +Question 19. Is every oriented planar graph 3-mixing ? +Acknowledgement +I am grateful to Fr´ed´eric Havet and Nicolas Nisse for stimulating discussions. +References +[1] Pierre Aboulker and Guillaume Aubian. +Four proofs of the directed Brooks’ Theorem. +arXiv preprint +arXiv:2109.01600, 2021. +[2] Valentin Bartier. Combinatorial and Algorithmic aspects of Reconfiguration. PhD thesis, Universit´e Grenoble +Alpes, 2021. +[3] D. Bokal, G. Fijavz, M. Juvan, P.M. Kayll, and B. Mohar. The circular chromatic number of a digraph. J. +Graph Theory, 46(3):227–240, 2004. +[4] Marthe Bonamy, Matthew Johnson, Ioannis Lignos, Viresh Patel, and Daniel Paulusma. Reconfiguration +graphs for vertex colourings of chordal and chordal bipartite graphs. Journal of Combinatorial Optimization, +27(1):132–143, 2014. +12 + +[5] Nicolas Bousquet, Fr´ed´eric Havet, Nicolas Nisse, Lucas Picasarri-Arrieta, and Amadeus Reinald. Digraph +redicolouring. arXiv preprint arXiv:2301.03417, 2023. +[6] R. L. Brooks. On colouring the nodes of a network. Mathematical Proceedings of the Cambridge Philosoph- +ical Society, 37(2):194–197, 1941. +[7] Luis Cereceda, Jan Van den Heuvel, and Matthew Johnson. Mixing 3-colourings in bipartite graphs. Euro- +pean Journal of Combinatorics, 30(7):1593–1606, 2009. +[8] Luis Cereceda, Jan van den Heuvel, and Matthew Johnson. Finding paths between 3-colorings. Journal of +Graph Theory, 67(1):69–82, 2011. +[9] Carl Feghali, Matthew Johnson, and Dani¨el Paulusma. A reconfigurations analogue of Brooks’ Theorem and +its consequences. Journal of Graph Theory, 83(4):340–358, 2016. +[10] Ararat Harutyunyan and Bojan Mohar. Gallai’s theorem for list coloring of digraphs. SIAM Journal on +Discrete Mathematics, 25(1):170–180, 2011. +[11] Ararat Harutyunyan and Bojan Mohar. Strengthened Brooks' theorem for digraphs of girth at least three. The +Electronic Journal of Combinatorics, 18(1), October 2011. +[12] Jan van den Heuvel. The complexity of change, page 127–160. London Mathematical Society Lecture Note +Series. Cambridge University Press, 2013. +[13] Bojan Mohar. Eigenvalues and colorings of digraphs. Linear Algebra and its Applications, 432(9):2273– +2277, 2010. +[14] Victor Neumann-Lara. The dichromatic number of a digraph. J. Combin. Theory Ser. B., 33:265–270, 1982. +[15] Naomi Nishimura. Introduction to reconfiguration. Algorithms, 11(4), 2018. +13 + diff --git a/A9E4T4oBgHgl3EQfEwz_/content/tmp_files/load_file.txt b/A9E4T4oBgHgl3EQfEwz_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0e47c45f356dc95041f0a154d64a4171a0d88b53 --- /dev/null +++ b/A9E4T4oBgHgl3EQfEwz_/content/tmp_files/load_file.txt @@ -0,0 +1,568 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf,len=567 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content='04881v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content='CO] 12 Jan 2023 Strengthening the Directed Brooks’ Theorem for oriented graphs and consequences on digraph redicolouring * Lucas Picasarri-Arrieta Universit´e Cˆote d’Azur, CNRS, I3S, INRIA, Sophia Antipolis, France lucas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content='picasarri-arrieta@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content='fr Abstract Let D = (V, A) be a digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We define ∆max(D) as the maximum of {max(d+(v), d−(v)) | v ∈ V } and ∆min(D) as the maximum of {min(d+(v), d−(v)) | v ∈ V }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' It is known that the dichromatic number of D is at most ∆min(D) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' In this work, we prove that every digraph D which has dichromatic number exactly ∆min(D) + 1 must contain the directed join of ←→ Kr and ←→ Ks for some r, s such that r + s = ∆min(D) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' In particular, every oriented graph ⃗G with ∆min( ⃗G) ≥ 2 has dichromatic number at most ∆min( ⃗G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let ⃗G be an oriented graph of order n such that ∆min( ⃗G) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Given two 2-dicolourings of ⃗G, we show that we can transform one into the other in at most n steps, by recolouring one vertex at each step while maintaining a dicolouring at any step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Furthermore, we prove that, for every oriented graph ⃗G on n vertices, the distance between two k-dicolourings is at most 2∆min( ⃗G)n when k ≥ ∆min( ⃗G) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We then extend a theorem of Feghali to digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We prove that, for every digraph D with ∆max(D) = ∆ ≥ 3 and every k ≥ ∆ + 1, the k-dicolouring graph of D consists of isolated vertices and at most one further component that has diameter at most c∆n2, where c∆ = O(∆2) is a constant depending only on ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content='1 Graph (re)colouring Given a graph G = (V, E), a k-colouring of G is a function c : V −→ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=', k} such that, for every edge xy ∈ E, we have c(x) ̸= c(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' So for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=', k}, c−1(i) induces an independent set on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' The chromatic number of G, denoted by χ(G), is the smallest k such that G admits a k-colouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' The maximum degree of G, denoted by ∆(G), is the degree of the vertex with the greatest number of edges incident to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' A simple greedy procedure shows that, for any graph G, χ(G) ≤ ∆(G) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' The celebrated theorem of Brooks [6] characterizes the graphs for which equality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Theorem 1 (Brooks, [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' A connected graph G satisfies χ(G) = ∆(G) + 1 if and only if G is an odd cycle or a complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' For any k ≥ χ(G), the k-colouring graph of G, denoted by Ck(G), is the graph whose vertices are the k- colourings of G and in which two k-colourings are adjacent if they differ by the colour of exactly one vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' A path between two given colourings in Ck(G) corresponds to a recolouring sequence, that is a sequence of pairs composed of a vertex of G, which is going to receive a new colour, and a new colour for this vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If Ck(G) is connected, we say that G is k-mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' A k-colouring of G is k-frozen if it is an isolated vertex in Ck(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' The graph G is k-freezable if it admits a k-frozen colouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' In the last fifteen years, since the papers of Cereceda, van den Heuvel and Johnson [8, 7], graph recolouring has been studied by many researchers in graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We refer the reader to the PhD thesis of Bartier [2] for a complete overview on graph recolouring and to the surveys of van Heuvel [12] and Nishimura [15] for reconfiguration problems in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Feghali [9] proved the following analogue of Brooks’ Theorem for graphs recolouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Research supported by research grant DIGRAPHS ANR-19-CE48-0013 and by the French government, through the EUR DS4H Invest- ments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-17-EURE-0004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' 1 Theorem 2 (Feghali, [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let G = (V, E) be a connected graph with ∆(G) = ∆ ≥ 3, k ≥ ∆ + 1, and α, β two k-colourings of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then at least one of the following holds: α is k-frozen, or β is k-frozen, or there is a recolouring sequence of length at most c∆|V |2 between α and β, where c∆ = O(∆) is a constant depending on ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content='2 Digraph (re)dicolouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' In this paper, we are looking for extensions of the previous results on graphs colouring and recolouring to digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let D be a digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' A digon is a pair of arcs in opposite directions between the same vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' A simple arc is an arc which is not in a digon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' For any two vertices x, y ∈ V (D), the digon {xy, yx} is denoted by [x, y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' The digon graph of D is the undirected graph with vertex set V (D) in which uv is an edge if and only if [u, v] is a digon of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' An oriented graph is a digraph with no digon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' The bidirected graph associated to a graph G, denoted by ←→ G , is the digraph obtained from G, by replacing every edge by a digon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' The underlying graph of D, denoted by UG(D), is the undirected graph G with vertex set V (D) in which uv is an edge if and only if uv or vu is an arc of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let v be a vertex of a digraph D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' The out-degree (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' in-degree) of a vertex v, denoted by d+(v) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' d−(v)), is the number of arcs leaving (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' entering) v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We define the maximum degree of v as dmax(v) = max{d+(v), d−(v)}, and the minimum degree of v as dmin(v) = min{d+(v), d−(v)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We can then define the cor- responding maximum degrees of D: ∆max(D) = maxv∈V (D)(dmax(v)) and ∆min(D) = maxv∈V (D)(dmin(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' A digraph D is ∆-diregular if, for every vertex v ∈ V (D), d−(v) = d+(v) = ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' In 1982, Neumann-Lara [14] introduced the notions of dicolouring and dichromatic number, which generalize the ones of colouring and chromatic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' A k-dicolouring of D is a function c : V (D) → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=', k} such that c−1(i) induces an acyclic subdigraph in D for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' The dichromatic number of D, denoted by ⃗χ(D), is the smallest k such that D admits a k-dicolouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' There is a one-to-one correspondence between the k-colourings of a graph G and the k-dicolourings of the associated bidirected graph ←→ G , and in particular χ(G) = ⃗χ(←→ G ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Hence every result on graph colourings can be seen as a result on dicolourings of bidirected graphs, and it is natural to study whether the result can be extended to all digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' The directed version of Brooks’ Theorem has been first proved by Mohar in [13], but people discovered a flaw in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Harutyunyan and Mohar then gave a stronger result in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Finally, Aboulker and Aubian gave four new proofs of the following theorem in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Theorem 3 (DIRECTED BROOKS’ THEOREM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let D be a connected digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then ⃗χ(D) ≤ ∆max(D) + 1 and equality holds if and only if one of the following occurs: D is a directed cycle, or D is a bidirected odd cycle, or D is a bidirected complete graph (of order at least 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' It is easy to prove, by induction on |V (D)|, that every digraph D can be dicoloured with ∆min(D) + 1 colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Hence, one can wonder if Brooks’ Theorem can be extended to digraphs using ∆min(D) instead of ∆max(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Unfortunately, Aboulker and Aubian [1] proved that, given a digraph D, deciding whether D is ∆min(D)-dicolourable is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Thus, unless P=NP, we cannot expect an easy characterization of digraphs satisfying ⃗χ(D) = ∆min(D) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let the maximum geometric mean of a digraph D be ˜∆(D) = max{ � d+(v)d−(v)|v ∈ V (D)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' By definition we have ∆min(D) ≤ ˜∆(D) ≤ ∆max(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Restricted to oriented graphs, Harutyunyan and Mohar [11] have strengthened Theorem 3 by proving the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' 2 Theorem 4 (Harutyunyan and Mohar [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' There is an absolute constant ∆1 such that every oriented graph ⃗G with ˜∆(⃗G) ≥ ∆1 has ⃗χ(⃗G) ≤ (1 − e−13) ˜∆(⃗G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' In Section 2, we give another strengthening of Theorem 3 on a large class of digraphs which contains oriented graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' The directed join of H1 and H2, denoted by H1 ⇒ H2, is the digraph obtained from disjoint copies of H1 and H2 by adding all arcs from the copy of H1 to the copy of H2 (H1 or H2 may be empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let D be a digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If D is not ∆min(D)-dicolourable, then one of the following holds: ∆min(D) ≤ 1, or ∆min(D) = 2 and D contains ←→ K2, or ∆min(D) ≥ 3 and D contains ←→ Kr ⇒ ←→ Ks, for some r, s such that r + s = ∆min(D) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' In particular, the following is a direct consequence of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let D be a digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If ⃗χ(D) = ∆min(D) + 1, then D contains the complete bidirected graph on � ∆min+1 2 � vertices as a subdigraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Moreover, since an oriented graph does not contain any digon, Corollary 6 implies the following: Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let ⃗G be an oriented graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If ∆min(⃗G) ≥ 2, then ⃗χ(⃗G) ≤ ∆min(⃗G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Corollary 6 is best possible: if we restrict D to not contain the complete bidirected graph on � ∆min+1 2 � + 1, then we show that deciding whether ⃗χ(D) ≤ ∆min(D) remains NP-complete (Theorem 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' For any k ≥ ⃗χ(D), the k-dicolouring graph of D, denoted by Dk(D), is the graph whose vertices are the k-dicolourings of D and in which two k-dicolourings are adjacent if they differ by the colour of exactly one vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Observe that Ck(G) = Dk(←→ G ) for any bidirected graph ←→ G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' A redicolouring sequence between two dicolourings is a path between these dicolourings in Dk(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' The digraph D is k-mixing if Dk(D) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' A k-dicolouring of D is k-frozen if it is an isolated vertex in Dk(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' The digraph D is k-freezable if it admits a k-frozen dicolouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' A vertex v is blocked to its colour in a dicolouring α if, for every colour c ̸= α(v), recolouring v to c in α creates a monochromatic directed cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Digraph redicolouring was first introduced in [5], where the authors generalized different results on graph re- colouring to digraphs, and proved some specific results on oriented graphs redicolouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' In particular, they studied the k-dicolouring graph of digraphs with bounded degeneracy or bounded maximum average degree, and they show that finding a redicolouring sequence between two given k-dicolouring of a digraph is PSPACE-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Dealing with the maximum degree of a digraph, they proved that, given an orientation of a subcubic graph ⃗G on n ver- tices, its 2-dicolouring graph D2(⃗G) is connected and has diameter at most 2n and they asked if this bound can be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We answer this question in Section 3 by proving the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let ⃗G be an oriented graph of order n such that ∆min(⃗G) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then D2(⃗G) is connected and has diameter exactly n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' In particular, if ⃗G is an orientation of a subcubic graph, then ∆min(⃗G) ≤ 1 (because d+(v) + d−(v) ≤ 3 for every vertex v), and so D2(⃗G) has diameter exactly n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Furthermore, we prove the following as a consequence of Corollary 7 and Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let ⃗G be oriented graph of order n with ∆min(⃗G) = ∆ ≥ 1, and let k ≥ ∆ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then Dk(⃗G) is connected and has diameter at most 2∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Corollary 9 does not hold for digraphs in general: indeed, ←→ Pn, the bidirected path on n vertices, satisfies ∆min(←→ Pn) = 2 and D3(←→ Pn) = C3(Pn) has diameter Ω(n2), as proved in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' In Section 4, we extend Theorem 2 to digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let D = (V, A) be a connected digraph with ∆max(D) = ∆ ≥ 3, k ≥ ∆ + 1, and α, β two k-dicolourings of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then at least one of the following holds: 3 α is k-frozen, or β is k-frozen, or there is a redicolouring sequence of length at most c∆|V |2 between α and β, where c∆ = O(∆2) is a constant depending only on ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Furthermore, we prove that a digraph D is k-freezable only if D is bidirected and its underlying graph is k-freezable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Thus, an obstruction in Theorem 10 is exactly the bidirected graph of an obstruction in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' 2 Strengthening of Directed Brooks’ Theorem for oriented graphs A digraph D is k-dicritical if ⃗χ(D) = k and for every vertex v ∈ V (D), ⃗χ(D − v) < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Observe that every digraph with dichromatic number at least k contains a k-dicritical subdigraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let F2 be {←→ K2}, and for each ∆ ≥ 3, we define F∆ = {←→ Kr ⇒ ←→ Ks | r, s ≥ 0 and r + s = ∆ + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' A digraph D is F∆-free if it does not contain F as a subdigraph, for any F ∈ F∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Theorem 5 can then be reformulated as follows: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let D be a digraph with ∆min(D) = ∆ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If D is F∆-free, then ⃗χ(D) ≤ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let D be a digraph such that ∆min(D) = ∆ ≥ 2 and ⃗χ(D) = ∆ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We will show that D contains some F ∈ F∆ as a subdigraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let (X, Y ) be a partition of V (D) such that for each x ∈ X, d+(x) ≤ ∆, and for each y ∈ Y , d−(y) ≤ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We define the digraph ˜D as follows: V ( ˜D) = V (D), A( ˜D) = A(D⟨X⟩) ∪ A(D⟨Y ⟩) ∪ {xy, yx | xy ∈ A(D), x ∈ X, y ∈ Y }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content='1: ⃗χ( ˜D) ≥ ∆ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Proof of claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Assume for a contradiction that there exists a ∆-dicolouring c of ˜D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then D, coloured with c, must contain a monochromatic directed cycle C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Now C is not contained in X nor Y , for otherwise C would be a monochromatic directed cycle of D⟨X⟩ or D⟨Y ⟩ and so a monochromatic directed cycle of ˜D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Thus C contains an arc xy from X to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' But then, [x, y] is a monochromatic digon in ˜D, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' ♦ Since ⃗χ( ˜D) ≥ ∆ + 1, there is a (∆ + 1)-dicritical subdigraph H of ˜D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' By dicriticality of H, for every vertex v ∈ V (H), d+ H(v) ≥ ∆ and d− H(v) ≥ ∆, for otherwise a ∆-dicolouring of H − v could be extended to H by choosing for v a colour which is not appearing in its out-neighbourhood or in its in-neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We define XH as X ∩ V (H) and YH as Y ∩ V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Note that both H⟨XH⟩ and H⟨YH⟩ are subdigraphs of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content='2: H is ∆-diregular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Proof of claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let ℓ be the number of digons between XH and YH in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Observe that, by definition of X and H, for each vertex x ∈ XH, d+ H(x) = ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Note also that, in H, ℓ is exactly the number of arcs leaving XH and exactly the number of arcs entering XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We get: ∆|XH| = � x∈XH d+ H(x) = ℓ + |A(H⟨XH⟩)| = � x∈XH d− H(x) which implies, since H is dicritical, d+ H(x) = d− H(x) = ∆ for every vertex x ∈ XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Using a symmetric argument, we prove that ∆|YH| = � y∈YH d+ H(y), implying d+ H(y) = d− H(y) = ∆ for every vertex y ∈ YH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' ♦ 4 Since H is ∆-diregular, then in particular ∆max(H) = ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Hence, because ⃗χ(H) = ∆ + 1, by Theorem 3, either ∆ = 2 and H is a bidirected odd cycle, or ∆ ≥ 3 and H is the bidirected complete graph on ∆ + 1 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If ∆ = 2 and H is a bidirected odd cycle, then at least one digon of H belongs to H⟨XH⟩ or H⟨YH⟩, for oth- erwise H would be bipartite (with bipartition (XH, YH)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Since both H⟨XH⟩ and H⟨YH⟩ are subdigraphs of D, this shows, as desired, that D contains a copy of ←→ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If k ≥ 3 and H is the bidirected complete graph on ∆ + 1 vertices, let AH be all the arcs from YH to XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then D⟨V (H)⟩ \\ AH is a subdigraph of D which belongs to F∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Now we will justify that Corollary 6 is best possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' To do so, we prove that given a digraph D which does not contain the bidirected complete graph on � ∆min(D)+1 2 � + 1 vertices, deciding if it is ∆min(D)-dicolourable is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We shall use a reduction from k-DICOLOURABILITY which is defined as follows: k-DICOLOURABILITY Input: A digraph D Question: Is D k-dicolourable ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' k-DICOLOURABILITY is NP-complete for every fixed k ≥ 2 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' It remains NP-complete when we restrict to digraphs D with ∆min(D) = k [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' For all k ≥ 2, k-DICOLOURABILITY remains NP-complete when restricted to digraphs D satisfying ∆min(D) = k and not containing the bidirected complete graph on � k+1 2 � + 1 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let D = (V, A) be an instance of k-DICOLOURABILITY for some fixed k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then we build D′ = (V ′, A′) as follows: For each vertex x ∈ V , we associate a copy of S− x ⇒ S+ x where S− x is the bidirected complete graph on � k+1 2 � vertices, and S+ x is the bidirected complete graph on � k+1 2 � vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' For each arc xy ∈ A, we associate all possible arcs x+y− in A′, such that x+ ∈ S+ x and y− ∈ S− y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' First observe that ∆min(D′) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let v be a vertex of D′, if v belongs to some S+ x , then d−(v) = k, otherwise it belongs to some S− x and then d+(v) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then observe that D′ does not contain the bidirected complete graph on � k+1 2 � + 1 vertices since every digon in D′ is contained in some S+ x or S− x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Thus we only have to prove that ⃗χ(D) ≤ k if and only if ⃗χ(D′) ≤ k to get the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let us first prove that ⃗χ(D) ≤ k implies ⃗χ(D′) ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Assume that ⃗χ(D) ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let φ : V −→ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' , k} be a k-dicolouring of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let φ′ be the k-dicolouring of D′ defined as follows: for each vertex x ∈ V , choose arbitrarily x− ∈ S− x , x+ ∈ S+ x , and set φ′(x−) = φ′(x+) = φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then choose a distinct colour for every other vertex v in S− x ∪ S+ x , and set φ′(v) to this colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We get that φ′ must be a k-dicolouring of D′: for each x ∈ V , every vertex but x− in S− x must be a sink in its colour class, and every vertex but x+ in S+ x must be a source in its colour class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Thus if D′, coloured with φ′, contains a monochromatic directed cycle C′, then C′ must be of the form x− 1 x+ 1 x− 2 x+ 2 · · · x− ℓ x+ ℓ x− 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' But then C = x1x2 · · · xℓx1 is a monochromatic directed cycle in D coloured with φ: a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Reciprocally, let us prove that ⃗χ(D′) ≤ k implies ⃗χ(D) ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Assume that ⃗χ(D′) ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let φ′ : V ′ −→ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=', k} be a k-dicolouring of D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let φ be the k-dicolouring of D defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' For each vertex x ∈ V , we know that |S+ x ∪ S− x | = k + 1, thus there must be two vertices x+ and x− in S+ x ∪ S− x such that φ′(x+) = φ′(x−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Moreover, since both S+ x and S− x are bidirected, one of these two vertices belongs to S+ x and the other one belongs to S− x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We assume without loss of generality x+ ∈ S+ x and x− ∈ S− x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then we set φ(x) = φ′(x+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We get that φ must be a k- dicolouring of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If D, coloured with φ, contains a monochromatic directed cycle C = x1x2 · · · xℓx1, then C′ = x− 1 x+ 1 x− 2 x+ 2 · · · x− ℓ x+ ℓ x− 1 is a monochromatic directed cycle in D′ coloured with φ′, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' 5 3 Redicolouring oriented graphs In this section, we restrict to oriented graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We first prove Theorem 8, let us restate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let ⃗G be an oriented graph of order n such that ∆min(⃗G) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then D2(⃗G) is connected and has diameter exactly n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Observe that, if D2(⃗G) is connected, then its diameter must be at least n: for any 2-dicolouring α, we can define its mirror ¯α where, for every vertex v ∈ V (⃗G), α(v) ̸= ¯α(v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' then every redicolouring sequence between α and ¯α has length at least n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let C be a directed cycle of length at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then D2(C) is connected and has diameter exactly n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let α and β be any two 2-dicolourings of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let x = diff(α, β) = |{v ∈ V (C) | α(v) ̸= β(v)}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' By induction on x ≥ 0, let us show that there exists a path of length at most x from α to β in D2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' This clearly holds for x = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=', α = β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Assume x > 0 and the result holds for x − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let v ∈ V (C) be such that α(v) ̸= β(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If v can be recoloured in β(v), then we recolour it and reach a new 2-dicolouring α′ such that diff(α′, β) = x−1 and the result holds by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Else if v cannot be recoloured, then recolouring v must create a monochromatic directed cycle, which must be C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then there must be a vertex v′, different from v, such that β(v) = α(v′) ̸= β(v′), and v′ can be recoloured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We recolour it and reach a new 2-dicolouring α′ such that diff(α′, β) = x − 1 and the result holds by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We are now ready to prove Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let α and β be any two 2-dicolourings of ⃗G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We will show that there exists a redicolouring sequence of length at most n between α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We may assume that ⃗G is strongly connected, otherwise we consider each strongly connected component independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' This implies in particular that ⃗G does not contain any sink nor source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let (X, Y ) be a partition of V (⃗G) such that, for every x ∈ X, d+(x) = 1, and for every y ∈ Y , d−(y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Assume first that ⃗G⟨X⟩ contains a directed cycle C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Since every vertex in X has exactly one out-neighbour, there is no arc leaving C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Thus, since ⃗G is strongly connected, ⃗G must be exactly C, and the result holds by Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Using a symmetric argument, we get the result when ⃗G⟨Y ⟩ contains a directed cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Assume now that both ⃗G⟨X⟩ and ⃗G⟨Y ⟩ are acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Thus, since every vertex in X has exactly one out- neighbour, ⃗G⟨X⟩ is the union of disjoint and independent in-trees, that are oriented trees in which all arcs are directed towards the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We denote by Xr the set of roots of these in-trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Symmetrically, ⃗G⟨Y ⟩ is the union of disjoint and independent out-trees (oriented trees in which all arcs are directed away from the root), and we denote by Yr the set of roots of these out-trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Set Xℓ = X \\ Xr and Yℓ = Y \\ Yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Observe that the arcs from X to Y form a perfect matching directed from Xr to Yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We denote by Mr this perfect matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Observe also that there can be any arc from Y to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Now we define X1 r and Y 1 r two subsets of Xr and Yr respectively, depending on the two 2-dicolourings α and β, as follows: X1 r = {x | xy ∈ Mr, α(x) = β(y) ̸= α(y) = β(x)} Y 1 r = {y | xy ∈ Mr, α(x) = β(y) ̸= α(y) = β(x)} Set X2 r = Xr \\ X1 r and Y 2 r = Yr \\ Y 1 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We denote by M 1 r (respectively M 2 r ) the perfect matching from X1 r to Y 1 r (respectively from X2 r to Y 2 r ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Figure 1 shows a partitioning of V (⃗G) into X1 r , X2 r, Xℓ, Y 1 r , Y 2 r , Yℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content='1: There exists a redicolouring sequence of length sα from α to some 2-dicolouring α′ and a redicolouring sequence of length sβ from β to some 2-dicolouring β′ such that each of the following holds: (i) For any arc xy ∈ Mr, α′(x) ̸= α′(y) and β′(x) ̸= β′(y), (ii) For any arc xy ∈ M 2 r , α′(x) = β′(x) (and so α′(y) = β′(y) by (i)), and (iii) sα + sβ ≤ |X2 r| + |Y 2 r |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' 6 X1 r X2 r Xℓ Y 1 r Y 2 r Yℓ ⃗G dicoloured with α X1 r X2 r Xℓ Y 1 r Y 2 r Yℓ ⃗G dicoloured with β Figure 1: The partitioning of V (⃗G) into X1 r, X2 r , Xℓ, Y 1 r , Y 2 r , Yℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Proof of claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We consider the arcs xy of M 2 r one after another and do the following recolourings depending on the colours of x and y in both α and β to get α′ and β′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If α(x) = α(y) = β(x) = β(y), then we recolour x in both α and β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Else if α(x) = α(y) ̸= β(x) = β(y), then we recolour x in α and we recolour y in β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Else if α(x) = β(x) ̸= α(y) = β(y), then we do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Else if α(x) ̸= α(y) = β(x) = β(y), then we recolour x in β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Finally if α(y) ̸= α(x) = β(x) = β(y), then we recolour y in β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Each of these recolourings is valid because, when a vertex in X2 r (respectively Y 2 r ) is recoloured, it gets a colour different from its only out-neighbour (respectively in-neighbour).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let α′ and β′ be the the two resulting 2- dicolourings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' By construction, α′ and β′ agree on X2 r ∪ Y 2 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' For each arc xy ∈ M 2 r , either α(x) = α′(x) or α(y) = α′(y), and the same holds for β and β′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' This implies that sα + sβ ≤ 2|M 2 r | = |X2 r | + |Y 2 r |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' ♦ Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content='2: There exists a redicolouring sequence from α′ to some 2-dicolouring ˜α of length s′ α and a redicolouring sequence from β′ to some 2-dicolouring ˜β of length s′ β such that each of the following holds: (i) ˜α and ˜β agree on V (⃗G) \\ (X1 r ∪ Y 1 r ), (ii) α′ and ˜α agree on Xr ∪ Yr, (iii) β′ and ˜β agree on Xr ∪ Yr, (iv) Xℓ ∪ Yℓ is monochromatic in ˜α (and in ˜β by (i)), and (v) s′ α + s′ β ≤ |Xℓ| + |Yℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Proof of claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Observe that in both 2-dicolourings α′ and β′, we are free to recolour any vertex of Xℓ ∪ Yℓ since there is no monochromatic arc from X to Y and both ⃗G⟨X⟩ and ⃗G⟨Y ⟩ are acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let n1 (respectively n2) be the number of vertices in Xℓ ∪ Yℓ that are coloured 1 (respectively 2) in both α′ and β′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Without loss of generality, assume that n1 ≤ n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then we set each vertex of Xℓ ∪ Yℓ to colour 2 in both α′ and β′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let ˜α and ˜β the resulting 2-dicolouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then s′ α + s′ β is exactly |Xℓ| + |Yℓ| + n1 − n2 ≤ |Xℓ| + |Yℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' ♦ 7 Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content='3: There is a redicolouring sequence between ˜α and ˜β of length |X1 r| + |Y 1 r |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Proof of claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' By construction of ˜α and ˜β, we only have to exchange the colours of x and y for each arc xy ∈ M 1 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Without loss of generality, we may assume that the colour of all vertices in Xℓ ∪ Yℓ by ˜α and ˜β is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We first prove that, by construction, we can recolour any vertex of X1 r ∪Y 1 r from 1 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Assume not, then there is such a vertex x ∈ X1 r ∪Y 1 r such that recolouring x from 1 to 2 creates a monochromatic directed cycle C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Since both ⃗G⟨X⟩ and ⃗G⟨Y ⟩ are acyclic, C must contain an arc of Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Since Mr does not contain any monochromatic arc in ˜α, then this arc must be incident to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Now observe that colour 2, in ˜α, induces an independent set on both ⃗G⟨X⟩ and ⃗G⟨Y ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' This implies that C must contain at least 2 arcs in Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' This is a contradiction since recolouring x creates exactly one monochromatic arc in Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then, for each arc xy ∈ M 1 r , we can first recolour the vertex coloured 1 and then the vertex coloured 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Note that we maintain the invariant that colour 2 induces an independent set on both ⃗G⟨X⟩ and ⃗G⟨Y ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We get a redicolouring sequence from ˜α to ˜β in exactly 2|M 1 r | = |X1 r| + |Y 1 r | steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' ♦ Combining the three claims, we finally proved that there exists a redicolouring sequence between α and β of length at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' In the following, when α is a dicolouring of a digraph D, and H is a subdigraph of D, we denote by α|H the restriction of α to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We will prove Corollary 9, let us restate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let ⃗G be an oriented graph of order n with ∆min(⃗G) = ∆ ≥ 1, and let k ≥ ∆ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then Dk(⃗G) is connected and has diameter at most 2∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We will show the result by induction on ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Assume first that ∆ = 1, let k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let α be any k-dicolouring of ⃗G and γ be any 2-dicolouring of ⃗G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' To ensure that Dk(⃗G) is connected and has diameter at most 2n, it is sufficient to prove that there is a redicolouring sequence between α and γ of length at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let H be the digraph induced by the set of vertices coloured 1 or 2 in α, and let J be V (⃗G) \\ V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' By Theorem 8, since ∆min(H) ≤ ∆min(⃗G) ≤ 1, we know that there exists a redicolouring sequence, in H, from α|H to γ|H of length at most |V (H)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' This redicolouring sequence extends in ⃗G because it only uses colours 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let α′ be the obtained dicolouring of ⃗G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Since α′(v) = γ(v) for every v ∈ H, we can recolour each vertex in J to its colour in γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' This shows that there is a redicolouring sequence between α and γ of length at most |V (H)| + |J| = |V (⃗G)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' This ends the case ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Assume now that ∆ ≥ 2 and let k ≥ ∆ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let α and β be two k-dicolourings of ⃗G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' By Corollary 7, we know that ⃗χ(⃗G) ≤ ∆ ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We first show that there is a redicolouring sequence of length at most 2n from α to some (k − 1)-dicolouring γ of ⃗G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' From α, whenever it is possible we recolour each vertex coloured 1, 2 or k with a colour of {3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' , k − 1} (when k = 3 we do nothing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let ˜α be the obtained dicolouring, and let M be the set of vertices coloured in {3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' , k − 1} by ˜α (when k = 3, M is empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We get that H = ⃗G − M satisfies ∆min(H) ≤ 2, since every vertex in H has at least one in-neighbour and one out-neighbour coloured c for every c ∈ {3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=', k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' By Corollary 7, there exists a 2-dicolouring γ|H of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' From ˜α|H, whenever it is possible, we recolour a vertex coloured 1 or 2 to colour k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let ˆα be the resulting dicolouring, and ˆH be the subdigraph of H induced by the vertices coloured 1 or 2 in ˆα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We get that ∆min( ˆH) ≤ 1 since every vertex in ˆH has, in ⃗G, at least one in-neighbour and one out-neighbour coloured c for every c ∈ {3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=', k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' In at most |V ( ˆH)| steps, using Theorem 8, we can recolour the vertices of V ( ˆH) to their colour in γ|H (using only colours 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then we can recolour each vertex coloured k to its colour in γ|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' This results in a redicolouring sequence of length at most 2n from α to some (k − 1)-dicolouring γ of ⃗G , since colour k is not used in the resulting dicolouring (recall that M is coloured with {3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=', k − 1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Now, from β, whenever it is possible we recolour each vertex to colour k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let ˜β be the obtained k-dicolouring, and let N be the set of vertices coloured k in ˜β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We get that J = ⃗G − N satisfies ∆min(J) ≤ ∆ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Thus, by induction, there exists a redicolouring sequence from ˜β|J to γ|J, in at most 2(∆ − 1)|V (J)| steps (using only colours {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' , k − 1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Since N is coloured k in ˜β, this extends to a redicolouring sequence in ⃗G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Now, since γ does not use colour k, we can recolour each vertex in N to its colour in γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We finally get a redicolouring sequence from β to γ of length at most 2(∆ − 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Concatenating the redicolouring sequence from α to γ and the one from γ to β, we get a redicolouring sequence from α to β in at most 2∆n steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' 8 4 An analogue of Brook’s theorem for digraph redicolouring Let us restate Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let D be a connected digraph with ∆max(D) = ∆ ≥ 3, k ≥ ∆ + 1, and α, β two k-dicolourings of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then at least one of the following holds: α is k-frozen, or β is k-frozen, or there is a redicolouring sequence of length at most c∆|V |2 between α and β, where c∆ = O(∆2) is a constant depending only on ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' An L-assignment of a digraph D is a function which associates to every vertex a list of colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' An L- dicolouring of D is a dicolouring α where, for every vertex v of D, α(v) ∈ L(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' An L-redicolouring sequence is a redicolouring sequence γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' , γr, such that for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' , r}, γi is an L-dicolouring of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let D = (V, A) be a digraph and L be a list-assignment of D such that, for every vertex v ∈ V , |L(v)| ≥ dmax(v) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let α be an L-dicolouring of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If u ∈ V is blocked in α, then for each colour c ∈ L(u) different from α(u), u has exactly one out-neighbour u+ c and one in-neighbour u− c coloured c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Moreover, if u+ c ̸= u− c , there must be a monochromatic directed path from u+ c to u− c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' In particular, u is not incident to a monochromatic arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Since u is blocked to its colour in α, for each colour c ∈ L(u) different from α(u), recolouring u to c must create a monochromatic directed cycle C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let v be the out-neighbour of u in C and w be the in-neighbour of u in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then α(v) = α(w) = c, and there is a monochromatic directed path (in C) from v to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' This implies that, for each colour c ∈ L(u) different from α(u), u has at least one out-neighbour and at least one in-neighbour coloured c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Since |L(u)| ≥ dmax(u) + 1, then |L(u)| = dmax(u) + 1, and u must have exactly one out-neighbour and exactly one in-neighbour coloured c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' In particular, u cannot be incident to a monochromatic arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let D = (V, A) be a digraph such that for every vertex v ∈ V , N +(v) \\ N −(v) ̸= ∅ and N −(v) \\ N +(v) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let L be a list assignment of D, such that for every vertex v ∈ V , |L(v)| ≥ dmax(v) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then for any pair of L-dicolourings α, β of D, there is an L-redicolouring sequence of length at most (|V | + 3)|V |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let x = diff(α, β) = |{v ∈ V | α(v) ̸= β(v)}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We will show by induction on x that there is an L- redicolouring sequence from α to β of length at most (|V | + 3)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' The result clearly holds for x = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' α = β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let v ∈ V be such that α(v) ̸= β(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We denote α(v) by c and β(v) by c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If v can be recoloured to c′, then we recolour it and we get the result by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Assume now that v cannot be recoloured to c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Whenever v is contained in a directed cycle C of length at least 3, such that every vertex of C but v is coloured c′, we do the following: we choose w a vertex of C different from v, such that β(w) ̸= c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We know that such a w exists, for otherwise C would be a monochromatic directed cycle in β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Now, since w is incident to a monochromatic arc in C, and because |L(w)| ≥ dmax(w) + 1, by Lemma 13, we know that w can be recoloured to some colour different from c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Thus we recolour w to this colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Observe that it does not increase x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' After repeating this process, maybe v cannot be recoloured to c′ because it is adjacent by a digon to some vertices coloured c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We know that these vertices are not coloured c′ in β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Thus, whenever such a vertex can be recoloured, we recolour it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' After this, let η be the obtained dicolouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If v can be recoloured to c′ in η, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Otherwise, there must be some vertices, blocked to colour c′ in η, adjacent to v by a digon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let S be the set of such vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Observe that, by Lemma 13, for every vertex s ∈ S, c belongs to L(s), for otherwise s would not be blocked in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We distinguish two cases, depending on the size of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' 9 If |S| ≥ 2, then by Lemma 13, v can be recoloured to a colour c′′, different from both c and c′, because v is adjacent by a digon with two neighbours coloured c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Hence we can successively recolour v to c′′, and every vertex of S to c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' This does not create any monochromatic directed cycle because for each s ∈ S, since s is blocked in η, by Lemma 13 v must be the only neighbour of s coloured c in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We can finally recolour v to c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If |S| = 1, let w be the only vertex in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If v can be recoloured to any colour (different from c′ since w is coloured c′), then we first recolour v, allowing us to recolour w to c, because v is the single neighbour of w coloured c in η by Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We finally can recolour v to c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Assume then that v is blocked to colour c in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let us fix w+ ∈ N +(w) \\ N −(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Since w is blocked to c′ in η, by Lemma 13, there exists exactly one vertex w− ∈ N −(w) \\ N +(w) such that η(w+) = η(w−) = c′′ and there must be a monochromatic directed path from w+ to w−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Since v is blocked to colour c in η, either vw− /∈ A or w+v /∈ A, otherwise, by Lemma 13, there must be a monochromatic directed path from w− to w+, which is blocking v to its colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' But since there is also a monochromatic directed path from w+ to w− (blocking w) there would be a monochromatic directed cycle, a contradiction (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' w v w+ w− Figure 2: The vertices v, w, w+ and w−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We distinguish the two possible cases: – if vw− /∈ A, then we start by recolouring w− with a colour that does not appear in its in-neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' This is possible because w− has a monochromatic entering arc, and because |L(w−)| ≥ dmax(w−)+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We first recolour w with c′′, since c′′ does not appear in its in-neighbourhood anymore (w− was the only one by Lemma 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Next we recolour v with c′: this is possible because v does not have any out-neighbour coloured c′ since w was the only one by Lemma 13 and w− is not an out-neighbour of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We can finally recolour w to colour c and w− to c′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' After all these operations, we exchanged the colours of v and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' – if w+v /∈ A, then we use a symmetric argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Observe that we found an L-redicolouring sequence from α to a α′, in at most |V |+3 steps, such that diff(α′, β) < diff(α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Thus by induction, we get an L-redicolouring sequence of length at most (|V | + 3)x between α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We are now able to prove Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' The idea of the proof is to divide the digraph D into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' One of them is bidirected and we will use Theorem 2 as a black box on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' In the other part, we know that each vertex is incident to at least two simple arcs, one leaving and one entering, and we will use Lemma 14 on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Proof of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let D = (V, A) be a connected digraph with ∆max(D) = ∆, k ≥ ∆ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let α and β be two k-dicolourings of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Assume that neither α nor β is k-frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We first make a simple observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' For any simple arc xy ∈ A, we may assume that N +(y) \\ N −(y) ̸= ∅ and N −(x) \\ N +(x) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If this is not the case, then every directed cycle containing xy must contain a digon, implying that the k-dicolouring graph of D is also the k-dicolouring graph of D \\ {xy}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then we may look for a redicolouring sequence in D \\ {xy}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' 10 Let X = {v ∈ V | N +(v) = N −(v)} and Y = V \\ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Observe that D⟨X⟩ is bidirected, and thus the dicolourings of D⟨X⟩ are exactly the colourings of UG(D⟨X⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We first show that α|D⟨X⟩ and β|D⟨X⟩ are not frozen k-colourings of D⟨X⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If Y is empty, then D⟨X⟩ = D and α|D⟨X⟩ and β|D⟨X⟩ are not k-frozen by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Otherwise, since D is connected, there exists x ∈ X such that, in D⟨X⟩, d+(x) = d−(x) ≤ ∆ − 1, implying that x is not blocked in any dicolouring of D⟨X⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Thus, by Theorem 2, there is a redicolouring sequence γ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' , γ′ r in D⟨X⟩ from α|D⟨X⟩ to β|D⟨X⟩, where r ≤ c∆|X|2, and c∆ = O(∆) is a constant depending on ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We will show that, for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' , r − 1}, if γi is a k-dicolouring of D which agrees with γ′ i on X, then there exist a k-dicolouring γi+1 of D that agrees with γ′ i+1 on X and a redicolouring sequence from γi to γi+1 of length at most ∆ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Observe that α agrees with γ′ 1 on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Now assume that there is such a γi, which agrees with γ′ i on X, and let vi ∈ X be the vertex for which γ′ i(vi) ̸= γ′ i+1(vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We denote by c (respectively c′) the colour of vi in γ′ i (respectively γ′ i+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If recolouring vi to c′ in γi is valid then we have the desired γi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Otherwise, we know that vi is adjacent with a digon (since vi is only adjacent to digons) to some vertices (at most ∆) coloured c′ in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Whenever such a vertex can be recoloured to a colour different from c′, we recolour it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Let ηi be the reached k-dicolouring after these operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If vi can be recoloured to c′ in ηi we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If not, then the neighbours of vi coloured c′ in Y are blocked to colour c′ in ηi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We denote by S the set of these neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We distinguish two cases: If |S| ≥ 2, then by Lemma 13, vi can be recoloured to a colour c′′, different from both c and c′, because vi has two neighbours with the same colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Then we successively recolour vi to c′′, and every vertex of S to c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' This does not create any monochromatic directed cycle because, by Lemma 13, for each s ∈ S, vi is the only neighbour of s coloured c in ηi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We can finally recolour vi to c′ to reach the desired γi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If |S| = 1, let y be the only vertex in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Since y belongs to Y and is blocked to its colour in ηi, by Lemma 13, we know that y has an out-neighbour y+ ∈ N +(y)\\N −(y) and an in-neighbour y− ∈ N −(y)\\N +(y) such that there is a monochromatic directed path from y+ to y−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Observe that both y+ and y− are recolourable in ηi by Lemma 13, because there are incident to a monochromatic arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' – If vi is not adjacent to y+, then we recolour y+ to any possible colour, and we recolour y to ηi(y+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We can finally recolour vi to c′ to reach the desired γi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' – If vi is not adjacent to y−, then we recolour y− to any possible colour, and we recolour y to ηi(y−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We can finally recolour vi to c′ to reach the desired γi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' – Finally if vi is adjacent to both y+ and y−, since ηi(y+) = ηi(y−), then vi can be recoloured to a colour c′′ different from c and c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' This allows us to recolour y to c, and we finally can recolour vi to c′ to reach the desired γi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We have shown that there is a redicolouring sequence of length at most (∆ + 2)c∆n2 from α to some α′ that agrees with β on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Now we define the list-assignment: for each y ∈ Y , L(y) = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' , k} \\ {β(x) | x ∈ N(y) ∩ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Observe that, for every y ∈ Y , |L(y)| ≥ k − |N +(y) ∩ X| ≥ ∆ + 1 − (∆ − d+ Y (y)) ≥ d+ Y (y) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Symmetrically, we get |L(y)| ≥ d− Y (y) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' This implies, in D⟨Y ⟩, |L(y)| ≥ dmax(y) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Note also that both α′ |D⟨Y ⟩ and β|D⟨Y ⟩ are L-dicolourings of D⟨Y ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Note finally that, for each y ∈ Y , N +(y) \\ N −(y) ̸= ∅ and N +(y) \\ N −(y) ̸= ∅ by choice of X and Y and by the initial observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' By Lemma 14, there is an L- redicolouring sequence in D⟨Y ⟩ between α′ |D⟨Y ⟩ and β|D⟨Y ⟩, with length at most (|Y | + 3)|Y |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' By choice of L, this extends directly to a redicolouring sequence from α′ to β on D of the same length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' The concatenation of the redicolouring sequence from α to α′ and the one from α′ to β leads to a redicolouring sequence from α to β of length at most c′ ∆|V |2, where c′ ∆ = O(∆2) is a constant depending on ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' 11 Remark 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If α is a k-frozen dicolouring of a digraph D, with k ≥ ∆max(D) + 1, then D must be bidirected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If D is not bidirected, then we choose v a vertex incident to a simple arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' If v cannot be recoloured in α, by Lemma 13, since v is incident to a simple arc, there exists a colour c for which v has an out-neighbour w and an in-neighbour u both coloured c, such that u ̸= w and there is a monochromatic directed path from w to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' But then, every vertex on this path is incident to a monochromatic arc, and it can be recoloured by Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Thus, α is not k-frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' This shows that an obstruction of Theorem 10 is exactly the bidirected graph of an obstruction of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' 5 Further research In this paper, we established some analogues of Brooks’ Theorem for the dichromatic number of oriented graphs and for digraph redicolouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Many open questions arise, we detail a few of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Restricted to oriented graphs, Mcdiarmid and Mohar (see [11]) conjectured that the Directed Brooks’ Theorem can be improved to the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Conjecture 16 (Mcdiarmid and Mohar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Every oriented graph ⃗G has ⃗χ(⃗G) = O � ∆max log(∆max) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Concerning digraph redicolouring, we believe that Corollary 9 and Theorem 10 can be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' We pose the following two conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Conjecture 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' There is an absolute constant c such that for every integer k and every oriented graph ⃗G such that k ≥ ∆min(⃗G) + 1, the diameter of Dk(⃗G) is bounded by cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Conjecture 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' There is an absolute constant d such that for every integer k and every digraph D with k ≥ ∆max(D) + 1, the diameter of Dk(D) is bounded by dn2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Given an orientation ⃗G of a planar graph, a celebrated conjecture from Neumann-Lara [14] states that the dichromatic number of ⃗G is at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' It is known that it must be 4-mixing because planar graphs are 5- degenerate [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' It is also known that there exists 2-freezable orientations of planar graphs [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Thus the following problem, stated in [5], remains open: Question 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Is every oriented planar graph 3-mixing ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Acknowledgement I am grateful to Fr´ed´eric Havet and Nicolas Nisse for stimulating discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' References [1] Pierre Aboulker and Guillaume Aubian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Four proofs of the directed Brooks’ Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' arXiv preprint arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content='01600, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' [2] Valentin Bartier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Combinatorial and Algorithmic aspects of Reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' PhD thesis, Universit´e Grenoble Alpes, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Bokal, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Fijavz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Juvan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Kayll, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Mohar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' The circular chromatic number of a digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Graph Theory, 46(3):227–240, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' [4] Marthe Bonamy, Matthew Johnson, Ioannis Lignos, Viresh Patel, and Daniel Paulusma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Journal of Combinatorial Optimization, 27(1):132–143, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' 12 [5] Nicolas Bousquet, Fr´ed´eric Havet, Nicolas Nisse, Lucas Picasarri-Arrieta, and Amadeus Reinald.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Digraph redicolouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' arXiv preprint arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content='03417, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Brooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' On colouring the nodes of a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Mathematical Proceedings of the Cambridge Philosoph- ical Society, 37(2):194–197, 1941.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' [7] Luis Cereceda, Jan Van den Heuvel, and Matthew Johnson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Mixing 3-colourings in bipartite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Euro- pean Journal of Combinatorics, 30(7):1593–1606, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' [8] Luis Cereceda, Jan van den Heuvel, and Matthew Johnson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Finding paths between 3-colorings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Journal of Graph Theory, 67(1):69–82, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' [9] Carl Feghali, Matthew Johnson, and Dani¨el Paulusma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' A reconfigurations analogue of Brooks’ Theorem and its consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Journal of Graph Theory, 83(4):340–358, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' [10] Ararat Harutyunyan and Bojan Mohar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Gallai’s theorem for list coloring of digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' SIAM Journal on Discrete Mathematics, 25(1):170–180, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' [11] Ararat Harutyunyan and Bojan Mohar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=" Strengthened Brooks' theorem for digraphs of girth at least three." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' The Electronic Journal of Combinatorics, 18(1), October 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' [12] Jan van den Heuvel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' The complexity of change, page 127–160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' London Mathematical Society Lecture Note Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Cambridge University Press, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' [13] Bojan Mohar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Eigenvalues and colorings of digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Linear Algebra and its Applications, 432(9):2273– 2277, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' [14] Victor Neumann-Lara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' The dichromatic number of a digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Theory Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=', 33:265–270, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' [15] Naomi Nishimura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Introduction to reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' Algorithms, 11(4), 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} +page_content=' 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf'} diff --git a/AdFJT4oBgHgl3EQfrS3C/content/tmp_files/2301.11608v1.pdf.txt b/AdFJT4oBgHgl3EQfrS3C/content/tmp_files/2301.11608v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9dd08c0afc570ffecfad66b3aab6af5cd1368f7d --- /dev/null +++ b/AdFJT4oBgHgl3EQfrS3C/content/tmp_files/2301.11608v1.pdf.txt @@ -0,0 +1,1275 @@ +A Multi-View Joint Learning Framework for Embedding Clinical Codes and Text +Using Graph Neural Networks +Lecheng Kong, Christopher King, Bradley Fritz, Yixin Chen +Washington University in St. Louis +One Brookings Drive +St. Louis, Missouri 63130, USA +{jerry.kong, christopherking, bafritz, ychen25}@wustl.edu +Abstract +Learning to represent free text is a core task in many clini- +cal machine learning (ML) applications, as clinical text con- +tains observations and plans not otherwise available for in- +ference. State-of-the-art methods use large language models +developed with immense computational resources and train- +ing data; however, applying these models is challenging be- +cause of the highly varying syntax and vocabulary in clinical +free text. Structured information such as International Clas- +sification of Disease (ICD) codes often succinctly abstracts +the most important facts of a clinical encounter and yields +good performance, but is often not as available as clinical text +in real-world scenarios. We propose a multi-view learning +framework that jointly learns from codes and text to com- +bine the availability and forward-looking nature of text and +better performance of ICD codes. The learned text embed- +dings can be used as inputs to predictive algorithms indepen- +dent of the ICD codes during inference. Our approach uses a +Graph Neural Network (GNN) to process ICD codes, and Bi- +LSTM to process text. We apply Deep Canonical Correlation +Analysis (DCCA) to enforce the two views to learn a similar +representation of each patient. In experiments using planned +surgical procedure text, our model outperforms BERT models +fine-tuned to clinical data, and in experiments using diverse +text in MIMIC-III, our model is competitive to a fine-tuned +BERT at a tiny fraction of its computational effort. +We also find that the multi-view approach is beneficial for +stabilizing inferences on codes that were unseen during train- +ing, which is a real problem within highly detailed coding +systems. We propose a labeling training scheme in which +we block part of the training code during DCCA to improve +the generalizability of the GNN to unseen codes. In experi- +ments with unseen codes, the proposed scheme consistently +achieves superior performance on code inference tasks. +1 +Introduction +An electronic health record (EHR) stores a patient’s com- +prehensive information within a healthcare system. It pro- +vides rich contexts for evaluating the patient’s status and fu- +ture clinical plans. The information in an EHR can be clas- +sified as structured or unstructured. Over the past decade, +ML techniques have been widely applied to uncover pat- +terns behind structured information such as lab results (Yu, +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +Beam, and Kohane 2018; Shickel et al. 2017; Goldstein et al. +2017). Recently, the surge of deep learning and large-scale +pre-trained networks has allowed unstructured data, mainly +clinical notes, to be effectively used for learning (Huang, Al- +tosaar, and Ranganath 2019; Lee et al. 2020; Si et al. 2019). +However, most methods focus on either structured or un- +structured data only. +A particularly informative type of structured data is the +International Classification of Diseases (ICD) codes. ICD is +an expert-identified hierarchical medical concept ontology +used to systematically organize medical concepts into cate- +gories and encode valuable domain knowledge about a pa- +tient’s diseases and procedures. +Because ICD codes are highly specific and unambigu- +ous, ML models that use ICD codes to predict procedure +outcomes often yield more accurate results than those do +not (Deschepper et al. 2019; Liu et al. 2020a). However, +the availability of ICD codes is not always guaranteed. For +example, billing ICD codes are generated after the clinical +encounter, meaning that we cannot use the ICD codes to +predict post-operative outcomes before the surgery. A more +subtle but crucial drawback of using ICD codes is that there +might be unseen codes during inference. When a future pro- +cedure is associated with a code outside the trained subset, +most existing models using procedure codes cannot accu- +rately represent the case. Shifts in coding practices can also +cause data during inference to not overlap the trained set. +On the other hand, unstructured text data are readily and +consistently available. Clinical notes are generated as free +text and potentially carry a doctor’s complete insight about +a patient’s condition, including possible but not known di- +agnoses and planned procedures. Unfortunately, the clinical +text is a challenging natural language source, containing am- +biguous abbreviations, input errors, and words and phrases +rarely seen in pre-training sources. It is consequently diffi- +cult to train a robust model that predicts surgery outcomes +from the large volume of free texts. Most current models +rely on large-scale pre-trained models (Huang, Altosaar, and +Ranganath 2019; Lee et al. 2020). Such methods require +a considerable corpus of relevant texts to fine-tune, which +might not be available at a particular facility. Hence, mod- +els that only consider clinical texts suffer from poor perfor- +mance and incur huge computation costs. +To overcome the problems of models using only text or +arXiv:2301.11608v1 [cs.CL] 27 Jan 2023 + +codes, we propose to learn from the ICD codes and clini- +cal text in a multi-view joint learning framework. We ob- +serve that despite having different formats, the text and code +data are complementary and broadly describe the same un- +derlying facts about the patient. This enables each learner +(view) to use the other view’s representation as a regulariza- +tion function where less information is present. Under our +framework, even when one view is missing, the other view +can perform inference independently and maintain the effec- +tive data representation learned from the different perspec- +tives, which allows us to train reliable text models without a +vast corpus and computation cost required by other text-only +models. +Specifically, we make the following contributions in this +paper. (1) We propose a multi-view learning framework us- +ing Deep Canonical Correlation Analysis (DCCA) for ICD +codes and clinical notes. (2) We propose a novel tree-like +structure to encode ICD codes by relational graph and ap- +ply Relational Graph Convolution Network (RGCN) to em- +bed ICD codes. (3) We use a two-stage Bi-LSTM to en- +code lengthy clinical texts. (4) To solve the unseen code pre- +diction problem, we propose a labeling training scheme in +which we simulate unseen node prediction during training. +Combined with the DCCA optimization process, the training +scheme teaches the RGCN to discriminate between unseen +and seen codes during inference and achieves better perfor- +mance than plain RGCN. +2 +Related Works +Deep learning on clinical notes. Many works focus on +applying deep learning to learn representations of clini- +cal texts for downstream tasks. Early work (Boag et al. +2018) compared the performance of classic NLP meth- +ods including bag-of-words (Zhang, Jin, and Zhou 2010), +Word2Vec (Mikolov et al. 2013), and Long-Short-Term- +Memory (LSTM) (Hochreiter and Schmidhuber 1997) on +clinical prediction tasks. These methods solely learn from +the training text, but as the clinical texts are very noisy, they +either tend to overfit the data or fail to uncover valuable pat- +terns behind the text. Inspired by large-scale pre-trained lan- +guage models such as BERT (Devlin et al. 2018), a series of +works developed transformer models pre-trained on medical +notes, including ClinicalBERT (Huang, Altosaar, and Ran- +ganath 2019), BioBERT (Lee et al. 2020), and PubBERT +(Alsentzer et al. 2019). These models fine-tune general lan- +guage models on a large corpus of clinical texts and achieve +superior performance. Despite the general nature of these +models, the fine-tuning portion may not translate well to new +settings. For example, PubBERT is trained on the clinical +texts of a single tertiary hospital, and the colloquial terms +used and procedures typically performed may not map to +different hospitals. BioBERT is trained on Pubmed abstracts +and articles, which also is likely poorly representative of the +topics and terms used to, for example, describe a planned +surgery. +Some other models propose to use joint learning models +to learn from the clinical text, and structured data (e.g., mea- +sured blood pressure and procedure codes) (Wei et al. 2016; +Zhang et al. 2020a). Since the structured data are less noisy, +these models can produce better and more stable results. +However, most assume the co-existence of text and struc- +tured data at the inference time, while procedure codes for a +patient are frequently incomplete until much later. +Machine learning and procedure codes. Procedure +codes are a handy resource for EHR data mining. Most +works focus on automatic coding, using machine learning +models to predict a patient’s diagnostic codes from clini- +cal notes (Pascual, Luck, and Wattenhofer 2021; Li and Yu +2020). Some other works directly use the billing code to pre- +dict clinical outcomes (Liu et al. 2020a; Deschepper et al. +2019), whereas our work focuses on using the high correla- +tion of codes and text data to augment the performance of +each. Most of these works exploit the code hierarchies by +human-defined logic based on domain knowledge. In con- +trast, our proposed framework uses GNN and can encode +arbitrary relations between codes. +Graph neural networks. A series of works (Xu et al. +2018; Gilmer et al. 2017) summarize GNN structures in +which each node iteratively aggregates neighbor nodes’ em- +bedding and summarizes information in a neighborhood. +The resulting node embeddings can be used to predict down- +stream tasks. RGCN (Schlichtkrull et al. 2018) generalizes +GNN to heterogeneous graphs where nodes and edges can +have different types. Our model utilizes such heterogeneous +properties on our proposed hierarchy graph encoding. Some +works (Liu et al. 2020b; Choi et al. 2020) applied GNN to +model interaction between EHRs, whereas our model uses +GNN on the code hierarchy. +Privileged information. Our approach is related to the +Learning Under Privileged Information (LUPI) (Vapnik and +Vashist 2009) paradigm, where the privileged information +is only accessible during training (in this case, billing code +data). Many works have applied LUPI to other fields like +computer vision (Lambert, Sener, and Savarese 2018) and +metric learning (Fouad et al. 2013). +3 +Methods +Admissions with ICD codes and clinical text can be repre- +sented as D = {(C1, A1, y1), ..., (Cn, An, yn)}, where Ci +is a set of ICD codes for admission i, Ai is a set of clin- +ical texts, and yi is the desired task label (e.g. mortality, +re-admission, etc.). The ultimate goal is to minimize task- +appropriate losses L defined as: +min +fC,gC +� +i +L(fC(gC(Ci)), yi) +(1) +and +min +fA,gA +� +i +L(fA(gA(Ai)), yi), +(2) +where gC and gA embed codes and texts to vector repre- +sentations respectively, and fC and fA map representations +to the task labels. Note that (gC, fC) and (gA, fA) should +operate independently during inference, meaning that even +when one type of data is missing, we can still make accurate +predictions. +In this section, we first propose a novel ICD ontology +graph encoding method and describe how we use Graph + +Figure 1: Overall multi-view joint learning framework. +Blue boxes/arrows represent the text prediction pipeline, and +green represents the code prediction pipeline. Dashed boxes +and arrows denote processes only happening during training. +By removing the dashed parts, text and code pipelines can +predict tasks independently. +Neural Network (GNN) to parameterize gC. We then de- +scribe the two-stage Bi-LSTM (gA) to embed lengthy clini- +cal texts. We then describe how to use DCCA on the repre- +sentation from gC and gA to generate representations that are +less noisy and more informative, so the downstream models +fC and fA are able to make accurate predictions. Figure 1 +shows the overall architecture of our multi-view joint learn- +ing framework. +3.1 +ICD Ontology as Graphs +The ICD ontology has a hierarchical scheme. We can rep- +resent it as a tree graph as shown in Figure 2, where each +node is a medical concept and a node’s children are finer di- +visions of the concept. All top-level nodes are connected to a +root node. In this tree graph, only the leaf nodes correspond +to observable codes in the coding system, all other nodes are +the hierarchy of the ontology. This representation is widely +adopted by many machine learning systems (Zhang et al. +2020b; Li, Ma, and Gao 2021) as a refinement of the earlier +approach of grouping together all codes at the top level of the +hierarchy. A tree graph is ideal for algorithms based on mes- +sage passing. It allows pooling of information within disjoint +groups, and encodes a compact set of neighbors. However, +it (1) ignores the granularity of different levels of classifica- +tion, and (2) cannot encode similarities of nodes that are dis- +tant from each other. This latter point comes about because +a tree system may split on factors that are not the most rele- +Figure 2: Top: Conventional encoding of ICD ontology. Bot- +tom Left: ICD ontology encoded with relations. Relation +types for different levels are denoted by different colors. +Bottom Right: Jump connection creates additional edges to +leaf nodes’ predecessors, denoted by dashed color lines. +vant for a given task, such as the same procedure in an arm +versus a leg, or because cross-system concepts are empiri- +cally very correlated in medical syndromes, such as kidney +failure and certain endocrine disorders. +To overcome the aforementioned problems, we propose +to augment the tree graph with edge types and jump connec- +tions. Unlike conventional tree graphs, where all edges have +the same edge type, we use different edge types for connec- +tions between different levels in the tree graph as shown in +the bottom left of Figure 2. For example, ICD-10 codes have +seven characters and hence eight levels in the graph (includ- +ing the root level). The edges between the root node and its +children have edge Type 1, and the edges between the sev- +enth level and the last level (actual code level) have edge +Type 7. Different edge types not only encode whether two +procedures are related but also encode the level of similarity +between codes. +With multiple edge types introduced to the graph, we are +able to further extend the graph structure by jump connec- +tions. For each leaf node, we add one additional edge be- +tween the node and each of its predecessors up to the root +node, as shown in the bottom right of Figure 2. The edge +type depends on the level that the predecessor resides. For +example, in the ICD-10 tree graph, a leaf node will have +seven additional connections to its predecessors. Its edge to +the root node will have Type 8 (the first seven types are used +to represent connections between levels), and its edge to the +third level node will have Type 10. Jump connections signifi- +cantly increase the connectivity of the graph. Meanwhile, we +still maintain the good hierarchical information of the origi- + +Adm 1: Posterior Cervical Decompression. +Adm 1: 0QSH06Z +Adm 2: Thoracic Laminectomy for. +Adm 2: 00CU0ZZ,009U3ZX,02HV33Z +Adm 3: ... +Adm 3: ... +Word2Vec +Code to Node Index +Two-Stage LSTM +RGCN +1 +Codes Embedding +Text Embedding +Generated from +Sum/Max Pooling +Text Projection Matrix +DCCA +Code Projection Matrix +Projected Text +Projected Codes +Embedding +Embedding +MLP +MLP +Downstream Task Predictior +Downstream Task PredictionRoot node r connects +to all level-1 ontology +Bottom level nodes +represent actual codes +0 +8 +0RGAOT0 +ORGA0T1 +5A09357 +5A09358 +Relation-Augmented Graph +Jump Connection Graphnal tree graph because the jump connections are represented +by a different set of edge types. Using jump connection helps +uncover relationships between codes that are not presented +in the ontology. For example, the relationship between ane- +mia and renal failure can be learned using jump connec- +tion even though these diverge at the root node in ICD-9 +and ICD-10. Moreover, GNNs suffer from over-smoothing, +where all node representations converge to the same value +when the GNN has too many layers (Li, Han, and Wu 2018). +If we do not employ jump connections, the maximal distance +between one leaf node to another is twice the number of +levels in the graph. To capture the connection between the +nodes, we will need a GNN with that many layers, which +is computationally expensive and prone to over-smoothing. +Jump connections make the distance between two leaf nodes +two, and this ensures that the GNN is able to embed any +correlation between two nodes. We will discuss this in more +detail in Section 3.2. +3.2 +Embedding ICD Codes using GNN +We use GNN to embed medical concepts in the ICD ontol- +ogy. Let G = {V, E, R} be a graph, where V is its set of the +vertex (medical concepts in the ICD graph), E ⊆ {V ×V } is +its set of edges (connects medical concept to its sub-classes), +and R is the set of edge type in the graph (edges in different +levels and jump connection). As each ICD code corresponds +to one node in the graph, we use code and node interchange- +ably. +We adopt RGCN (Schlichtkrull et al. 2018), which itera- +tively updates a node’s embedding from its neighbor nodes. +Specifically, the kth layer of RGCN on node u ∈ V is: +h(k+1) +u += σ +� +�� +r∈R +� +v∈N r +u +1 +cu,r +W (k) +r +h(k) +v ++ W (k)h(k) +u +� +� (3) +where N r +i is the set of neighbors of i that connects to i by re- +lation r, h(k) +i +is the embedding of node i after k GNN layers, +h0 +i is a randomly initialized trainable embedding, W (k) +r +is a +linear transformation on embeddings of nodes in N r +i , W (k) +updates the embedding of u, and σ is a nonlinear activation +function. We have c = |N r +i | as a normalization factor. +After T iterations, h(T ) +u +can be used to learn down- +stream tasks. Since a patient can have a set of codes, Ci = +{vi1, vi2, vi3, ...} ⊆ V , we use sum and max pooling to sum- +marize Ci in an embedding function gC: +gC(Ci) = +� +v∈Ci +h(T ) +v +⊕ max({h(T ) +v +|v ∈ Ci}), +(4) +where max is the element-wise maximization, and ⊕ rep- +resents vector concatenation. Summation more accurately +summarizes the codes’ information, while maximization +provides regularization and stability in DCCA, which we +will discuss in Section 3.4. +Training RGCN helps embed the ICD codes into vectors +based on the defined ontology. Nodes that are close together +in the graph will be assigned similar embeddings because +of their similar neighborhood. Moreover, distant nodes that +appear together frequently in the health record can also be +assigned correlated embeddings because the jump connec- +tion keeps the maximal distance between two nodes at two. +Consider a set of codes C = {u, v}, because of the sum- +mation in the code pooling, using a 2-layer RGCN, we will +have non-zero gradients of hT +u and hT +v with respect to h0 +v +and h0 +u, respectively, which connects the embeddings of u +and v. In contrast, applying RGCN on a graph without jump +connections will result in zero gradients when the distance +between u and v is greater than two. +3.3 +Embedding Clinical Notes using Bi-LSTM +Patients can have different numbers of clinical texts in each +encounter. Where applicable, we sort the texts in an en- +counter in ascending order by time, and have a set of texts +Ai = (ai1, ai2, ..., ain). In our examples, we concatenate +the texts together to a single document Hi, and we have +Hi = CAT(Ai) = � +j={1...n} aij. We leave to future work +the possibility of further modeling the collection. +The concatenated text might be very lengthy with over +ten thousands word tokens, and RNN suffers from dimin- +ishing gradients with LSTM-type modifications. While at- +tention mechanisms are effective for arbitrary long-range +dependence, they require large sample size and expensive +computational resources. Hence, following previously suc- +cessful approach (Huang et al. 2019), we adopt a two-stage +model which stacks a low-frequency RNN on a local RNN. +Given Hi, we first split it into blocks of equal size b, Hi = +{Hi1, Hi2, ..., HiK}. The last block HiK is padded to length +b. The two-stage model first generates block-wise text em- +beddings by +lHik = LSTM({w(Hik1), w(Hik2), ..., w(Hikb)}), +(5) +where w(·) is a Word2Vec (Mikolov et al. 2013) trainable +embedding function. The representation of Ai is given by +gA(Ai) = LSTM({lHi1, ..., lHiK}). +(6) +The two-stage learning scheme minimizes the effect of di- +minishing gradients while maintaining the temporal order of +the text. +3.4 +DCCA between Graph and Text Data +As previously mentioned, ICD codes may not be available at +the time when models would be most useful, but are struc- +tured and easier to analyze, while the clinical text is read- +ily available but very noisy. Despite different data formats, +they usually describe the same information: the main diag- +noses and treatments for an encounter. Borrowing ideas from +multi-view learning, we can use them to supplement each +other. Many existing multi-view learning methods require +the presence of both views during inference and are not able +to adapt to the applications we envision. Specifically, we use +DCCA (Andrew et al. 2013; Wang et al. 2015) on gA(Ai) +and gC(Ci) to learn a joint representation. DCCA solves the + +following optimization problem, +max +gC,gA,U,V +1 +N tr(U T M T +C MAV ) +s.t. +U T ( 1 +N M T +C MC + rCI)U = I, +V T ( 1 +N M T +AMA + rAI)V = I, +uT +i M T +C MAvj = 0, +∀i ̸= j, +1 ≤ i, j ≤ L +MC = stack{gC(Ci)|∀i}, +MA = stack{gA(Ai)|∀i}, +(7) +where MC and MA are the matrices stacked by vector rep- +resentations of codes and texts, (rC, rA) > 0 are regulariza- +tion parameters. U = [u1, ..., uL] and V = [v1, ..., vL] maps +GNN and Bi-LSTM output to maximally correlated embed- +ding, and L is a hyper-parameter controlling the number of +correlated dimensions. We use gC(Ci)U, gA(Ai)V as the fi- +nal embedding of codes and texts. By maximizing their cor- +relation, we force the weak learner (usually the LSTM) to +learn a similar representation as the strong learner (usually +the GNN) and to filter out inputs unrelated to the structured +data. Hence, when a record’s codes can yield correct results, +its text embedding is highly correlated with that of the codes, +and the text should also be likely to produce correct predic- +tions. +During development, we found that a batch of ICD data +often contains many repeated codes with the same embed- +ding and that a SUM pooling tended to obtain a less than +full rank embedding matrix MC and MA, which causes in- +stability in solving the optimization problem. A nonlinear +max pooling function helps prevent this. +The above optimization problem suggests full-batch train- +ing. However, the computation graph will be too large for +the text and code data. Following (Wang et al. 2015), we use +large mini-batches to train the model, and from the experi- +mental results, they sufficiently represent the overall distri- +bution. After training, we stack MC, MA again from all data +output and obtain U and V as fixed projection matrix from +equation (7). +After obtaining the projection matrices and embedding +models, we attach two MLPs (fA and fC) to the embedding +models as the classifier, and train/fine-tune fA (fC) and gA +(gC) together in an end-to-end fashion with respect to the +learning task using the loss functions in (1) and (2). +4 +Predicting Unseen Codes +In the previous section, we discuss the formulation of ICD +ontology and how we can use DCCA to generate embed- +dings that share representations across views. In this section, +we will demonstrate another use case for DCCA-regularized +embeddings. In real-world settings, the set of codes that re- +searchers observe in training is usually a small subset of the +entire ICD ontology. In part, this is due to the extreme speci- +ficity of some ontologies, with ICD-10-PCS having 87,000 +distinct procedures and ICD-10-CM 68,000 diagnostic pos- +sibilities before considering that some codes represent a fur- +ther modification of another entity. In even large training +samples, some codes will likely be seen zero or a small num- +ber of times in training. Traditional models using indepen- +dent code embedding are expected to function poorly on rare +codes and have arbitrary output on previously unseen nodes, +even if similar entities are contained in the training data. +Our proposed model and the graph-embedded hierarchy +can naturally address the above challenge. Its two features +enable predictions of novel codes at inference: +• Relational embedding. By embedding the novel code +in the ontology graph, we are able to exploit the repre- +sentation of its neighbors. For example, a rare diagnostic +procedure’s embedding is highly influenced by other pro- +cedures that are nearby in the ontology. +• Jump connection. While other methods also exploit +the proximity defined by the hierarchy, as we suggested +above, codes can be highly correlated but remain distant +in the graph. Jump connections increase the graph con- +nectivity; hence, our model can seek the whole hierarchy +for potential connection to the missing code. Because the +connections across different levels are assigned different +relation types, our GNN can also differentiate the likeli- +hood of connections across different levels and distances. +Meanwhile, during inference, the potential problem is that +the model does not automatically differentiate between the +novel and the previously seen codes. Because the model +never uses novel codes to generate any gC(Ci), the embed- +dings of the seen and novel nodes experience different gra- +dient update processes and hence are from different distri- +butions. Nevertheless, during inference, the model will treat +them as if they are from the same distribution. However, +such transferability and credibility of novel node embed- +dings are not guaranteed, and applying them homogeneously +may result in untrustworthy predictions. +Hence, we propose a labeling training scheme to teach +the model how to handle novel nodes during inference. Let +G = {V, E, R} be the ICD graph and U be the set of unique +nodes in the training set, U ⊆ V . We select a random subset +Us from U to form the seen nodes during training, and Uu = +V \ Us be treated as unseen nodes. We augment the initial +node embeddings with 1-0 labels, formally, +h0+ +u += h0 +u ⊕ 1 +∀u ∈ Us +h0+ +v += h0 +v ⊕ 0 +∀v ∈ V \ Us +(8) +Note that we still use h0 +u as the trainable node embedding, +while the input to the RGCN is augmented to h0+ +u . We fur- +ther extract data that only contain the seen nodes to form the +seen data: Ds = {(Ci, Ai, yi)|c ∈ Us∀c ∈ Ci}. +We, again, use DCCA on Ds to maximize the correlation +between the text representation and the code representation. +After obtaining the projection matrix, we train on the en- +tire dataset D to minimize the prediction loss. Note that D +contains nodes that do not appear in the DCCA process and +are labeled differently from the seen nodes. The different la- +bels allow the RGCN to tell whether a node is unseen during +the DCCA process. If unseen nodes hurt the prediction, it +will be reflected in the prediction loss. Intuitively, if unseen +nodes are less credible, data with more 0-labeled nodes will + +Method +Local Data +MIMIC-III +DEL +DIA +TH +D30 +MORT +R30 +Corr. +17.3 ± 1.3 +16.8 ± 2.6 +16.8 ± 2.6 +16.8 ± 2.6 +10.4 ± 1.7 +12.7 ± 2.3 +BERT +65.2 ± 0.6 +76.3 ± 1.2 +62.1 ± 1.1 +74.6 ± 1.8 +88.4 ± 1.8 +69.2 ± 1.9 +ClinicalBERT +66.3 ± 0.5 +77.0 ± 0.9 +62.7 ± 0.8 +74.9 ± 1.5 +90.5 ± 1.3 +71.4 ± 1.8 +Bi-LSTM +64.6 ± 0.2 +76.8 ± 1.8 +61.3 ± 1.2 +73.9 ± 1.9 +87.3 ± 1.7 +68.4 ± 2.6 +DCCA+Bi-LSTM +66.9 ± 0.8 +78.9 ± 1.1 +61.6 ± 1.1 +76.5 ± 1.3 +87.2 ± 1.6 +71.1 ± 1.4 +RGCN +76.4 ± 1.2 +97.2 ± 1.1 +75.9 ± 3.0 +91.5 ± 1.0 +90.4 ± 1.0 +68.6 ± 1.4 +DCCA+RGCN +78.9 ± 1.3 +98.4 ± 0.9 +77.6 ± 1.2 +91.5 ± 1.3 +90.5 ± 1.5 +67.2 ± 2.5 +RGCN+Bi-LSTM +79.5 ± 1.7 +97.1 ± 1.4 +75.6 ± 0.8 +90.8 ± 0.8 +91.3 ± 1.2 +69.5 ± 1.2 +DCCA+RGCN+Bi-LSTM +78.7 ± 2.3 +98.2 ± 1.3 +77.1 ± 2.9 +91.0 ± 0.9 +90.1 ± 1.3 +71.2 ± 1.0 +Table 1: DCCA Joint Learning and baseline AUROC (%). Top 4 lines use clinical notes only during inference, middle 2 ICD +codes only, and bottom 2 both. Corr = Sum of correlation of latent representations over 20 dimensions. +Method +Local Data +MIMIC-III +DEL +DIA +TH +D30 +MORT +R30 +RGCN +74.6 ± 1.2 +87.3 ± 13.1 +67.4 ± 6.9 +82.8 ± 3.7 +84.5 ± 3.6 +60.4 ± 2.8 +RGCN+Labling +73.2 ± 0.6 +87.4 ± 14.9 +68.5 ± 3.4 +83.8 ± 2.1 +85.7 ± 3.6 +61.3 ± 2.3 +DCCA+RGCN +74.9 ± 1.0 +89.1 ± 12.5 +70.8 ± 0.9 +83.5 ± 1.9 +85.1 ± 4.1 +61.7 ± 2.6 +DCCA+RGCN+Labeling +75.3 ± 1.1 +95.4 ± 0.7 +70.6 ± 3.2 +84.4 ± 1.4 +86.4 ± 4.2 +63.4 ± 2.8 +Table 2: Ablation Study of the Labeling Training Scheme under Unseen Code Setting in AUROC (%). +have poor prediction results; GNN can capture this charac- +teristic and reflect it in the prediction by assigning less pos- +itive/negative scores to queries with more 0-labeled nodes. +The labeling training scheme essentially blocks a part of the +training code during DCCA and thus obtains embeddings +for Us and Uu from different distributions. And we train on +the entire training dataset so that the model learns to handle +seen and unseen codes heterogeneously. This setup mimics +the actual inference scenario. Note that despite being differ- +ent, the distributions of seen and unseen node embeddings +can be similar and overlapped. Thus, the additional 1-0 la- +beling is necessary to differentiate them. +5 +Experimental Results +Datasets. We use two datasets to evaluate the performance +of our framework: Proprietary Dataset. This dataset con- +tains medical records of 38,551 admissions at the local Hos- +pital from 2018 to 2021. Each entry is also associated with +a free text procedural description and a set of ICD-10 pro- +cedure codes. We aim to use our framework to predict a set +of post-operative outcomes, including delirium (DEL), dial- +ysis (DIA), troponin high (TH), and death in 30 days (D30). +MIMIC-III dataset (Johnson et al. 2016). This dataset con- +tains medical records of 58,976 unique ICU hospital ad- +mission from 38,597 patients at the Beth Israel Deaconess +Medical Center between 2001 and 2012. Each admission +record is associated with a set of ICD-9 diagnoses codes +and multiple clinical notes from different sources, includ- +ing case management, consult, ECG, discharge summary, +general nursing, etc. We aim to predict two outcomes from +the codes and texts: (1) In-hospital mortality (MORT). We +use admissions with hospital expire flag=1 in the MIMIC- +III dataset as the positive data and sample the same number +of negative data to form the final dataset. All clinical notes +generated on the last day of admission are filtered out to +avoid directly mentioning the outcome. We use all clinical +notes ordered by time and take the first 2,500-word tokens +as the input text. (2) 30-day readmission (R30). We follow +(Huang, Altosaar, and Ranganath 2019), label admissions +where a patient is readmitted within 30 days as positive, and +sample an equal number of negative admissions. Newborn +and death admissions are filtered out. We only use clinical +notes of type Discharge Summary and take the first 2,500- +word tokens as the input text. Sample sizes can be found in +Table 3. +Effectiveness of DCCA training. We split the dataset +with a train/validation/test ratio of 8:1:1 and use 5-fold +cross-validation to evaluate our model. GNN and Bi-LSTM +are optimized in the DCCA process using the training set. +The checkpoint model with the best validation correlation +is picked to compute the projection matrix only from the +training dataset. Then we attach an MLP head to the tar- +get prediction model (either the GNN or the Bi-LSTM) and +fine-tune the model in an end-to-end fashion to minimize the +prediction loss. +For this task, we compare our framework to popular pre- +trained models ClinicalBERT and BERT. We also compare +it to the base GNN and Bi-LSTM to show the effective- +ness of our proposed framework. We additionally provide +experimental results where both text and code embedding + +are used to make predictions. We compare our model with a +vanilla multi-view model without DCCA. For all baselines, +we report their Area Under Receiver Operating Characteris- +tic (AUROC) as evaluation metrics, and Average Precision +(AP) can be found in Appendix A. For all datasets, we set +L, the number of correlated dimensions to 20, and report the +total amount of correlation obtained (Corr). +Table 1 shows the main results. For clinical notes predic- +tion, we can see that the codes augmented model can con- +sistently outperform the base Bi-LSTM, with an average rel- +ative performance increase of 2.4% on the proprietary data +and 1.6% on the MIMIC-III data. Our proposed method out- +performs BERT on most tasks and achieves very competi- +tive performance compared to that of ClinicalBERT. Note +that our model only trains on the labeled EHR data without +unsupervised training on extra data like BERT and Clini- +calBERT do. ClinicalBERT has been previously trained and +fine-tuned on the entire MIMIC dataset, including the dis- +charge summaries, and therefore these results may overesti- +mate its performance. +For ICD code prediction, we see that DCCA brings a 1.5% +performance increase on the proprietary data. Since the +codes model significantly outperforms the language model +on all tasks, the RGCN is a much stronger learner and has +less information to learn from the text model. Comparing the +results of the proprietary and the MIMIC datasets, we can +see that DCCA brings a more significant performance boost +to the proprietary dataset, presumably because of the larger +amount of correlation obtained in the proprietary dataset +(85% versus 58%). Moreover, an important difference in +these datasets is the ontology used: MIMIC-III uses ICD-9 +and the proprietary dataset uses ICD-10. The ICD-9 ontol- +ogy tree has a height of four, which is much smaller than that +of ICD-10 and is more coarsely classified. This may also ex- +plain the smaller performance gains in MIMIC-III. +The combined model with DCCA only brings a slight per- +formance boost compared to the one without because the +amount of information for the models to learn is equiva- +lent. Nevertheless, the DCCA model encourages the two +views’ embeddings to agree and allows independent predic- +tion. In contrast, a vanilla multi-view model does not help +the weaker learner learn from the stronger learner. +Unseen Codes Experiments. We identify the set of +unique codes in the dataset. We split the codes into k-fold +and ran k experiments on each split. For each experiment, +we pick one fold as the unseen code set. Data that contain +at least one unseen code are used as the evaluation set. The +evaluation set is split into two halves as the valid and test +sets. The rest of the data forms the training set. We pick an- +other fold from the code split as the DCCA unseen code +set. Training set data that do not contain any DCCA unseen +code form the DCCA training set. Then, the entire training +set is used for task fine-tuning. Because the distribution of +codes is not uniform, the number of data for each split is not +equal across different folds. We use k=10 for the proprietary +dataset and k=20 for the MIMIC-III dataset to generate a +reasonable data division. We provide average split sizes in +Appendix C. +For this task, we compare our method with the base GNN, +# Admission +# Pos. Samples +# Unique codes +DEL +11,064 +5,367 +5,637 +DIA +38,551 +1,387 +9,320 +TH +38,551 +1,235 +9,320 +D30 +38,551 +1,444 +9,320 +MORT +5,926 +2,963 +4,448 +R30 +10,998 +5,499 +3,645 +Table 3: Statistics of different datasets and tasks. +base GNN augmented with the same labeling training strat- +egy, and DCCA-optimized GNN to demonstrate the out- +standing performance of our framework. Similarly, we re- +port AUROC and include AP in Appendix A. +Table 2 summarizes the results of the unseen codes exper- +iments. Note that all test data contain at least one code that +never appears in the training process. In such a more diffi- +cult inference scenario, comparing the plain RGCN with the +DCCA-augmented RGCN, we see a 2.2% average relative +performance increase on the proprietary dataset. With the +labeling learning method, we can further improve the per- +formance gain to 4.2%. On the MIMIC-III dataset, the per- +formance boost of our model over the plain RGCN is 3.6%, +demonstrating our method’s ability to differentiate seen and +unseen codes. We also notice that DCCA alone only slightly +improves the performance on the MIMIC-III dataset (1.4%). +We suspect that while the labeling training scheme helps dis- +tinguish seen and unseen codes, the number of data used +in the DCCA process is also reduced. As MORT and R30 +datasets are smaller and a small DCCA training set may not +faithfully represent the actual data distribution, the regular- +ization effect of DCCA diminishes. +6 +Conclusions +Predicting patient outcomes from EHR data is an essen- +tial task in clinical ML. Conventional methods that solely +learn from clinical texts suffer from poor performance, and +those that learn from codes have limited application in real- +world clinical settings. In this paper, we propose a multi- +view framework that jointly learns from the clinical notes +and ICD codes of EHR data using Bi-LSTM and GNN. We +use DCCA to create shared information but maintain each +view’s independence during inference. This allows accurate +prediction using clinical notes when the ICD codes are miss- +ing, which is commonly the case in pre-operative analysis. +We also propose a label augmentation method for our frame- +work, which allows the GNN model to make effective infer- +ences on codes that are not seen during training, enhancing +generalizability. Experiments are conducted on two different +datasets. Our methods show consistent effectiveness across +tasks. In the future, we plan to incorporate more data types in +the EHR and combine them with other multi-view learning +methods to make more accurate predictions. + +References +Alsentzer, E.; Murphy, J. 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International +journal of machine learning and cybernetics, 1(1): 43–52. + +A +Average Precision Score Results +AP results demonstrate a similar pattern to AUROC results, +where DCCA augmented model can consistently outper- +form the base model while achieving very competitive re- +sults compared to ClinicalBERT for the text data as shown +in Table 5. The proposed labeling training scheme can also +consistently improve our model’s performance on the un- +seen codes experiments, as shown in Table 6. +B +Hyperparameters +We use grid search for hyperparameter tuning. For missing +view experiments on text, we fix the number of RGCN layers +to be 3. We use 32 for all hidden dimensions as we found that +varying hidden size has minimal impact on the performance +of the data. Text and Code represent the hyperparameters +used for text and code inference tasks. Table 7 summarizes +the set of hyperparameters used for tuning. +C +Unseen Code Sample Size +We use 10-fold code split for the local data and 20-fold code +split for the MIMIC-III data so that the split sizes are reason- +able for training. We report the average number of samples +for all tasks in Table 4. +DCCA Train +Full Train +Test +DEL +3,458.9 +4,624.5 +3,219.4 +DIA +19,305.4 +23,717.8 +7,416.3 +TH +19,305.4 +23,717.8 +7,416.3 +D30 +19,305.4 +23,717.8 +7,416.3 +MORT +4,603.1 +6,148.2 +2,424.7 +R30 +2,528.1 +3,264.0 +1,330.8 +Table 4: Average Split Size in Unseen Codes Experiment +. +D +Data And Implementation +We adopted the local dataset because it is the only dataset we +have access to that uses both clinical free texts and ICD-10 +codes. The implementation details of the MIMIC-III dataset +experiments can be found in the supplementary material +(code). + +Local Data +MIMIC-III +DEL +DIA +TH +D30 +MORT +R30 +Corr. +17.3 ± 1.3 +16.8 ± 2.6 +16.8 ± 2.6 +16.8 ± 2.6 +10.4 ± 1.7 +12.7 ± 2.3 +BERT +65.4 ± 0.7 +23.6 ± 1.2 +6.5 ± 1.2 +15.1 ± 1.6 +85.9 ± 1.9 +67.7 ± 1.6 +ClinicalBERT +66.0 ± 0.6 +23.5 ± 0.7 +7.0 ± 0.8 +15.8 ± 2.1 +88.6 ± 1.2 +70.1 ± 2.2 +LSTM +64.4 ± 0.5 +22.1 ± 1.6 +6.3 ± 0.9 +14.3 ± 0.7 +85.4 ± 1.4 +66.2 ± 2.8 +DCCA+LSTM +65.4 ± 0.4 +24.6 ± 0.8 +6.3 ± 1.4 +15.9 ± 1.0 +84.9 ± 1.6 +70.4 ± 2.1 +RGCN +74.2 ± 1.7 +91.9 ± 2.1 +11.7 ± 0.3 +60.4 ± 4.8 +90.1 ± 1.7 +67.4 ± 2.2 +DCCA+RGCN +76.6 ± 1.7 +90.6 ± 1.6 +14.8 ± 0.5 +60.8 ± 4.9 +90.2 ± 1.2 +68.4 ± 1.9 +RGCN+Bi-LSTM +78.6 ± 2.6 +90.3 ± 1.6 +12.6 ± 0.7 +62.1 ± 1.5 +90.2 ± 1.4 +66.8 ± 1.5 +DCCA+RGCN+Bi-LSTM +77.2 ± 2.1 +91.6 ± 2.0 +15.8 ± 0.6 +61.7 ± 1.2 +89.6 ± 1.7 +67.6 ± 1.6 +Table 5: Effect of DCCA Joint Learning Compared to Different Baselines in AP (%). +Method +Local Data +MIMIC-III +DEL +DIA +TH +D30 +MORT +R30 +RGCN +72.6 ± 1.5 +82.1 ± 9.4 +9.2 ± 4.1 +53.4 ± 7.1 +87.6 ± 3.6 +63.7 ± 3.1 +RGCN+Labling +73.2 ± 0.9 +83.6 ± 9.6 +9.1 ± 4.7 +54.2 ± 9.1 +88.5 ± 3.9 +65.4 ± 3.6 +DCCA+RGCN +73.8 ± 1.2 +85.3 ± 8.1 +12.6 ± 1.3 +53.6 ± 8.2 +88.7 ± 3.0 +65.0 ± 2.9 +DCCA+RGCN+Labeling +74.5 ± 1.1 +89.4 ± 1.3 +12.7 ± 3.0 +53.5 ± 6.9 +89.9 ± 3.1 +65.1 ± 3.4 +Table 6: Ablation Study of the Labeling Training Scheme under Unseen Code Setting in AP (%). +Hyperparameter +Local-Text +Local-Code +MIMIC-III-Text +MIMIC-III-Code +GNN +#layers +3 +{2,3,4} +3 +{2,3,4} +LSTM +block size(b) +- +- +30 +30 +MLP +#layers +2 +2 +1 +1 +dropout +{0,0.2,0.4} +{0,0.2,0.4} +{0.2,0.4,0.6,0.8} +{0.2,0.4,0.6,0.8} +DCCA +learning rate +0.001 +0.001 +0.001 +0.001 +batch size +1024 +1024 +400 +400 +Task +learning rate +{1e-3,1e-4,1e-5} +{1e-3,1e-4,1e-5} +{1e-3,1e-4,1e-5} +{1e-3,1e-4,1e-5} +batch size +256 +256 +32 +32 +Table 7: Hyperparameters used for tuning. + diff --git 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='kong, christopherking, bafritz, ychen25}@wustl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='edu Abstract Learning to represent free text is a core task in many clini- cal machine learning (ML) applications, as clinical text con- tains observations and plans not otherwise available for in- ference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' State-of-the-art methods use large language models developed with immense computational resources and train- ing data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' however, applying these models is challenging be- cause of the highly varying syntax and vocabulary in clinical free text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Structured information such as International Clas- sification of Disease (ICD) codes often succinctly abstracts the most important facts of a clinical encounter and yields good performance, but is often not as available as clinical text in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We propose a multi-view learning framework that jointly learns from codes and text to com- bine the availability and forward-looking nature of text and better performance of ICD codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' The learned text embed- dings can be used as inputs to predictive algorithms indepen- dent of the ICD codes during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Our approach uses a Graph Neural Network (GNN) to process ICD codes, and Bi- LSTM to process text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We apply Deep Canonical Correlation Analysis (DCCA) to enforce the two views to learn a similar representation of each patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' In experiments using planned surgical procedure text, our model outperforms BERT models fine-tuned to clinical data, and in experiments using diverse text in MIMIC-III, our model is competitive to a fine-tuned BERT at a tiny fraction of its computational effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We also find that the multi-view approach is beneficial for stabilizing inferences on codes that were unseen during train- ing, which is a real problem within highly detailed coding systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We propose a labeling training scheme in which we block part of the training code during DCCA to improve the generalizability of the GNN to unseen codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' In experi- ments with unseen codes, the proposed scheme consistently achieves superior performance on code inference tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 1 Introduction An electronic health record (EHR) stores a patient’s com- prehensive information within a healthcare system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' It pro- vides rich contexts for evaluating the patient’s status and fu- ture clinical plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' The information in an EHR can be clas- sified as structured or unstructured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Over the past decade, ML techniques have been widely applied to uncover pat- terns behind structured information such as lab results (Yu, Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Beam, and Kohane 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Shickel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Goldstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Recently, the surge of deep learning and large-scale pre-trained networks has allowed unstructured data, mainly clinical notes, to be effectively used for learning (Huang, Al- tosaar, and Ranganath 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Si et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' However, most methods focus on either structured or un- structured data only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' A particularly informative type of structured data is the International Classification of Diseases (ICD) codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' ICD is an expert-identified hierarchical medical concept ontology used to systematically organize medical concepts into cate- gories and encode valuable domain knowledge about a pa- tient’s diseases and procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Because ICD codes are highly specific and unambigu- ous, ML models that use ICD codes to predict procedure outcomes often yield more accurate results than those do not (Deschepper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' However, the availability of ICD codes is not always guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' For example, billing ICD codes are generated after the clinical encounter, meaning that we cannot use the ICD codes to predict post-operative outcomes before the surgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' A more subtle but crucial drawback of using ICD codes is that there might be unseen codes during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' When a future pro- cedure is associated with a code outside the trained subset, most existing models using procedure codes cannot accu- rately represent the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Shifts in coding practices can also cause data during inference to not overlap the trained set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' On the other hand, unstructured text data are readily and consistently available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Clinical notes are generated as free text and potentially carry a doctor’s complete insight about a patient’s condition, including possible but not known di- agnoses and planned procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Unfortunately, the clinical text is a challenging natural language source, containing am- biguous abbreviations, input errors, and words and phrases rarely seen in pre-training sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' It is consequently diffi- cult to train a robust model that predicts surgery outcomes from the large volume of free texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Most current models rely on large-scale pre-trained models (Huang, Altosaar, and Ranganath 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Such methods require a considerable corpus of relevant texts to fine-tune, which might not be available at a particular facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Hence, mod- els that only consider clinical texts suffer from poor perfor- mance and incur huge computation costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' To overcome the problems of models using only text or arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='11608v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='CL] 27 Jan 2023 codes, we propose to learn from the ICD codes and clini- cal text in a multi-view joint learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We ob- serve that despite having different formats, the text and code data are complementary and broadly describe the same un- derlying facts about the patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' This enables each learner (view) to use the other view’s representation as a regulariza- tion function where less information is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Under our framework, even when one view is missing, the other view can perform inference independently and maintain the effec- tive data representation learned from the different perspec- tives, which allows us to train reliable text models without a vast corpus and computation cost required by other text-only models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Specifically, we make the following contributions in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' (1) We propose a multi-view learning framework us- ing Deep Canonical Correlation Analysis (DCCA) for ICD codes and clinical notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' (2) We propose a novel tree-like structure to encode ICD codes by relational graph and ap- ply Relational Graph Convolution Network (RGCN) to em- bed ICD codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' (3) We use a two-stage Bi-LSTM to en- code lengthy clinical texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' (4) To solve the unseen code pre- diction problem, we propose a labeling training scheme in which we simulate unseen node prediction during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Combined with the DCCA optimization process, the training scheme teaches the RGCN to discriminate between unseen and seen codes during inference and achieves better perfor- mance than plain RGCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2 Related Works Deep learning on clinical notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Many works focus on applying deep learning to learn representations of clini- cal texts for downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Early work (Boag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2018) compared the performance of classic NLP meth- ods including bag-of-words (Zhang, Jin, and Zhou 2010), Word2Vec (Mikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2013), and Long-Short-Term- Memory (LSTM) (Hochreiter and Schmidhuber 1997) on clinical prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' These methods solely learn from the training text, but as the clinical texts are very noisy, they either tend to overfit the data or fail to uncover valuable pat- terns behind the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Inspired by large-scale pre-trained lan- guage models such as BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2018), a series of works developed transformer models pre-trained on medical notes, including ClinicalBERT (Huang, Altosaar, and Ran- ganath 2019), BioBERT (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2020), and PubBERT (Alsentzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' These models fine-tune general lan- guage models on a large corpus of clinical texts and achieve superior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Despite the general nature of these models, the fine-tuning portion may not translate well to new settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' For example, PubBERT is trained on the clinical texts of a single tertiary hospital, and the colloquial terms used and procedures typically performed may not map to different hospitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' BioBERT is trained on Pubmed abstracts and articles, which also is likely poorly representative of the topics and terms used to, for example, describe a planned surgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Some other models propose to use joint learning models to learn from the clinical text, and structured data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=', mea- sured blood pressure and procedure codes) (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Since the structured data are less noisy, these models can produce better and more stable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' However, most assume the co-existence of text and struc- tured data at the inference time, while procedure codes for a patient are frequently incomplete until much later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Machine learning and procedure codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Procedure codes are a handy resource for EHR data mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Most works focus on automatic coding, using machine learning models to predict a patient’s diagnostic codes from clini- cal notes (Pascual, Luck, and Wattenhofer 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Li and Yu 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Some other works directly use the billing code to pre- dict clinical outcomes (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Deschepper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2019), whereas our work focuses on using the high correla- tion of codes and text data to augment the performance of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Most of these works exploit the code hierarchies by human-defined logic based on domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' In con- trast, our proposed framework uses GNN and can encode arbitrary relations between codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' A series of works (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Gilmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2017) summarize GNN structures in which each node iteratively aggregates neighbor nodes’ em- bedding and summarizes information in a neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' The resulting node embeddings can be used to predict down- stream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' RGCN (Schlichtkrull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2018) generalizes GNN to heterogeneous graphs where nodes and edges can have different types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Our model utilizes such heterogeneous properties on our proposed hierarchy graph encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Some works (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2020) applied GNN to model interaction between EHRs, whereas our model uses GNN on the code hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Privileged information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Our approach is related to the Learning Under Privileged Information (LUPI) (Vapnik and Vashist 2009) paradigm, where the privileged information is only accessible during training (in this case, billing code data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Many works have applied LUPI to other fields like computer vision (Lambert, Sener, and Savarese 2018) and metric learning (Fouad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 3 Methods Admissions with ICD codes and clinical text can be repre- sented as D = {(C1, A1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=', (Cn, An, yn)}, where Ci is a set of ICD codes for admission i, Ai is a set of clin- ical texts, and yi is the desired task label (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' mortality, re-admission, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' The ultimate goal is to minimize task- appropriate losses L defined as: min fC,gC � i L(fC(gC(Ci)), yi) (1) and min fA,gA � i L(fA(gA(Ai)), yi), (2) where gC and gA embed codes and texts to vector repre- sentations respectively, and fC and fA map representations to the task labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Note that (gC, fC) and (gA, fA) should operate independently during inference, meaning that even when one type of data is missing, we can still make accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' In this section, we first propose a novel ICD ontology graph encoding method and describe how we use Graph Figure 1: Overall multi-view joint learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Blue boxes/arrows represent the text prediction pipeline, and green represents the code prediction pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Dashed boxes and arrows denote processes only happening during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' By removing the dashed parts, text and code pipelines can predict tasks independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Neural Network (GNN) to parameterize gC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We then de- scribe the two-stage Bi-LSTM (gA) to embed lengthy clini- cal texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We then describe how to use DCCA on the repre- sentation from gC and gA to generate representations that are less noisy and more informative, so the downstream models fC and fA are able to make accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Figure 1 shows the overall architecture of our multi-view joint learn- ing framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 ICD Ontology as Graphs The ICD ontology has a hierarchical scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We can rep- resent it as a tree graph as shown in Figure 2, where each node is a medical concept and a node’s children are finer di- visions of the concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' All top-level nodes are connected to a root node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' In this tree graph, only the leaf nodes correspond to observable codes in the coding system, all other nodes are the hierarchy of the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' This representation is widely adopted by many machine learning systems (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Li, Ma, and Gao 2021) as a refinement of the earlier approach of grouping together all codes at the top level of the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' A tree graph is ideal for algorithms based on mes- sage passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' It allows pooling of information within disjoint groups, and encodes a compact set of neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' However, it (1) ignores the granularity of different levels of classifica- tion, and (2) cannot encode similarities of nodes that are dis- tant from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' This latter point comes about because a tree system may split on factors that are not the most rele- Figure 2: Top: Conventional encoding of ICD ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Bot- tom Left: ICD ontology encoded with relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Relation types for different levels are denoted by different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Bottom Right: Jump connection creates additional edges to leaf nodes’ predecessors, denoted by dashed color lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' vant for a given task, such as the same procedure in an arm versus a leg, or because cross-system concepts are empiri- cally very correlated in medical syndromes, such as kidney failure and certain endocrine disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' To overcome the aforementioned problems, we propose to augment the tree graph with edge types and jump connec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Unlike conventional tree graphs, where all edges have the same edge type, we use different edge types for connec- tions between different levels in the tree graph as shown in the bottom left of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' For example, ICD-10 codes have seven characters and hence eight levels in the graph (includ- ing the root level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' The edges between the root node and its children have edge Type 1, and the edges between the sev- enth level and the last level (actual code level) have edge Type 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Different edge types not only encode whether two procedures are related but also encode the level of similarity between codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' With multiple edge types introduced to the graph, we are able to further extend the graph structure by jump connec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' For each leaf node, we add one additional edge be- tween the node and each of its predecessors up to the root node, as shown in the bottom right of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' The edge type depends on the level that the predecessor resides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' For example, in the ICD-10 tree graph, a leaf node will have seven additional connections to its predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Its edge to the root node will have Type 8 (the first seven types are used to represent connections between levels), and its edge to the third level node will have Type 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Jump connections signifi- cantly increase the connectivity of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Meanwhile, we still maintain the good hierarchical information of the origi- Adm 1: Posterior Cervical Decompression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Adm 1: 0QSH06Z Adm 2: Thoracic Laminectomy for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Adm 2: 00CU0ZZ,009U3ZX,02HV33Z Adm 3: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Adm 3: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='Word2Vec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='Code to Node Index ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='Two-Stage LSTM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='RGCN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='Codes Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='Text Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='Generated from ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='Sum/Max Pooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='Text Projection Matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='DCCA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='Code Projection Matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='Projected Text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='Projected Codes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='MLP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='MLP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='Downstream Task Predictior ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='Downstream Task PredictionRoot node r connects ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='to all level-1 ontology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='Bottom level nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='represent actual codes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='0RGAOT0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='ORGA0T1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5A09357 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5A09358 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='Relation-Augmented Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='Jump Connection Graphnal tree graph because the jump connections are represented ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='by a different set of edge types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Using jump connection helps uncover relationships between codes that are not presented in the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' For example, the relationship between ane- mia and renal failure can be learned using jump connec- tion even though these diverge at the root node in ICD-9 and ICD-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Moreover, GNNs suffer from over-smoothing, where all node representations converge to the same value when the GNN has too many layers (Li, Han, and Wu 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' If we do not employ jump connections, the maximal distance between one leaf node to another is twice the number of levels in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' To capture the connection between the nodes, we will need a GNN with that many layers, which is computationally expensive and prone to over-smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Jump connections make the distance between two leaf nodes two, and this ensures that the GNN is able to embed any correlation between two nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We will discuss this in more detail in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 Embedding ICD Codes using GNN We use GNN to embed medical concepts in the ICD ontol- ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Let G = {V, E, R} be a graph, where V is its set of the vertex (medical concepts in the ICD graph), E ⊆ {V ×V } is its set of edges (connects medical concept to its sub-classes), and R is the set of edge type in the graph (edges in different levels and jump connection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' As each ICD code corresponds to one node in the graph, we use code and node interchange- ably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We adopt RGCN (Schlichtkrull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2018), which itera- tively updates a node’s embedding from its neighbor nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Specifically, the kth layer of RGCN on node u ∈ V is: h(k+1) u = σ � �� r∈R � v∈N r u 1 cu,r W (k) r h(k) v + W (k)h(k) u � � (3) where N r i is the set of neighbors of i that connects to i by re- lation r, h(k) i is the embedding of node i after k GNN layers, h0 i is a randomly initialized trainable embedding, W (k) r is a linear transformation on embeddings of nodes in N r i , W (k) updates the embedding of u, and σ is a nonlinear activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We have c = |N r i | as a normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' After T iterations, h(T ) u can be used to learn down- stream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Since a patient can have a set of codes, Ci = {vi1, vi2, vi3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='} ⊆ V , we use sum and max pooling to sum- marize Ci in an embedding function gC: gC(Ci) = � v∈Ci h(T ) v ⊕ max({h(T ) v |v ∈ Ci}), (4) where max is the element-wise maximization, and ⊕ rep- resents vector concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Summation more accurately summarizes the codes’ information, while maximization provides regularization and stability in DCCA, which we will discuss in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Training RGCN helps embed the ICD codes into vectors based on the defined ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Nodes that are close together in the graph will be assigned similar embeddings because of their similar neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Moreover, distant nodes that appear together frequently in the health record can also be assigned correlated embeddings because the jump connec- tion keeps the maximal distance between two nodes at two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Consider a set of codes C = {u, v}, because of the sum- mation in the code pooling, using a 2-layer RGCN, we will have non-zero gradients of hT u and hT v with respect to h0 v and h0 u, respectively, which connects the embeddings of u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' In contrast, applying RGCN on a graph without jump connections will result in zero gradients when the distance between u and v is greater than two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 Embedding Clinical Notes using Bi-LSTM Patients can have different numbers of clinical texts in each encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Where applicable, we sort the texts in an en- counter in ascending order by time, and have a set of texts Ai = (ai1, ai2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=', ain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' In our examples, we concatenate the texts together to a single document Hi, and we have Hi = CAT(Ai) = � j={1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='n} aij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We leave to future work the possibility of further modeling the collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' The concatenated text might be very lengthy with over ten thousands word tokens, and RNN suffers from dimin- ishing gradients with LSTM-type modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' While at- tention mechanisms are effective for arbitrary long-range dependence, they require large sample size and expensive computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Hence, following previously suc- cessful approach (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2019), we adopt a two-stage model which stacks a low-frequency RNN on a local RNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Given Hi, we first split it into blocks of equal size b, Hi = {Hi1, Hi2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=', HiK}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' The last block HiK is padded to length b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' The two-stage model first generates block-wise text em- beddings by lHik = LSTM({w(Hik1), w(Hik2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=', w(Hikb)}), (5) where w(·) is a Word2Vec (Mikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2013) trainable embedding function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' The representation of Ai is given by gA(Ai) = LSTM({lHi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=', lHiK}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' (6) The two-stage learning scheme minimizes the effect of di- minishing gradients while maintaining the temporal order of the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 DCCA between Graph and Text Data As previously mentioned, ICD codes may not be available at the time when models would be most useful, but are struc- tured and easier to analyze, while the clinical text is read- ily available but very noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Despite different data formats, they usually describe the same information: the main diag- noses and treatments for an encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Borrowing ideas from multi-view learning, we can use them to supplement each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Many existing multi-view learning methods require the presence of both views during inference and are not able to adapt to the applications we envision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Specifically, we use DCCA (Andrew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2015) on gA(Ai) and gC(Ci) to learn a joint representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' DCCA solves the following optimization problem, max gC,gA,U,V 1 N tr(U T M T C MAV ) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' U T ( 1 N M T C MC + rCI)U = I, V T ( 1 N M T AMA + rAI)V = I, uT i M T C MAvj = 0, ∀i ̸= j, 1 ≤ i, j ≤ L MC = stack{gC(Ci)|∀i}, MA = stack{gA(Ai)|∀i}, (7) where MC and MA are the matrices stacked by vector rep- resentations of codes and texts, (rC, rA) > 0 are regulariza- tion parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' U = [u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=', uL] and V = [v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=', vL] maps GNN and Bi-LSTM output to maximally correlated embed- ding, and L is a hyper-parameter controlling the number of correlated dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We use gC(Ci)U, gA(Ai)V as the fi- nal embedding of codes and texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' By maximizing their cor- relation, we force the weak learner (usually the LSTM) to learn a similar representation as the strong learner (usually the GNN) and to filter out inputs unrelated to the structured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Hence, when a record’s codes can yield correct results, its text embedding is highly correlated with that of the codes, and the text should also be likely to produce correct predic- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' During development, we found that a batch of ICD data often contains many repeated codes with the same embed- ding and that a SUM pooling tended to obtain a less than full rank embedding matrix MC and MA, which causes in- stability in solving the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' A nonlinear max pooling function helps prevent this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' The above optimization problem suggests full-batch train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' However, the computation graph will be too large for the text and code data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Following (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2015), we use large mini-batches to train the model, and from the experi- mental results, they sufficiently represent the overall distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' After training, we stack MC, MA again from all data output and obtain U and V as fixed projection matrix from equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' After obtaining the projection matrices and embedding models, we attach two MLPs (fA and fC) to the embedding models as the classifier, and train/fine-tune fA (fC) and gA (gC) together in an end-to-end fashion with respect to the learning task using the loss functions in (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 4 Predicting Unseen Codes In the previous section, we discuss the formulation of ICD ontology and how we can use DCCA to generate embed- dings that share representations across views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' In this section, we will demonstrate another use case for DCCA-regularized embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' In real-world settings, the set of codes that re- searchers observe in training is usually a small subset of the entire ICD ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' In part, this is due to the extreme speci- ficity of some ontologies, with ICD-10-PCS having 87,000 distinct procedures and ICD-10-CM 68,000 diagnostic pos- sibilities before considering that some codes represent a fur- ther modification of another entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' In even large training samples, some codes will likely be seen zero or a small num- ber of times in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Traditional models using indepen- dent code embedding are expected to function poorly on rare codes and have arbitrary output on previously unseen nodes, even if similar entities are contained in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Our proposed model and the graph-embedded hierarchy can naturally address the above challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Its two features enable predictions of novel codes at inference: Relational embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' By embedding the novel code in the ontology graph, we are able to exploit the repre- sentation of its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' For example, a rare diagnostic procedure’s embedding is highly influenced by other pro- cedures that are nearby in the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Jump connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' While other methods also exploit the proximity defined by the hierarchy, as we suggested above, codes can be highly correlated but remain distant in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Jump connections increase the graph con- nectivity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' hence, our model can seek the whole hierarchy for potential connection to the missing code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Because the connections across different levels are assigned different relation types, our GNN can also differentiate the likeli- hood of connections across different levels and distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Meanwhile, during inference, the potential problem is that the model does not automatically differentiate between the novel and the previously seen codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Because the model never uses novel codes to generate any gC(Ci), the embed- dings of the seen and novel nodes experience different gra- dient update processes and hence are from different distri- butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Nevertheless, during inference, the model will treat them as if they are from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' However, such transferability and credibility of novel node embed- dings are not guaranteed, and applying them homogeneously may result in untrustworthy predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Hence, we propose a labeling training scheme to teach the model how to handle novel nodes during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Let G = {V, E, R} be the ICD graph and U be the set of unique nodes in the training set, U ⊆ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We select a random subset Us from U to form the seen nodes during training, and Uu = V \\ Us be treated as unseen nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We augment the initial node embeddings with 1-0 labels, formally, h0+ u = h0 u ⊕ 1 ∀u ∈ Us h0+ v = h0 v ⊕ 0 ∀v ∈ V \\ Us (8) Note that we still use h0 u as the trainable node embedding, while the input to the RGCN is augmented to h0+ u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We fur- ther extract data that only contain the seen nodes to form the seen data: Ds = {(Ci, Ai, yi)|c ∈ Us∀c ∈ Ci}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We, again, use DCCA on Ds to maximize the correlation between the text representation and the code representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' After obtaining the projection matrix, we train on the en- tire dataset D to minimize the prediction loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Note that D contains nodes that do not appear in the DCCA process and are labeled differently from the seen nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' The different la- bels allow the RGCN to tell whether a node is unseen during the DCCA process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' If unseen nodes hurt the prediction, it will be reflected in the prediction loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Intuitively, if unseen nodes are less credible, data with more 0-labeled nodes will Method Local Data MIMIC-III DEL DIA TH D30 MORT R30 Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 BERT 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 ClinicalBERT 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 Bi-LSTM 64.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 DCCA+Bi-LSTM 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 78.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 RGCN 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='0 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='0 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 DCCA+RGCN 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 RGCN+Bi-LSTM 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 DCCA+RGCN+Bi-LSTM 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='0 Table 1: DCCA Joint Learning and baseline AUROC (%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Top 4 lines use clinical notes only during inference, middle 2 ICD codes only, and bottom 2 both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Corr = Sum of correlation of latent representations over 20 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Method Local Data MIMIC-III DEL DIA TH D30 MORT R30 RGCN 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 ± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 RGCN+Labling 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 DCCA+RGCN 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='0 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 DCCA+RGCN+Labeling 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 Table 2: Ablation Study of the Labeling Training Scheme under Unseen Code Setting in AUROC (%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' have poor prediction results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' GNN can capture this charac- teristic and reflect it in the prediction by assigning less pos- itive/negative scores to queries with more 0-labeled nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' The labeling training scheme essentially blocks a part of the training code during DCCA and thus obtains embeddings for Us and Uu from different distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' And we train on the entire training dataset so that the model learns to handle seen and unseen codes heterogeneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' This setup mimics the actual inference scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Note that despite being differ- ent, the distributions of seen and unseen node embeddings can be similar and overlapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Thus, the additional 1-0 la- beling is necessary to differentiate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 5 Experimental Results Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We use two datasets to evaluate the performance of our framework: Proprietary Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' This dataset con- tains medical records of 38,551 admissions at the local Hos- pital from 2018 to 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Each entry is also associated with a free text procedural description and a set of ICD-10 pro- cedure codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We aim to use our framework to predict a set of post-operative outcomes, including delirium (DEL), dial- ysis (DIA), troponin high (TH), and death in 30 days (D30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' MIMIC-III dataset (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' This dataset con- tains medical records of 58,976 unique ICU hospital ad- mission from 38,597 patients at the Beth Israel Deaconess Medical Center between 2001 and 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Each admission record is associated with a set of ICD-9 diagnoses codes and multiple clinical notes from different sources, includ- ing case management, consult, ECG, discharge summary, general nursing, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We aim to predict two outcomes from the codes and texts: (1) In-hospital mortality (MORT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We use admissions with hospital expire flag=1 in the MIMIC- III dataset as the positive data and sample the same number of negative data to form the final dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' All clinical notes generated on the last day of admission are filtered out to avoid directly mentioning the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We use all clinical notes ordered by time and take the first 2,500-word tokens as the input text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' (2) 30-day readmission (R30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We follow (Huang, Altosaar, and Ranganath 2019), label admissions where a patient is readmitted within 30 days as positive, and sample an equal number of negative admissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Newborn and death admissions are filtered out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We only use clinical notes of type Discharge Summary and take the first 2,500- word tokens as the input text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Sample sizes can be found in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Effectiveness of DCCA training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We split the dataset with a train/validation/test ratio of 8:1:1 and use 5-fold cross-validation to evaluate our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' GNN and Bi-LSTM are optimized in the DCCA process using the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' The checkpoint model with the best validation correlation is picked to compute the projection matrix only from the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Then we attach an MLP head to the tar- get prediction model (either the GNN or the Bi-LSTM) and fine-tune the model in an end-to-end fashion to minimize the prediction loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' For this task, we compare our framework to popular pre- trained models ClinicalBERT and BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We also compare it to the base GNN and Bi-LSTM to show the effective- ness of our proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We additionally provide experimental results where both text and code embedding are used to make predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We compare our model with a vanilla multi-view model without DCCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' For all baselines, we report their Area Under Receiver Operating Characteris- tic (AUROC) as evaluation metrics, and Average Precision (AP) can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' For all datasets, we set L, the number of correlated dimensions to 20, and report the total amount of correlation obtained (Corr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Table 1 shows the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' For clinical notes predic- tion, we can see that the codes augmented model can con- sistently outperform the base Bi-LSTM, with an average rel- ative performance increase of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4% on the proprietary data and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6% on the MIMIC-III data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Our proposed method out- performs BERT on most tasks and achieves very competi- tive performance compared to that of ClinicalBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Note that our model only trains on the labeled EHR data without unsupervised training on extra data like BERT and Clini- calBERT do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' ClinicalBERT has been previously trained and fine-tuned on the entire MIMIC dataset, including the dis- charge summaries, and therefore these results may overesti- mate its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' For ICD code prediction, we see that DCCA brings a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5% performance increase on the proprietary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Since the codes model significantly outperforms the language model on all tasks, the RGCN is a much stronger learner and has less information to learn from the text model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Comparing the results of the proprietary and the MIMIC datasets, we can see that DCCA brings a more significant performance boost to the proprietary dataset, presumably because of the larger amount of correlation obtained in the proprietary dataset (85% versus 58%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Moreover, an important difference in these datasets is the ontology used: MIMIC-III uses ICD-9 and the proprietary dataset uses ICD-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' The ICD-9 ontol- ogy tree has a height of four, which is much smaller than that of ICD-10 and is more coarsely classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' This may also ex- plain the smaller performance gains in MIMIC-III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' The combined model with DCCA only brings a slight per- formance boost compared to the one without because the amount of information for the models to learn is equiva- lent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Nevertheless, the DCCA model encourages the two views’ embeddings to agree and allows independent predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' In contrast, a vanilla multi-view model does not help the weaker learner learn from the stronger learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Unseen Codes Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We identify the set of unique codes in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We split the codes into k-fold and ran k experiments on each split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' For each experiment, we pick one fold as the unseen code set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Data that contain at least one unseen code are used as the evaluation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' The evaluation set is split into two halves as the valid and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' The rest of the data forms the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We pick an- other fold from the code split as the DCCA unseen code set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Training set data that do not contain any DCCA unseen code form the DCCA training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Then, the entire training set is used for task fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Because the distribution of codes is not uniform, the number of data for each split is not equal across different folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We use k=10 for the proprietary dataset and k=20 for the MIMIC-III dataset to generate a reasonable data division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We provide average split sizes in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' For this task, we compare our method with the base GNN, # Admission # Pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Samples # Unique codes DEL 11,064 5,367 5,637 DIA 38,551 1,387 9,320 TH 38,551 1,235 9,320 D30 38,551 1,444 9,320 MORT 5,926 2,963 4,448 R30 10,998 5,499 3,645 Table 3: Statistics of different datasets and tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' base GNN augmented with the same labeling training strat- egy, and DCCA-optimized GNN to demonstrate the out- standing performance of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Similarly, we re- port AUROC and include AP in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Table 2 summarizes the results of the unseen codes exper- iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Note that all test data contain at least one code that never appears in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' In such a more diffi- cult inference scenario, comparing the plain RGCN with the DCCA-augmented RGCN, we see a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2% average relative performance increase on the proprietary dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' With the labeling learning method, we can further improve the per- formance gain to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' On the MIMIC-III dataset, the per- formance boost of our model over the plain RGCN is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6%, demonstrating our method’s ability to differentiate seen and unseen codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We also notice that DCCA alone only slightly improves the performance on the MIMIC-III dataset (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We suspect that while the labeling training scheme helps dis- tinguish seen and unseen codes, the number of data used in the DCCA process is also reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' As MORT and R30 datasets are smaller and a small DCCA training set may not faithfully represent the actual data distribution, the regular- ization effect of DCCA diminishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 6 Conclusions Predicting patient outcomes from EHR data is an essen- tial task in clinical ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Conventional methods that solely learn from clinical texts suffer from poor performance, and those that learn from codes have limited application in real- world clinical settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' In this paper, we propose a multi- view framework that jointly learns from the clinical notes and ICD codes of EHR data using Bi-LSTM and GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We use DCCA to create shared information but maintain each view’s independence during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' This allows accurate prediction using clinical notes when the ICD codes are miss- ing, which is commonly the case in pre-operative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We also propose a label augmentation method for our frame- work, which allows the GNN model to make effective infer- ences on codes that are not seen during training, enhancing generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Experiments are conducted on two different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Our methods show consistent effectiveness across tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' In the future, we plan to incorporate more data types in the EHR and combine them with other multi-view learning methods to make more accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' References Alsentzer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Murphy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Boag, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=';' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' and Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2020a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Combining structured and unstructured data for predictive models: a deep learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' BMC medical informatics and decision making, 20(1): 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' King, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Avidan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' and Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2020b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Hierarchical attention propagation for healthcare represen- tation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, 249–256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Jin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' and Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Understanding bag-of-words model: a statistical framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' International journal of machine learning and cybernetics, 1(1): 43–52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' A Average Precision Score Results AP results demonstrate a similar pattern to AUROC results, where DCCA augmented model can consistently outper- form the base model while achieving very competitive re- sults compared to ClinicalBERT for the text data as shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' The proposed labeling training scheme can also consistently improve our model’s performance on the un- seen codes experiments, as shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' B Hyperparameters We use grid search for hyperparameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' For missing view experiments on text, we fix the number of RGCN layers to be 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We use 32 for all hidden dimensions as we found that varying hidden size has minimal impact on the performance of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Text and Code represent the hyperparameters used for text and code inference tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Table 7 summarizes the set of hyperparameters used for tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' C Unseen Code Sample Size We use 10-fold code split for the local data and 20-fold code split for the MIMIC-III data so that the split sizes are reason- able for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' We report the average number of samples for all tasks in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' DCCA Train Full Train Test DEL 3,458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 4,624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 3,219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 DIA 19,305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 23,717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 7,416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 TH 19,305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 23,717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 7,416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 D30 19,305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 23,717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 7,416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 MORT 4,603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 6,148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 2,424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 R30 2,528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 3,264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='0 1,330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 Table 4: Average Split Size in Unseen Codes Experiment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' D Data And Implementation We adopted the local dataset because it is the only dataset we have access to that uses both clinical free texts and ICD-10 codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' The implementation details of the MIMIC-III dataset experiments can be found in the supplementary material (code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Local Data MIMIC-III DEL DIA TH D30 MORT R30 Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 BERT 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 ClinicalBERT 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 LSTM 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 DCCA+LSTM 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='0 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 RGCN 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 DCCA+RGCN 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 RGCN+Bi-LSTM 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 DCCA+RGCN+Bi-LSTM 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 Table 5: Effect of DCCA Joint Learning Compared to Different Baselines in AP (%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content=' Method Local Data MIMIC-III DEL DIA TH D30 MORT R30 RGCN 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 RGCN+Labling 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='7 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='5 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='9 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4 ± 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8} {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='4,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='6,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='8} DCCA learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} +page_content='001 batch size 1024 1024 400 400 Task learning rate {1e-3,1e-4,1e-5} {1e-3,1e-4,1e-5} {1e-3,1e-4,1e-5} {1e-3,1e-4,1e-5} batch size 256 256 32 32 Table 7: Hyperparameters used for tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf'} diff --git a/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf b/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9c157ed8774b6a011062f7011984ee5a401debf9 --- /dev/null +++ b/C9AyT4oBgHgl3EQfSPck/content/2301.00080v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1881b3b5bc4619c7dba7467de32ea62b5f12e0550a1f1b24675c5a8944771362 +size 1545665 diff 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b/CNFQT4oBgHgl3EQf-jfx/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f5389ca76f2c87d43b35c1f08315651d21cf2130e455dbbeb6f34b832d8d2550 +size 55889 diff --git a/DtE0T4oBgHgl3EQfQgBB/content/tmp_files/2301.02193v1.pdf.txt b/DtE0T4oBgHgl3EQfQgBB/content/tmp_files/2301.02193v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1cc15e1915f9ae16a43140d8566f3e44befe097 --- /dev/null +++ b/DtE0T4oBgHgl3EQfQgBB/content/tmp_files/2301.02193v1.pdf.txt @@ -0,0 +1,2495 @@ +arXiv:2301.02193v1 [physics.app-ph] 5 Jan 2023 +Universal scaling between wave speed and size +enables nanoscale high-performance reservoir +computing based on propagating spin-waves +Satoshi Iihama,1,2 Yuya Koike,2,3,5 Shigemi Mizukami,2,4 Natsuhiko Yoshinaga2,5∗ +1Frontier Research Institute for Interdisciplinary Sciences (FRIS), Tohoku University, +Sendai, 980-8578, Japan +2WPI Advanced Institute for Materials Research (AIMR), Tohoku University, +Katahira 2-1-1, Sendai, 980-8577, Japan +3Department of Applied Physics, Tohoku University, +Sendai, 980-8579, Japan +4Center for Science and Innovation in Spintronics (CSIS), Tohoku University, +Sendai, 980-8577, Japan +5MathAM-OIL, AIST, Sendai, 980-8577, Japan +∗To whom correspondence should be addressed; E-mail: yoshinaga@tohoku.ac.jp. +Neuromorphic computing using spin waves is promising for high-speed nanoscale +devices, but the realization of high performance has not yet been achieved. +Here we show, using micromagnetic simulations and simplified theory with +response functions, that spin-wave physical reservoir computing can achieve +miniaturization down to nanoscales keeping high computational power com- +parable with other state-of-art systems. We also show the scaling of system +sizes with the propagation speed of spin waves plays a key role to achieve high +performance at nanoscales. +1 + +Introduction +Non-local magnetization dynamics in a nanomagnet, spin-waves, can be used for processing in- +formation in an energy-efficient manner since spin-waves carry information in a magnetic ma- +terial without Ohmic losses (1). The wavelength of the spin-wave can be down to the nanometer +scale, and the spin-wave frequency becomes several GHz to THz frequency, which are promis- +ing properties for nanoscale and high-speed operation devices. Recently, neuromorphic com- +puting using spintronics technology has attracted great attention for the development of future +low-power consumption artificial intelligence (2). Spin-waves can be created by various means +such as magnetic field, spin-transfer torque, spin-orbit torque, voltage induced change in mag- +netic anisotropy and can be detected by the magnetoresistance effect (3). Therefore, neuromor- +phic computing using spin waves may have a potential of realisable devices. +Reservoir computing (RC) is a promising neuromorphic computation framework. RC is a +variant of recurrent neural networks (RNNs) and has a single layer, referred to as a reservoir, to +transform an input signal into an output (4). In contrast with the conventional RNNs, RC does +not update the weights in the reservoir. Therefore, by replacing the reservoir of an artificial +neural network with a physical system, for example, magnetization dynamics, we may realize a +neural network device to perform various tasks, such as time-series prediction (4,5), short-term +memory (6, 7), pattern recognition, and pattern generation. Several physical RC has been pro- +posed: spintronic oscillators (8,9), optics (10), photonics (11,12), fluids, soft robots, and others +(see reviews (13–15)). Among these systems, spintronic RC has the advantage in its potential +realization of nanoscale devices at high speed of GHz frequency with low power consumption, +which may outperform conventional electric computers in future. So far, spintronic RC has +been considered using spin-torque oscillators (8, 9), magnetic skyrmion (16), and spin waves +in garnet thin films (17–19). However, the current performance of spintronic RC still remains +2 + +poor compared with the Echo State Network (ESN) (6, 7), idealized RC systems. The biggest +issue is a lack of our understanding of how to achieve high performance in the RC systems. +To achieve high performance, the reservoir has to have a large degree of freedom, N. How- +ever, in practice, it is difficult to increase the number of physical nodes, Np, because it requires +more wiring of multiple inputs. In this respect, wave-based computation in continuum media +has attracting features. The dynamics in the continuum media have large, possibly infinite, de- +grees of freedom. In fact, several wave-based computations have been proposed (20, 21). The +challenge is to use the advantages of both wave-based computation and RC to achieve high- +performance computing of time-series data. For spin wave-based RC, so far, the large degrees +of freedom are extracted only by using a large number of input and/or output nodes (19, 22). +Here, to propose a realisable spin wave RC, we use an alternative route; we extract the informa- +tion from the continuum media using a small number of physical nodes. +Along this direction, using Nv virtual nodes for the dynamics with delay was proposed to +increase N in (23). This idea was applied in optical fibres with a long delay line (11) and a net- +work of oscillators with delay (24). Nevertheless, the mechanism of high performance remains +elusive, and no unified understanding has been made. The increase of N = NpNv with Nv does +not necessarily improve performance. In fact, RC based-on STO struggles with insufficient per- +formance both in experiments (9) and simulations (25). The photonic RC requires a large size +of devices due to the long delay line (11,12). +In this work, we show nanoscale and high-speed RC based on spin wave propagation with a +small number of inputs can achieve performance comparable with the ESN and other state-of-art +RC systems. More importantly, by using a simple theoretical model, we clarify the mechanism +of the high performance of spin wave RC. We show the scaling between wave speed and system +size to make virtual nodes effective. +3 + +Results +Reservoir computing using wave propagation +The basic task of RC is to transform an input signal Un to an output Yn for the discrete step n = +1, 2, . . . , T at the time tn. For example, for speech recognition, the input is an acoustic wave, +and the output is a word corresponding to the sound. Each word is determined not only by the +instantaneous input but also by the past history. Therefore, the output is, in general, a function +of all the past input, Yn = g ({Um}n +m=1) as in Fig. 1(a). The RC can also be used for time-series +prediction by setting the output as Yn = Un+1 (4). In this case, the state at the next time step +is predicted from all the past data; namely, the effect of delay is included. The performance of +the input-output transformation g can be characterized by how much past information does g +have, and how much nonlinear transformation does g perform. We will discuss that the former +is expressed by memory capacity (MC) (26), whereas the latter is measured by information +processing capacity(IPC) (27). +We propose physical computing based on a propagating wave (see Fig. 1(b,c)). Time series +of an input signal Un can be transformed into an output signal Yn (Fig. 1(a)). As we will discuss +below, this transformation requires large linear and nonlinear memories; for example, to predict +Yn, we need to memorize the information of Un−2 and Un−1. The input signal is injected in +the first input node and propagates in the device to the output node spending a time τ1 as in +Fig. 1(b). Then, the output may have past information at tn − τ1 corresponding to the step +n − m1. The output may receive the information from another input at different time tn − τ2. +The sum of the two peices of information is mixed and transformed as Un−m1Un−m2 either by +nonlinear readout or by nonlinear dynamics of the reservoir (see also Sec. B in Supplementary +Information). We will demonstrate the wave propagation can indeed enhances memory capacity +and learning performance of the input-output relationship. +4 + +Figure 1: +Illustration of physical reservoir computing and reservoir based on propa- +gating spin-wave network. (a) Schematic illustration of output function prediction by using +time-series data. Output signal Y is transformed by past information of input signal U. (b) +Schematic illustration of reservoir computing with multiple physical nodes. The output signal +at physical node A contains past input signals in other physical nodes, which are memorized by +the reservoir. (c) Schematic illustration of reservoir computing based on propagating spin-wave. +Propagating spin-wave in ferromagnetic thin film (m ∥ ez) is excited by spin-transfer torque at +multiple physical nodes with reference magnetic layer (m ∥ ex). x-component of magnetization +is detected by the magnetoresistance effect at each physical node. +5 + +a +(b) +Y(t) α U(t - T1) · U(t - t2) +output +input +g((U(t))) +U(t - T1) +X(t) α U(t - T1) + U(t - T2) +input +reservoir +U(t - T2) +Spin injector and detector +(c) +Ferromagnetic thin filmBefore explaining our learning strategy, we discuss how to achieve accurate learning of the +input-output relationship Yn = g ({Um}n +m=1) from the data. Here, the output may be dependent +on a whole sequence of the input {Um}n +m=1 = (U1, . . . , Un). Even when both Un and Yn are +one-variable time-series data, the input-output relationship g(·) may be T-variable polynomials, +where T is the length of the time series. Formally, g(·) can be expanded in a polynomial +series (Volterra series) such that g ({Um}n +m=1) = � +k1,k2,··· ,kt βk1,k2,··· ,ktUk1 +1 Uk2 +2 · · · Ukn +n with the +coefficients βk1,k2,··· ,kn. Therefore, even for the linear input-output relationship, we need T +coefficients in g(·), and as the degree of powers in the polynomials increases, the number of +the coefficients increases exponentially. This observation implies that a large number of data is +required to estimate the input-output relationship. Nevertheless, we may expect a dimensional +reduction of g(·) due to its possible dependence on the time close to t and on the lower powers. +Still, our physical computers should have degrees of freedom N ≫ 1, if not exponentially large. +The reservoir computing framework is used to handle time-series data of the input U and the +output Y (6). In this framework, the input-output relationship is learned through the reservoir +dynamics X(t), which in our case, is magnetization at the detectors. The reservoir state at a +time tn is driven by the input at the nth step corresponding to tn as +X(tn+1) = f (X(tn), Un) +(1) +with nonlinear (or possibly linear) function f(·). The output is approximated by the readout +operator ψ(·) as +ˆYn = ψ (X(tn)) . +(2) +Our study uses the nonlinear readout ψ (X(t)) = W1X(t) + W2X2(t) (5, 28). The weight +matrices W1 and W2 are estimated from the data of the reservoir dynamics X(t) and the true +output Yn, where X(t) is obtained by (1). With the nonlinear readout, the RC with linear +6 + +dynamics can achieve nonlinear transformation, as Fig.1(b). We stress that the system also +works with linear readout when the RC has nonlinear dynamics. We discuss this case in Sec.B. +Spin wave reservoir computing +We consider a magnetic device of a thin rectangular system with cylindrical injectors (see +Fig.1(c)). The size of the device is L × L × D. Under the uniform external magnetic field, +the magnetization is along the z direction. Electric current is injected at the Np injectors with +the radius a and the same height with the device. The spin-torque by the current drives mag- +netization m(x, t) and propagating spin-waves as schematically shown in Fig.1(c). The actual +demonstration of the spin-wave reservoir computing is shown in Fig. 2. We demonstrate the +spin-wave RC using two methods: the micromagnetic simulations and the theoretical model +using a response function. +In the micromagnetic simulations, we analyze the Landau-Lifshitz-Gilbert (LLG) equation +with the effective magnetic field Heff = Hext+Hdemag+Hexch consists of the external field, de- +magnetization, and the exchange interaction (see Theoretical analysis using response function +in Methods). The spin waves are driven by Slonczewski spin-transfer torque (29). The driving +term is proportional to the DC current j(t) at the nanocontact. We inject the DC current propor- +tional to the input time series U with a pre-processing filter. From the resulting spatially inho- +mogeneous magnetization m(x, t), we measure the averaged magnetization at ith nanocontact +mi(t). We use the method of time multiplexing with Nv virtual nodes (23). We choose the x- +component of magnetization mx,i as a reservoir state, namely, Xn = {mx,i(tn,k)}i∈[1,Np],k∈[1,Nv] +(see (14) in Methods for its concrete form). For the output transformation, we use ψ(mi,x) = +W1,imi,x + W2,im2 +i,x. Therefore, the dimension of our reservoir is 2NpNv. The nonlinear output +transformation can enhance the nonlinear transformation in reservoir (5), and it was shown that +even under the linear reservoir dynamics, RC can learn any nonlinearity (28, 30). In Sec. B in +7 + +0 +0.5 +0 +0.5 +0 +1.6 +3.2 +4.8 +6.4 +8 +0 +0.5 +・ +・ +・ +�i +n +n+1 +n+2 +n+3 +n+4 +0 +0.5 +Time step +higher damping region +cylindrical region to apply +spin-transfer torque +(a) +(b) +Training +� +Input, � +Time +Binary mask, �i +�n,� +�n +�� +�n +�� +�n�� +� +�� +Time (ns) +・ +・ +・ +Masked input, +Time step +Input, � +Time step +Output, � +Time step +� � +Time step +Time +� +0 +� +�n +�n+1 +�n,3 +�n,5 +�n,7 +(t) +(t) +Figure 2: +Dimension of spin-wave reservoir and prediction of NARMA10 task. +(a) +Input signals U are multiplied by binary mask Bi(t) and transformed into injected current +j(t) = 2jc ˜Ui(t) for the ith physical node. Current is injected into each physical node with +the cylindrical region to apply spin-transfer torque and to excite spin-wave. Higher damping +regions in the edges of the rectangle are set to avoid reflection of spin-waves. (b) Prediction of +NARMA10 task. x-component of magnetization at each physical and virtual node are collected +and output weights are trained by linear regression. +8 + +0.02 +0.01 +0 +-0.01 +Node (1, 1) +-0.02 +0.02 +0.01 +0 +-0.01 +(2,1) +-0.02 +0.02 +0.01 +0 +-0.01 +(Ny,Np +-0.02 +1000 +2000 +3000 +4000 +5000 +6000- OQutput, +Predicted +0.8 +0.6 +0.4 +0.2 +Error +0 +6000 +7000 +8000 +9000 +10000 +110000.8 +0.6 +0.4 +0.2 +0 +1000 +2000 +3000 +4000 +5000 +60000.5 +1000 +2000 +3000 +4000 +5000 +60001000nm +1000nm +Q +500nm +4nmSupplementary Information, we also discuss the linear readout, but including the z-component +of magnetization X = (mx, mz). In this case, mz plays a similar role to m2 +x. The performance +of the RC is measured by three tasks: MC, IPC, and NARMA10. The weights in the readout +are trained by reservoir variable X and the output Y (Fig.2(b), see also Methods). +To understand the mechanism of high performance of learning by spin wave propagation, +we also consider a simplified model using the response function of the spin wave dynamics. By +linearizing the magnetization around m = (0, 0, 1) without inputs, we may express the linear +response of the magnetization at the ith readout mi = mx,i + imy,i to the input as(see Methods) +mi(t) = +Np +� +j=1 +� +dt′Gij(t, t′)U(j)(t′). +(3) +Here, U(j)(t) is the input time series at jth nanocontact. The response function has a self part +Gii, that is, input and readout nanocontacts are the same, and the propagation part Gij, where +the distance between the input and readout nanocontacts is |Ri − Rj|. We use the quadratic +nonlinear readout, which has a structure +m2 +i (t) = +Np +� +j1=1 +Np +� +j2=1 +� +dt1 +� +dt2G(2) +ij1j2(t, t1, t2)U(j1)(t1)U(j2)(t2). +(4) +The response function of the nonlinear readout is G(2) +ij1j2(t, t1, t2) ∝ Gij1(t, t1)Gij2(t, t2). The +same structure as (4) appears when we use a second-order perturbation for the input (see Meth- +ods). In general, we may include the cubic and higher-order terms of the input. This expansion +leads to the Volterra series of the output in terms of the input time series, and suggests how the +spin wave RC works (see Sec. A.1 in Supplementary Information for more details). Once the +magnetization at each nanocontact is computed, we may estimate MC and IPC. +Figure 3 shows the results of the three tasks. When the time scale of the virtual node θ is +small and the damping is small, the performance of spin wave RC is high. As Fig. 3(a) shows, +we achieve MC ≈ 60 and IPC ≈ 60. Accordingly, we achieve a small error in the NARMA10 +9 + +task, NRMSE ≈ 0.2 (Fig. 3(c)). Theses performances are comparable with state-of-the-art ESN +with the number of nodes ∼ 100. When the damping is stronger, both MC and IPC become +smaller. Because the NARMA10 task requires the memory with the delay steps ≈ 10 and the +second order nonlinearity with the delay steps ≈ 10 (see Sec.A in Supplementary Information), +the NRMSE becomes larger when MC ≲ 10 and IPC ≲ 102/2. +The results of the micromagnetic simulations are semi-quantitatively reproduced by the the- +oretical model using the response function, as shown in Fig. 3(b). This result suggests that +the linear response function G(t, t′) captures the essential feature of delay t − t′ due to wave +propagation. +(a) +(b) +Non-linear, IPC +Linear, MC +0 +20 +40 +60 +80 +5 +2.5 +α = 5×10-4 +Frequency, 1/θ (GHz) +0 +20 +40 +60 +80 +α = 5×10-3 +Linear and non-linear memory capacity +0.2 +0.4 +0 +20 +40 +60 +80 +α = 5×10-2 +Distance of virtual nodes, θ (ns) +(c) +Normalized root mean square error, NRMSE +for NARMA10 task +0 +0.5 +1 +5 +2.5 +α = 5×10-4 + Training, + Test +Frequency, 1/θ (GHz) +0 +0.5 +1 +α = 5×10-3 + +0.2 +0.4 +0 +0.5 +1 +α = 5×10-2 +Distance of virtual nodes, θ (ns) +0 +20 +40 +60 +80 +5 +2.5 +α = 5×10-4 +Frequency, 1/θ (GHz) +0 +20 +40 +60 +80 +α = 5×10-3 +Linear and non-linear memory capacity +0.2 +0.4 +0 +20 +40 +60 +80 +α = 5×10-2 +Distance of virtual nodes, θ (ns) +Figure 3: Effect of virtual node distance on performance of spin-wave reservoir computing +obtained with 8 physical nodes and 8 virtual nodes. Memory capacity MC and information +processing capacity IPC obtained by (a) micromagnetics simulation and (b) response function +method plotted as a function of virtual node distance θ with different damping parameters α. +(c) Normalized root mean square error, NRMSE for NARMA10 task is plotted as a function of +θ with different α. +To confirm the high MC and IPC are due to spin-wave propagation, we perform micromag- +netic simulations with damping layers between nodes (Fig. 4(a)). The damping layers inhibit +spin wave propagation. The result of Fig. 4(b) shows that the memory capacity is substantially +lower than that without damping, particularly when θ is small. The NARMA10 task shows a +10 + +larger error (Fig. 4(d)). When θ is small, the suppression is less effective. This may be due to +incomplete suppression of wave propagation. +We also analyze the theoretical model with the response function by neglecting the inter- +action between two physical nodes, namely, Gij = 0 for i ̸= j. In this case, information +transmission between two physical nodes is not allowed. We obtain smaller MC and IPC than +the system with wave propagation, supporting our claim (see (Fig. 4(c))). +Our spin wave RC also works for the prediction of time-series data. In the study of (5), +the functional relationship between the state at t + ∆t and the states before t is learned by the +ESN. The trained ESN can estimate the state at t + ∆t from the past states, and therefore, it can +predict the dynamics without the data. In (5), the prediction for the chaotic time-series data was +demonstrated. Figure 5 shows the prediction using our spin wave RC for the Lorenz model. We +can demonstrate that the RC shows short-time prediction and, more importantly, reconstruct the +chaotic attractor. +Scaling of system size and wave speed +To clarify the mechanism of the high performance of our spin wave RC, we investigate MC +and IPC of the system with different characteristic length scales L and different wave propagat- +ing speed v. The characteristic length scale is controlled by the radius of the circle on which +inputs are located (see Fig. 2(a)). We use our theoretical model with the response function to +compute MC and IPC in the parameter space (v, R). This calculation can be done because the +computational cost of our model is much cheaper than numerical micromagnetic simulations. +Figure 6(a,b) shows that both MC and IPC have maximum when L ∝ v. To obtain a deeper +understanding of the result, we perform the same analyzes for the further simplified model, in +11 + +(� +� +(c) +(d) +Linear, MC +Non-linear, IPC +0 +20 +40 +60 +80 +5 +2.5 +α = 5 × 10-4 +C � +  +  +  +Frequency, 1/θ (GHz) +0.2 +0.4 +0 +20 +40 +60 +80 +No connection +Linear and non-linear memory capacity +Distance of virtual nodes, θ (ns) +Normalized root mean square error, NRMSE +for NARMA10 task +0 +0.5 +1 +5 +2.5 +   +!"#$ %&' + Training, + Test +Frequency, 1/θ (GHz) +0.2 +0.4 +0 +0.5 +1 + +No-connection +Distance of virtual nodes, θ (ns) +(a) +)*+ +-./ 123 +4 +56789 : +No-connection +0 +20 +40 +60 +80 +5 +2.5 +α = 5 × 10-4 +; <= >?@ ABD +EFG +H +I JK +Frequency, 1/θ (GHz) +0.2 +0.4 +0 +20 +40 +60 +80 +No connection (G +iL = 0 ) +Linear and non-linear memory capacity +Distance of virtual nodes, θ (ns) +Figure 4: +Effect of the network connection on the performance of reservoir computing. +(a) Schematic illustration of the network of physical nodes connected through propagating spin- +wave [left] and physical nodes with no connection [right]. Memory capacity MC and informa- +tion processing capacity IPC obtained using a connected network with 8 physical nodes [top] +and physical nodes with no connection [bottom] calculated by (a) micromagnetics simulation +and (b) response function method plotted as a function of virtual node distance θ. 8 virtual +nodes are used. (c) Normalized root mean square error, NRMSE for NARMA10 task obtained +by micromagnetics simulation is plotted as a function of θ with a connected network [top] and +physical nodes with no connection [bottom]. +12 + +ground truth +prediction +time +-10 +0 +10 +20 +30 +40 +5 +15 +25 +35 +45 +-20 +-10 +0 +10 +20 +-20 +-10 +10 +20 +0 +training +prediction +A1 +A1 +A3 +A3 +A1 +A2 +A3 +Figure 5: Prediction of time-series data for the Lorenz system using the RC with micro- +magnetic simulations. The parameters are θ = 0.4ns and α = 5.0×10−4. (a) The ground truth +(A1(t), A2(t), A3(t)) and the estimated time series ( ˆ +A1(t), ˆ +A1(t), ˆ +A3(t)) are shown in blue and +red, respectively. The training steps are during t < 0, whereas the prediction steps are during +t > 0. (b) The attractor in the A1A3 plane for the ground truth and during the prediction steps. +13 + +45 +40 +35 +30 +25 +20 +15 +10 +5 +-20 +-10 +0 +10 +2045 +40 +35 +30 +25 +20 +15 +10 +5 +-20 +-10 +0 +10 +20which the response function is replaced by the Gaussian function +Gij(t) = exp +� +− 1 +2w2 +� +t − Rij +v +�2� +(5) +where Rij is the distance between ith and jth physical nodes, and w is the width of the function. +Even in this simplified model, we obtain MC≈ 40 and IPC≈ 60, and also the maximum when +L ∝ v (Fig. 6(c,d)). From this result, the origin of the optimal ratio between the length and +speed becomes clearer; when L ≪ v, the response functions under different Rij overlap so +that different physical nodes cannot carry the information of different delay times. On the +other hand, when L ≫ v, the characteristic delay time L/v exceeds the maximum delay time +to compute MC and IPC, or exceeds the total length of the time series. Note that we set the +maximum delay time as 100, which is much longer than the value necessary for the NARMA10 +task. +The result suggests the universal scaling between the size of the system and the speed of +the RC based on wave propagation. Our system of the spin wave has a characteristic length +L ∼ 500 nm and a speed of v ∼ 200 m s−1. In fact, the reported photonic RC has characteristic +length scale of optical fibres close to the scaling in Fig. 7. +Discussion +Figure 7 shows reports of reservoir computing in literature with multiple nodes plotted as a +function of the length of nodes L and products of wave speed and delay time vτ0 for both +photonic and spintronic RC. For the spintronic RC, the dipole interaction is considered for wave +propagation in which speed is proportional to both saturation magnetization and thickness of the +film (31)(See supplementary information sec. C). For the photonic RC, the characteristic speed +is the speed of light, v ∼ 108 m s−1. Symbol size corresponds to MC taken from the literature +[See details of plots in supplementary information sec. D]. Plots are roughly on a broad oblique +14 + +(a) +(b) +(c) +(d) +wave speed (log m/s) +characteristic size (log nm) +1.0 +3.0 +2.0 +2.0 +3.0 +4.0 +MC +10 +20 +30 +40 +50 +60 +damping time +1.0 +wave speed (log m/s) +characteristic size (log nm) +1.0 +3.0 +2.0 +2.0 +3.0 +4.0 +IPC +10 +20 +30 +40 +50 +60 +damping time +1.0 +wave speed (log m/s) +characteristic size (log nm) +1.0 +3.0 +2.0 +2.0 +3.0 +4.0 +MC +10 +20 +30 +40 +1.0 +4.0 +5.0 +wave speed (log m/s) +characteristic size (log nm) +1.0 +3.0 +2.0 +2.0 +3.0 +4.0 +IPC +10 +20 +30 +40 +1.0 +4.0 +5.0 +60 +50 +time +response function +11( ) +G +t +12( ) +G +t +13( ) +G +t +14( ) +G +t +15( ) +G +t +time +dense +sparse +memorise +(e) +Figure 6: +Scaling between characteristic size and propagating wave speed obtained by +response function method. MC (a,c) and IPC (b,d) as a function of the characteristic length +scale between physical nodes R and the speed of wave propagation v. The results with the +response function for the dipole interaction (a,b) and for the Gaussian function (5) (c,d) are +shown. (e) Schematic illustration of the response function and its relation to wave propagation +between physical nodes. When the speed of the wave is too fast, all the response functions are +overlapped (dense regime), while the response functions cannot cover the time windows when +the speed of the wave is too slow (sparse regime). +15 + +line with a ratio L/(vτ0) ∼ 1. Therefore, the photonic RC requires a larger system size, as +long as the delay time of the input τ0 = Nvθ is the same order (τ0 = 0.3 − 3 ns in our spin +wave RC). As can be seen in Fig. 6, if one wants to reduce the length of physical nodes, one +must reduce wave speed or delay time; otherwise the information is dense, and the reservoir +cannot memorize many degrees of freedom (See Fig. 6(e)). Reducing delay time is challenging +since the experimental demonstration of the photonic reservoirs has already used the short delay +close to the instrumental limit. Also, reducing wave speed in photonics systems is challenging. +On the other hand, the wave speed of propagating spin-wave is much lower than the speed of +light and can be tuned by configuration, thickness and material parameters. If one reduces wave +speed or delay time over the broad line in Fig. 7, information becomes sparse and cannot be +used efficiently(See Fig. 6(e)). Therefore, there is an optimal condition for high-performance +RC. +The performance is comparable with other state of the art techniques, which are summa- +rized in Fig. 8. For example, for the spintronic RC, MC ≈ 30 (19) and NRMSE ≈ 0.2 (22) in +the NARMA10 task are obtained using Np ≈ 100 physical nodes. The spintronic RC with one +physical node but with 101 − 102 virtual nodes do not show high performance; MC is less than +10 (the bottom left points in Fig. 8). This fact suggests that the spintronic RC so far cannot use +virtual nodes effectively. On the other hand, for the photonic RC, comparable performances are +achieved using Nv ≈ 50 virtual nodes, but only one physical node. As we discussed, however, +the photonic RC requires mm system sizes. Our system achieves comparable performances +using ≲ 10 physical nodes, and the size is down to nanoscales keeping the 2 − 50 GHz compu- +tational speed. We also demonstrate that the spin wave RC can perform time-series prediction +and reconstruction of an attractor for the chaotic data. To our knowledge, this has not been done +in nanoscale systems. +Our results of micromagnetic simulations suggest that our system can be physically im- +16 + +0 +10 +20 +30 +40 +50 +60 +MC +This work ( +t +0 + = 1.6 ns) +This work ( +t +0 + = 0.16 ns) + Spintronic RC + ( 19 ) + ( 22 ) + Photonic RC + ( 46 ) + ( 47 ) +(48 ) +(12 ) + ( 50 ) +vt +0 (m) +Length, +L (m) +Dense +Sparse +Figure 7: Reports of reservoir computing using multiple nodes are plotted as a function +of the length between nodes and characteristic wave speed (v) times delay time (τ0) for +photonics system (open symbols) and spintronics system (solid symbols). The size of sym- +bols corresponds to memory capacity, which is taken from literature (12,19,22,32–35) and this +work. The gray scale represents memory capacity evaluated by using the response function +method [Eq. (5)]. +17 + +VJKhJK-Jh-J1 +10 +100 +0.1 +1 + This work (Calc., Nv = 8, θ = 0.2 ns) + This work (Calc., Nv = 8, θ = 0.02 ns) + Spintronic RC (Calc.) + (22) + Photonic RC + (45) (Nv = 50) +(48) (Nv = 50) + (49) (Nv = 50) +Normalized root mean square error, NRMSE +for NARMA10 task +Number of physical nodes, Np +1 +10 +100 +10 +100 +This work (Calc., Nv = 8, θ = 0.2 ns) +This work (Calc., Nv = 8, θ = 0.02 ns) + Spintronic RC (Calc.) +(44) +(19) +(22) + Spintronic RC (Exp.) +(9) (Nv = 250) +(51) (Nv = 40) + Photonic RC +(46) (Nv = 50) +(47) (Nv = 50) +(48) (Nv = 50) +Memory capacity, MC +Number of physical nodes, Np +(a) +(b) +Figure 8: Reservoir computing performance compared with different systems. (a) Memory +capacity, MC reported plotted as a function of physical nodes Np. (b) Normalized root mean +square error, NRMSE for NARMA10 task is plotted as a function of Np. Open blue symbols are +values reported using photonic RC while solid red symbols are values reported using spintronic +RC. MC and NRMSE for NARMA10 task are taken from Refs. (9,19,22,36,37) for spintronic +RC and Refs. (32–34,38,39) for photonic RC. +plemented. +All the parameters in this study are feasible using realistic materials (40–43). +Nanoscale propagating spin waves in a ferromagnetic thin film excited by spin-transfer torque +using nanometer electrical contacts have been observed (44–46). Patterning of multiple elec- +trical nanocontacts into magnetic thin films was demonstrated in mutually synchronized spin- +torque oscillators (46). In addition to the excitation of propagating spin-wave in a magnetic thin +film, its non-local magnetization dynamics can be detected by tunnel magnetoresistance effect +at each electrical contact, as schematically shown in Fig. 1(c), which are widely used for the +development of spintronics memory and spin-torque oscillators. In addition, virtual nodes are +effectively used in our system by considering the speed of propagating spin-wave and distance +of physical nodes; thus, high-performance reservoir computing can be achieved with the small +number of physical nodes, contrary to many physical nodes used in previous reports. This work +provides a way to realize nanoscale high-performance reservoir computing based on propagat- +ing spin-wave in a ferromagnetic thin film. +There is an interesting connection between our study to the recently proposed next-generation +18 + +RC (28, 47), in which the linear ESN is identified with the NVAR (nonlinear vectorial autore- +gression) method to estimate a dynamical equation from data. Our formula of the response func- +tion (3) results in the linear input-output relationship with a delay Yn+1 = anUn+an−1Un−1+. . . +(see Sec. A in Supplementary Information). More generally, with the nonlinear readout or with +higher-order response functions, we have the input-output relationship with delay and non- +linearity Yn+1 = anUn + an−1Un−1 + . . . + an,nUnYn + an,n−1UnUn−1 + . . . (see Sec. B in +Supplementary Information). These input-output relations are nothing but Volterra series of the +output as a function of the input with delay and nonlinearity (48). The coefficients of the ex- +pansion are associated with the response function. Therefore, the performance of RC falls into +the independent components of the matrix of the response function, which can be evaluated by +how much delay the response functions between two nodes cover without overlap. The results +would be helpful to a potential design of the network of the physical nodes. +We should note that the polynomial basis of the input-output relation in this study originates +from spin wave excitation around the stationary state mz = 1. When the input data has a hier- +archical structure, another basis may be more efficient than the polynomial expansion. Another +setup of magnetic systems may lead to a different basis. We believe that our study shows simple +but clear intuition of the mechanism of high-performance RC, that can lead to the exploration +of another setup for more practical application of the physical RC. +19 + +Materials and Methods +Micromagnetic simulations +We analyze the LLG equation using the micromagnetic simulator mumax3 (49). The LLG +equation for the magnetization M(x, t) yields +∂tM(x, t) = − +γµ0 +1 + α2M × Heff − +αγµ0 +Ms(1 + α2)M × (M × Heff) ++ +ℏPγ +4M2s eDJ(x, t)M × (M × mf) . +(6) +We consider the effective magnetic field as +Heff = Hext + Hdemag + Hexch, +(7) +Hext = H0ez +(8) +Hms = − 1 +4π +� +∇∇ +1 +|r − r′|dr′ +(9) +Hexch = 2Aex +µ0Ms +∆M, +(10) +where Hext is the external magnetic field, Hms is the magnetostatic interaction, and Hexch is the +exchange interaction with the exchange parameter Aex. +The size of our system is L = 1000 nm and D = 4 nm. The number of mesh points is +200 in the x and y directions, and 1 in the z direction. We consider Co2MnSi Heusler alloy +ferromagnet, which has a low Gilbert damping and high spin polarization with the parameter +Aex = 23.5 pJ/m, Ms = 1000 kA/m, and α = 5 × 10−4 (40,41,41–43). Out-of-plane magnetic +field µ0H0 = 1.5 T is applied so that magnetization is pointing out-of-plane. The spin-polarized +current field is included by the Slonczewski model (29) with polarization parameter P = 1 and +spin torque asymmetry parameter λ = 1 with the reduced Planck constant ℏ and the charge of +an electron e. The uniform fixed layer magnetization is mf = ex. We use absorbing boundary +layers for spin waves to ensure the magnetization vanishes at the boundary of the system (50). +We set the initial magnetization as m = ez. +20 + +The reference time scale in this system is τ0 = 1/γµ0Ms ≈ 5 ps, where γ is the gyromag- +netic ratio, µ0 is permeability, and Ms is saturation magnetization. The reference length scale is +the exchange length l0 ≈ 5 nm. The relevant parameters are Gilbert damping α, the time scale +of the input time series θ, and the characteristic length between the input nodes R0. +The injectors and detectors of spin are placed as cylindrical nanocontacts embedded in the +region with their radius a and height D. We set a = 20nm unless otherwise stated. The +input time series is uniform random noise Un ∈ U(0, 0.5). The injected density current is +set as j(tn) = 2jcUn with jc = 2 × 10−4/(πa2)A/m2. Under a given input time series of +the length T, we apply the current during the time θ, and then update the current at the next +step. The same input current with different filters is injected for different virtual nodes (see +Learning with reservoir computing). The total simulation time is, therefore, TθNv. +Learning with reservoir computing +Our RC architecture consists of reservoir state variables +X(t + ∆t) = f (X(t), U(t)) +(11) +and the readout +Yn = W · ˜˜X(tn). +(12) +In our spin wave RC, the reservoir state is chosen as x-component of the magnetization +X = +� +mx,1(tn), . . . , mx,i(tn), . . . , mx,Np(tn) +�T , +(13) +for the indices for the physical nodes i = 1, 2, . . . , Np. Here, Np is the number of physical +nodes, and each mx,i(tn) is a T-dimensional row vector with n = 1, 2, . . . , T. We use a time- +multiplex network of virtual nodes in RC (23), and use Nv virtual nodes with time interval θ. +21 + +The expanded reservoir state is expressed by NpNv × T matrix ˜X as (see Fig.2(b)) +˜X = (mx,1(tn,1), mx,1(tn,2), . . . , mx,1(tn,k), . . . , mx,1(tn,Nv), +. . . , mx,i(tn,1), mx,i(tn,2), . . . , mx,i(tn,k), . . . , mx,i(tn,Nv), . . . , +mx,Np(tn,1), mx,Np(tn,2), . . . , mx,Np(tn,k), . . . , mx,Np(tn,Nv) +�T , +(14) +where tn,k = ((n − 1)Nv − (k − 1))θ for the indices of the virtual nodes k = 1, 2, . . . , Nv. The +total number of rows is N = NpNv. We use the nonlinear readout by augmenting the reservoir +state as +˜˜X = +� +˜X +˜X ◦ ˜X +� +, +(15) +where ˜X(t) ◦ ˜X(t) is the Hadamard product of ˜X(t), that is, component-wise product. The +readout weight W is trained by the data of the output Y (t) +W = Y · ˜˜X† +(16) +where X† is pseudo-inverse of X. +In the time-multiplexing approach, the input time-series U = (U1, U2, . . . , UT) ∈ RT is +translated into piece-wise constant time-series ˜U(t) = Un with t = (n − 1)Nvθ + s under +k = 1, . . . , T and s = [0, Nvθ) (see Fig. 2(a)). This means that the same input remains during +the time period τ0 = Nvθ. To use the advantage of physical and virtual nodes, the actual input +Ji(t) at the ith physical node is ˜U(t) multiplied by τ0-periodic random binary filter Bi(t). Here, +Bi(t) ∈ {0, 1} is piece-wise constant during the time θ. At each physical node, we use different +realizations of the binary filter as in Fig. 2(a). +Unless otherwise stated, We use 1000 steps of the input time-series as burn-in. After these +steps, we use 5000 steps for training and 5000 steps for test for the MC, IPC, and NARMA10 +tasks. +22 + +NARMA task +The NARMA10 task is based on the discrete differential equation, +Yn+1 = αYn + βYn +9 +� +p=0 +Yn−p + γUnUn−9 + δ. +(17) +Here, Un is an input taken from the uniform random distribution U(0, 0.5), and yk is an output. +We choose the parameter as α = 0.3, β = 0.05, γ = 1.5, and δ = 0.1. In RC, the input is +U = (U1, U2, . . . , UT) and the output Y = (Y1, Y2, . . . , YT). The goal of the NARMA10 task +is to estimate the output time-series Y from the given input U. The training of RC is done by +tuning the weights W so that the estimated output ˆY (tn) is close to the true output Yn in terms +of squared norm | ˆY (tn) − Yn|2. +The performance of the NARMA10 task is measured by the deviation of the estimated time +series ˆY = W · ˜˜X from the true output Y. The normalized root-mean-square error (NRMSE) +is +NRMSE ≡ +�� +n( ˆY (tn) − Yn)2 +� +n Y 2 +n +. +(18) +Performance of the task is high when NRMSE ≈ 0. In the ESN, it was reported that NRMSE ≈ +0.4 for N = 50 and NRMSE ≈ 0.2 for N = 200 (51). The number of node N = 200 was used +for the speech recognition with ≈ 0.02 word error rate (51), and time-series prediction of sptio- +temporal chaos (5). Therefore, NRMSE ≈ 0.2 is considered as reasonably high performance in +practical application. We also stress that we use the same order of nodes (virtual and physical +nodes) N = 128 to achieve NRMSE ≈ 0.2. +Memory capacity and information processing capacity +Memory capacity (MC) is a measure of the short-term memory of RC. This was introduced +in (6). For the input Un of random time series taken from the uniform distribution, the network +23 + +is trained for the output Yn = Un−k. The MC is computed from +MCk = ⟨Un−k, W · X(tn)⟩2 +⟨U2 +n⟩⟨(W · X(tn))2⟩. +(19) +This quantity is decaying as the delay k increases, and MC is defined as +MC = +kmax +� +k=1 +MCk. +(20) +Here, kmax is a maximum delay, and in this study we set it as kmax = 100. The advantage of MC +is that when the input is independent and identically distributed (i.i.d.), and the output function +is linear, then MC is bounded by N, the number of internal nodes. +Information processing capacity (IPC) is a nonlinear version of MC (27). In this task, the +output is set as +Yn = +� +k +Pdk(Un−k) +(21) +where dk is non-negative integer, and Pdk(x) is the Legendre polynomials of x order dk. We +may define +IPCd0,d1,...,dT −1 = +⟨Yn, W · X(tn)⟩2 +⟨Y 2 +n ⟩⟨(W · X(tn))2⟩. +(22) +and then compute jth order IPC as We may define +IPCj = +� +dks.t.j=� +k dk +IPCd1,d2,...,dT . +(23) +When j = 1, the IPC is, in fact, equivalent to MC, because P0(x) = 1 and P1(x) = x. In +this case, Yn = Un−k for di = 1 when i = k and di = 0 otherwise. (23) takes the sum over +all possible delay k, which is nothing but MC. When j > 1, IPC captures all the nonlinear +transformation and delays up to the jth polynomial order. For example, when j = 2, the output +can be Yn = Un−k1Un−k2 or Yn = U2 +n−k + const. In this study, we focus on j = 2 because +24 + +the second-order nonlinearity is essential for the NARMA10 task (see Sec. A in Supplementary +Information). +The relevance of MC and IPC is clear by considering the Volterra series of the input-output +relation, +Yn = +� +k1,k2,··· ,kt +βk1,k2,··· ,knUk1 +1 Uk2 +2 · · · Ukn +n . +(24) +Instead of polynomial basis, we may use orthonormal basis such as the Legendre polynomials +Yn = +� +k1,k2,··· ,kn +βk1,k2,··· ,knPk1(U1)Pk2(U2) · · ·Pkn(Un). +(25) +Each term in (25) is characterized by the non-negative indices (k1, k2, . . . , kn). Therefore, the +terms corresponding to j = � +i ki = 1 in Yn have information on linear terms with time +delay. Similarly, the terms corresponding to j = � +i ki = 2 have information of second-order +nonlinearity with time delay. In this view, the estimation of the output Y (t) is nothing but the +estimation of the coefficients βk1,k2,...,kn. In RC, the readout of the reservoir state at ith node +(either physical or virtual node) can also be expanded as the Volterra series +˜˜X(i)(tn) = +� +k1,k2,··· ,kn +˜˜β(i) +k1,k2,··· ,knUk1 +1 Uk2 +2 · · · Ukn +n . +(26) +Therefore, MC and IPC are essentially a reconstruction of βk1,k2,··· ,kn from ˜˜β(i) +k1,k2,··· ,kn with +i ∈ [1, N]. This can be done by regarding βk1,k2,··· ,kn as a T + T(T − 1)/2 + · · · -dimensional +vector, and using the matrix M associated with the readout weights as +βk1,k2,··· ,kn = M · + + + + + + +˜˜β(1) +k1,k2,··· ,kn +˜˜β(2) +k1,k2,··· ,kn +... +˜˜β(N) +k1,k2,··· ,kn + + + + + + +. +(27) +MC corresponds to the reconstruction of βk1,k2,··· ,kn for � +i ki = 1, whereas the second-order +IPC is the reconstruction of βk1,k2,··· ,kn for � +i ki = 2. If all of the reservoir states are indepen- +25 + +dent, we may reconstruct N components in βk1,k2,··· ,kn. In realistic cases, the reservoir states are +not independent, and therefore, we can estimate only < N components in βk1,k2,··· ,kn. +Prediction of chaotic time-series data +Following (5), we perform the prediction of time-series data from the Lorenz model. The model +is a three-variable system of (A1(t), A2(t), A3(t)) yielding the following equation +dA1 +dt = 10(A2 − A1) +(28) +dA2 +dt = A1(28 − A3) − A2 +(29) +dA3 +dt = A1A2 − 8 +3A3. +(30) +The parameters are chosen such that the model exhibits chaotic dynamics. Similar to the other +tasks, we apply the different masks of binary noise for different physical nodes, B(l) +i (t) ∈ +{−1, 1}. Because the input time series is three-dimensional, we use three independent masks +for A1, A2, and A3, therefore, l ∈ {1, 2, 3}. The input for the ith physical node after the mask is +given as Bi(t) ˜Ui(t) = B(1) +i (t)A1(t)+B(2) +i (t)A2(t)+B(3) +i (t)A3(t). Then, the input is normalized +so that its range becomes [0, 0.5], and applied as an input current. Once the input is prepared, +we may compute magnetization dynamics for each physical and virtual node, as in the case of +the NARMA10 task. We note that here we use the binary mask of {−1, 1} instead of {0, 1} +used for other tasks. We found that the {0, 1} does not work for the prediction of the Lorenz +model, possibly because of the symmetry of the model. +The ground-truth data of the Lorenz time-series is prepared using the Runge-Kutta method +with the time step ∆t = 0.025. The time series is t ∈ [−60, 75], and t ∈ [−60, −50] is used for +relaxation, t ∈ (−50, 0] for training, and t ∈ (0, 75] for prediction. During the training steps, +we compute the output weight by taking the output as Y = (A1(t + ∆t), A2(t + ∆t), A3(t + +∆t)). After training, the RC learns the mapping (A1(t), A2(t), A3(t)) → (A1(t + ∆t), A2(t + +26 + +∆t), A3(t + ∆t)). For the prediction steps, we no longer use the ground-truth input but the +estimated data ( ˆ +A1(t), ˆ +A2(t), ˆ +A3(t)). Using the fixed output weights computed in the training +steps, the time evolution of the estimated time-series ( ˆ +A1(t), ˆ +A2(t), ˆ +A3(t)) is computed by the +RC. +Theoretical analysis using response function +We consider the Landau-Lifshitz-Gilbert equation for the magnetization field m(x, t), +∂tm(x, t) = −m × heff − m × (m × heff) + σ(x, t)m × (m × mf) +(31) +We normalize both the magnetic and effective fields by saturation magnetization as m = +M/Ms and heff = Heff/Ms. This normalization applies to all the fields including external +and anisotropic fields. We also normalize the current density as σ(x, t) = J(x, t)/j0 for the +current density J(x) and the unit of current density j0 = +4M2 +s eπa2Dµ0 +ℏP +. We apply the current +density at the nanocontact as +J(x, t) = 2jc ˜U(t) +Np +� +i=1 +χa(|x − Ri|) +(32) +Here χa(x) is a characteristic function χa(x) = 1 when x ≤ a and χa(x) = 0 otherwise. +We expand the solution of (31) around the uniform magnetization m(x, t) = (0, 0, 1) with- +out current injection as +m(x, t) = m0(x, t) + ǫm(1)(x, t) + O(ǫ2). +(33) +Here, m0(x, t) = (0, 0, 1) and ǫ ≪ 1 is a small parameter corresponding to the magnitude of the +input σ(x, t). The first-order term corresponds to a linear response of the magnetization to the +input σ, whereas the higher-order terms describe nonlinear responses, for example, m(2)(x, t) ∼ +σ(x1, t1)σ(x2, t2). Because our input is driven by the spin torque with fixed layer magnetization +in the x-direction, mf = ex, only mx and my appear in the first-order term O(ǫ). Deviation of +27 + +mz from mz = 1 appears in O(ǫ2). Therefore, for the first-order term m(1), we may define the +complex magnetization +m = mx + imy. +(34) +Here, we will show the magnetization is expressed by the response function Gij(t). The +input at the jth physical node affects the magnetization at the ith physical node as +mi(t) += +� +dτGii(t − τ)σi(τ) + � +i̸=j +� +dτGij(t − τ)σj(τ). +(35) +The input for the jth physical node is expressed by σj(t) = 2jcBj(t) ˜Uj(t). Because different +physical nodes have different masks discussed in Learning with reservoir computing in Meth- +ods. When the wave propagation is dominated by the exchange interaction, the response func- +tion for the same node is +Gii(t − τ) = 1 +2πe−˜h(α+i)(t−τ) +� +1 − e− +a2 +4(α+i)(t−τ) +� +(36) +and for different nodes, it becomes +Gij(t − τ) = a2 +2πe−˜h(α+i)(t−τ)e− +|Ri−Rj|2 +4(α+i)(t−τ) +1 +2(α + i)(t − τ). +(37) +When the wave propagation is dominated by the dipole interaction, the response function for +the same node is +Gii(t − τ) = 1 +2πe−˜h(α+i)(t−τ) −1 + +� +1 + +a2 +(d/4)2(α+i)2(t−τ)2 +� +1 + +a2 +(d/4)2(α+i)2(t−τ)2 +(38) +and for different nodes it becomes +Gij(t − τ) = a2 +2πe−˜h(α+i)(t−τ) +× +1 +(d/4)2(α + i)2(t − τ)2 +� +1 + +|Ri−Rj|2 +(d/4)2(α+i)2(t−τ)2 +�3/2. +(39) +28 + +Clearly, Gii(0) → 1 and Gij(0) → 0, while Gii(∞) → 0 and Gij(∞) → 0. +Once the magnetization is expressed in the form of (35), we may compute the reservoir state +X under the input U. Then, we may use the same method as in Learning with reservoir computing, +and estimate the output ˆY. Similar to the micromagnetic simulations, we evaluate the perfor- +mance by MC, IPC, and NARMA10 tasks. +We may extend the analyzes for the higher-order terms in the expansion of (33). In Sec.B in +Supplementary Materials, we show the second-order term m(2)(x, t) has only the z-component, +and moreover, it is dependent only on the first-order terms. As a result, the second-order term +is expressed as +m(2) +z (x, t) = −1 +2 +� +(m(1) +x )2 + (m(1) +y )2� +. +(40) +To compute the response functions, we linearize (31) for the complex magnetization m(x, t) +as +∂tm(x, t) = Lm + σ(x, t), +(41) +where the linear operator is expressed as +L = +� +−˜h + ∆ +� +(α + i) . +(42) +In the Fourier space, the linearized equation becomes +∂tmk(t) = Lkmk + σk(t), +(43) +with +Lk = − +� +˜h + k2� +(α + i) . +(44) +The solution of ((43)) is obtained as +mk(t) = +� +dτeLk(t−τ)σk(τ). +(45) +29 + +We have Np cylindrical shape inputs with radius a and the ith input is located at Ri. The input +function is expressed as +σ(x) = +Np +� +i=1 +χa (|x − Ri|) . +(46) +We are interested in the magnetization at the input mi(t) = m(x = Ri, t), which is +mi = +1 +(2π)2 +� +j +� +dτe−˜h(α+i)(t−τ) +� +dke−k2(α+i)(t−τ)eik·(Ri−Rj)2πaJ1(ka)σj(t) += a +2π +� +j +� +dτe−˜h(α+i)(t−τ) +� +dke−k2(α+i)(t−τ)J0 (k|Ri − Rj|) J1(ka)σj(t) +(47) +For the same node, |Ri − Rj| = 0, and we may compute the integral explicitly as (36). When +ka ≪ 1, we may assume J1(ka) ≈ ka/2, and finally, come up with (37). +When the thickness d of the material is thin, the dispersion relation becomes +Lk = −˜h(α + i) +�� +1 + k2 +˜h +� � +1 + k2 +˜h ++ βk +˜h +� +(48) +where +β = d +2. +(49) +We assume for k ≪ β +� +˜h, then the linearized operator becomes +Lk = −(α + i) +� +˜h + kd +4 +� +(50) +leading to (38) and (39). +Acknowledgements: +S. M. thanks to CSRN at Tohoku University. Numerical simulations in this work were carried +out in part by AI Bridging Cloud Infrastructure (ABCI) at National Institute of Advanced In- +dustrial Science and Technology (AIST), and by the supercomputer system at the information +30 + +initiative center, Hokkaido University, Sapporo, Japan. +Funding: +This work is support by JSPS KAKENHI grant numbers 21H04648, 21H05000 to S.M., by +JST, PRESTO Grant Number JPMJPR22B2 to S.I., X-NICS, MEXT Grant Number JPJ011438 +to S.M., and by JST FOREST Program Grant Number JPMJFR2140 to N.Y. +Author Contributions +S.M., N.Y., S.I. conceived the research. S.I., Y.K., N.Y. carried out simulations. N.Y., S.I. an- +alyzed the results. N.Y., S.I., S.M. wrote the manuscript. All the authors discussed the results +and analysis. +Competing Interests +The authors declare that they have no competing financial interests. +Data and materials availability: +All data are available in the main text or the supplementary materials. +A +Connection between the NARMA10 task and MC/IPC +In this section, we discuss the necessary properties of reservoir computing to achieve high per- +formance of the NARMA10 task. In short, the NARMA10 task is dominated by the memory of +31 + +nine step previous data and second-order nonlinearity. We discuss these properties in two meth- +ods. The first method is based on the extended Dynamic Mode Decomposition (DMD) (52) and +the higher-order DMD (53). The second method is a regression of the input-output relationship. +We will discuss the details of the two methods. Our results are consistent with previous studies; +the requirement of memory was discussed in (54), and the second-order nonlinear terms with a +time delay in (55). +The NARMA10 task is based on the discrete differential equation, +Yn+1 = αYn + βYn +9 +� +i=0 +Yn−i + γUnUn−9 + δ. +(51) +Here, Un is an input at the time step n taken from the uniform random distribution U(0, 0.5), +and Yn is an output. We choose the parameter as α = 0.3, β = 0.05, γ = 1.5, and δ = 0.1. +In the first method, we estimate the transition matrix A from the state variable Yn = +(Y1, Y2, . . . , Yn) to Yn+1 = (Y2, Y3, . . . , Yn+1) yielding +Yn+1 = A · Yn. +(52) +We may extend the notion of the state variable to contain delayed data and polynomials of the +output with time delay as +Yn = (Yn, Yn−1, . . . , Y1, YnYn, YnYn−1, . . . , Y1Y1) . +(53) +Including the delay terms following from the higher-order DMD (53), while the polynomial +nonlinear terms are used as a polynomial dictionary in the extended DMD (52). Here, (53) +contains all the combination of the second-order terms with time delay, Yn−i1Yn−i2 with the +integers i1 and i2 in 0 ≤ i1 ≤ l2 ≤ n − 1. We may straightforwardly include higher-order terms +in powers in (53). In the NARMA10 task, the output Yn+1 is also affected by the input Un. +Therefore, the extended DMD is generalized to include the control as (56) +Yn+1 = (A B) · +�Yn +Un +� +, +(54) +32 + +where the state variable corresponding to the input includes time delay and nonlinearity, and is +described as +Un = (Un, Un−1, . . . , U1, UnUn, UnUn−1, . . . , U1U1) . +(55) +We denote the generalized transition matrix as +Ξ = (A B) . +(56) +The idea of DMD is to estimate the transition matrix from the data. This is done by taking +pseudo inverse of the state variables as +ˆΞ = Yk+1 · +� +Yk +Uk +�† +. +(57) +Here, M† is the pseudoinverse of the matrix M. This is nothing but a least-square estimation +for the cost function of l.h.s minus r.h.s of (54). We may include the Tikhonov regularization +term. +Note that for the extended DMD (52) and the higher-order DMD (53), the transition matrix +Ξ is further decomposed into characteristic modes associated with its eigenvalues. The decom- +position gives us a dimensional reduction of the system. The estimation of the transition matrix +is also called nonlinear system identification, particularly, nonlinear autoregression with exoge- +nous inputs (NARX). In this work, we focus on the estimation of the input-output relationship, +and do not discuss the dimensional reduction. For time-series prediction, we estimate the func- +tion Yn+1 = f(Yn, Yn−1, . . . , Y1), and we do not need the input Un in (54). Even in this case, +we may consider a similar estimation of Ξ (in fact, A). This estimation is the method used in +the next-generation RC (47). +The second method is based on the Volterra series of the state variable Yn by the input Un. +In this method, we assume that the state variable is independent of its initial condition. Then, +33 + +we may express the state variable as +Yn = G · Un. +(58) +Note that Un includes the input and its polynomials with a time delay as in (55). Similar to the +first method, we estimate G by +ˆG = Yt · U† +t. +(59) +The estimated ˆG gives us information on which time delay and nonlinearity dominate the state +variable. +0 +5 +10 +15 +20 +25 +30 +delay +NRMSE +0 +0.2 +0.4 +0.6 +0.8 +1.0 +test +training +linear +(A) +(B) +(C) +(D) +(E) +(F) +0 +5 +10 +15 +20 +25 +30 +delay +NRMSE +0 +0.2 +0.6 +0.8 +1.0 +0.4 +second-order nonlinearity +0 +5 +10 +15 +20 +25 +30 +delay +NRMSE +0 +0.2 +0.6 +0.8 +1.0 +0.4 +third-order nonlinearity +0 +5 +10 +15 +20 +25 +30 +delay +NRMSE +0 +0.2 +0.4 +0.6 +0.8 +1.0 +linear +0 +5 +10 +15 +20 +25 +30 +delay +NRMSE +0 +0.2 +0.6 +0.8 +1.0 +0.4 +second-order nonlinearity +0 +5 +10 +15 +20 +25 +30 +delay +NRMSE +0 +0.2 +0.6 +0.8 +1.0 +0.4 +third-order nonlinearity +Figure 9: (A-C) the estimation based on the extended DMD, (D-F) the estimation based on the +Volterra series. The dictionary of each case is (A,D) first-order (linear) delay terms, (B,E) up to +second-order delay terms, and (C,F) up to third-order delay terms. +The results of the two estimation methods are shown in Fig. 9. Both approaches suggest +that memory of ≈ 10 steps is enough to get high performance, and further memory does not +improve the error. The second-order nonlinear term shows a reasonably small NRMSE of ≈ +0.01. Including the third-order nonlinearity improves the error, but there is a sign of overfitting +34 + +at a longer delay because the number of the state variables is too large. It should also be noted +that even with the linear terms, the NRMSE becomes ≈ 0.35. This result implies that although +NRMSE ≈ 0.35 is often considered good performance, nonlinearity of the data is not learned +at the error of this order. +A.1 +The MC and IPC tasks as Volterra series for linear and nonlinear +readout +In (3) and (4) in the main text, we show that the magnetization at the input region is expressed +by the response function. The magnetization at the time tn corresponding to the input Un at the +n step is expressed as +m(tn) = anUn + an−1Un−1 + · · · , +(60) +where the coefficients an can be computed from the response function. We first consider the +linear case, but we will generalize the expression for the nonlinear case. Because we use virtual +nodes, the input Un at the step n continues during the time period t ∈ [tn, tn+1) discretized by +Nv steps as (tn,1, tn,2, . . . , tn,Nv), and is multiplied by the filter of the binary noise (see Fig.2 and +Methods in the main text). Therefore, the magnetization is expressed by the response functions +G(t − t′) is formally expressed as +m(tn) = +Np +� +i +[(G(0) + G(θ) + · · · G(θ(Nv − 1))) σi(tn) ++ (G(θNv) + G(θ(Nv + 1)) + · · · G(θ(2Nv − 1))) σi(tn−1) ++ · · ·] , +(61) +where σi(tn) ∝ Un is the non-dimensionalized current injection at the time tn at the ith physical +node, which is proportional to Un. Therefore, (61) results in the expression of (60). Our input +is taken from a uniform random distribution. Therefore, the inner product of the reservoir state, +35 + +which is nothing but magnetization, and (delayed) input to learn MC is +⟨m(tn), Un⟩ = +T +� +n=1 +m(tn)Un = an⟨U2 +n⟩ + O(1/T). +(62) +Similarly, the variance of the magnetization is equal to the variance of the input with the coef- +ficient associated with m(tn). +We may express the MC and IPC tasks in a matrix form as +˜S ≈ W · G · (S ◦ Win) . +(63) +Here, S is the matrix associated with the original input, and ˜S is the delayed one. The output +weight is denoted by W, and Win is the matrix associated with the mask of binary noise. The +goal of MC and IPC tasks is to approximate the delayed input ˜S by the reservoir states G · S. +Here, the reservoir states are expressed by the response function G and input denoted by S. We +define delayed input ˜S ∈ RK×T +˜S = + + + + + +Un +Un+1 +Un+2 +· · · +Un−1 +Un +Un+1 +· · · +Un−2 +Un−1 +Un +· · · +... +... +... +... + + + + + . +(64) +Here, T is the number of the time series, and K is the total length of the delay that we consider. +The ith row shows the i−1 delayed time series. The input S ∈ RTNv×T to compute the reservoir +states are expressed as +S = + + + + + + + + + + + + +σ(tn) +σ(tn+1) +σ(tn+2) +· · · +... +... +... +... +σ(tn) +σ(tn+1) +σ(tn+2) +· · · +σ(tn−1) +σ(tn) +σ(tn+1) +· · · +... +... +... +... +σ(tn−2) +σ(tn−1) +σ(tn) +· · · +... +... +... +... + + + + + + + + + + + + +. +(65) +Note that σ(tn) ∝ Un upto constant. Due to time multiplexing, each row is repeated Nv times, +and then the time series is delayed in the next row. After multiplying the input filter Win, the +36 + +input is fed into the response function. The input filter Win ∈ RTNv×T is a stack of constant row +vectors with the length T. The Nv different realizations of row vectors are taken from binary +noise, and then the resulting Nv × T matrix is repeated T times in the row direction. This input +is multiplied by the coefficients of the Volterra series G ∈ RN×TNv +G = + + + + + +G(1)(0) +· · · +G(1)(θ(Nv − 1)) +G(1)(θNv) +· · · +G(1)(θ(2Nv − 1)) +· · · +G(2)(0) +· · · +G(2)(θ(Nv − 1)) +G(2)(θNv) +· · · +G(2)(θ(2Nv − 1)) +· · · +... +... +... +... +... +... +... +G(N)(0) +· · · +G(N)(θ(Nv − 1)) +G(N)(θNv) +· · · +G(N)(θ(2Nv − 1)) +· · · + + + + + +(66) +(63) implies that by choosing the appropriate W, we can get a canonical form of G. If +the canonical form has N × N identity matrix in the left part of W · G, then the reservoir +reproduces the time series up to N − 1 delay. This means that the rank of the matrix G, or the +number of independent rows, is the maximum number of steps of the delay. This is consistent +with the known fact that MC is bounded by the number of independent components of reservoir +variables (6). +Next we extend the Volterra series of the magnetization, including nonlinear terms. The +magnetization is expressed as +m(tn) = anσ(tn) + an−1σ(tn−1) + · · · + an,nσ(tn)σ(tn) + an,n−1σ(tn)σ(tn−1) + · · · . +(67) +The delayed input ˜S is rewritten as +˜S = + + + + + + + + + +Un +Un+1 +Un+2 +· · · +Un−1 +Un +Un+1 +· · · +... +... +... +... +UnUn +Un+1Un+1 +Un+2Un+2 +· · · +UnUn−1 +Un+1Un +Un+2Un+1 +· · · +... +... +... +... + + + + + + + + + +. +(68) +The matrix ˜S contains all the nonlinear combinations of the input series (Un, Un+1, · · ·). Ac- +cordingly, we should modify S and also G to include the nonlinear response functions. Note +that to guarantee the orthogonality, Legendre polynomials (or other orthogonal polynomials) +37 + +should be used instead of polynomials in powers. Nevertheless, up to the second order of +nonlinearity, which is relevant to consider the performance of NARMA10 (see Sec. A), the dif- +ference is only in the constant terms (P2(x) = x2 − 1 +2). Because we subtract the mean value of +the time series of all the input, output, and reservoir states, these constant terms do not change +our conclusion. With nonlinear terms, (66) is extended as G = (Glin, Gnonl). Still, the rank of +the matrix remains N at most. This is the reason why the total sum of IPC, including all the lin- +ear and nonlinear delays, is bounded by the number of independent reservoir variables. When +Gnonl = 0, the reservoir can memorize only the linear delay terms, but MC can be maximized +to be N. On the other hand, when Gnonl ̸= 0, it is possible that MC is less than N, but the +reservoir may have finite IPC. +When the readout is nonlinear, we use the reservoir state variable as +X = +� +M +M ◦ M +� +, +(69) +where ◦ is the Hadamard product. If M is linear in the input, G has a structure of +G = +� +Glin +0 +0 +Gnonlin +� +. +(70) +In this case, rank(G) = rank(Glin) + rank(Gnonlin). +B +Learning with multiple variables +In the main text, we use only mx for the readout as in (13)-(15). The readout is nonlinear and +has both the information of mx and m2 +x. In this section, we consider the linear readout, but +use both mx and mz for the output in micromagnetic simulations. We begin with the linear +readout only with mx. The results of the MC and IPC tasks are shown in Fig. 10(a,b). We +obtain a similar performance for the MC task with the result in the main text (Fig. 3). On the +other hand, the performance for the IPC task in Fig. 10(a) is significantly poorer than the result +38 + +in Fig. 3(a). This result demonstrates that the linear readout only with mx does not learn the +nonlinearity effectively. Note that in the theoretical model with the response function, the IPC +is exactly zero when we use the linear readout only with mx. The discrepancy arises from the +expansion (33) around m0 = (0, 0, 1) in the main text. Strictly speaking, the expansion should +be made around m0 under the constant input ⟨σ⟩ averaged over time at the input nanocontact. +This reference state is inhomogeneous in space, and is hard to compute analytically. Due to this +effect, mx in the micromagnetic simulations contain small nonlinearity. +Next, we consider the linear readout with mx and mz. As seen in Fig. 10(c,d), mz carries +nonlinear information, and enhances the IPC and learning performance of NARMA10 com- +pared with linear readout only with mx (Fig. 10 (a,b)). The performance is IPC ≈ 60 under +α = 5 × 10−4, which is comparable value with the results in the main text (Fig. 3(a,c)) where +the readout is (mx, m2 +x). Also, high performance for NARMA10 task, NRMSE ≈ 0.2, can be +obtained using variables (mx, mz). These results show that adding mz into the readout has a +similar effect to adding m2 +x. +Similarity between m2 +x and mz can be understood by using the theoretical formula with the +response function in the main text. We continue the expansion (33) at the second order, and +obtain +∂tm(2)(x, t) = − m(1) × ∆m(1) − αm(1) × +�� +˜hm(1) − ∆m(1)� +× ez +� ++ σ(x, t)m(1) × ey. +(71) +This result suggests that m(2) contains only the z component, and is slaved by m(1), which does +not have z component. Therefore, m(2) +z +can be computed as +m(2) +z (x, t) = −1 +2 +� +(m(1) +x )2 + (m(1) +y )2� +. +(72) +Because mx and my carry similar information, mz in the readout has a similar effect with m2 +x +in the readout. +39 + +C +Speed of propagating spin wave using dipole interaction +Propagating spin wave when magnetization is pointing along film normal is called magneto- +static forward volume mode, and its dispersion relation can be described by the following equa- +tion (31). +ω(k) = γµ0 +� +(H0 − Ms) +� +H0 − Ms +1 − e−kd +kd +� +. +(73) +Then, one can obtain the group velocity at k ∼ 0 as, +vg = dω +dk (k = 0) = γµ0Msd +4 +. +(74) +In the magneto-static spin wave driven by dipole interaction, group velocity is proportional to +both Ms and d. vg ∼ 200 m/s is obtained when the following parameters are used: µ0H = 1.5 +T, Ms = 1.0 × 106 A/m, d = 4 nm. The same estimation is used for calculating the speed of +information propagation for spin reservoirs in Refs. (19) and (22), which are used to plot Fig. 7 +in the main text. +D +Details of reservoir computing scaling compared with lit- +erature +In this section, details of Fig. 7 shown in the main text are described. MC and NRMSE for +NARMA10 tasks using photonic and spintronic RC are reported in Refs. (12,32–35,38,39) for +photonic RC and (9,19,22,25,36,37,57,58) for spintronic RC. Table 1 and 2 shows reports of +MC for photonic and spintronic RC with different length scales, which are plotted in Fig. 7 in +the main text. +40 + +Table 1: Report of photonic RC with different length scales used in Fig. 7 in the main text +Reports +Length, L +Time interval, τ0 +vτ0 +N +MC +Duport et al. (32) +1.6 km +8 µs +2.4 km +50 +21 +Dejonckheere et al. (33) +1.6 km +8 µs +2.4 km +50 +37 +Vincker et al. (34) +230 m +1.1 µs +340 m +50 +21 +Takano et al. (12) +11 mm +200 ps +60 mm +31 +1.5 +Sugano et al. (35) +10 mm +240 ps +72 mm +240 +10 +Note: speed of light, v = 3 × 108 m/s is used. +Table 2: Report of spin reservoirs with different length scales used in Fig. 7 in the main text +Reports +L +τ0 +v +vτ0 +N +MC +Nakane et al. (19) +5 µm +2 ns +2.4 km/s +4.8 µm +72 +21 +Dale et al. (22) +50 nm +10 ps +200 m/s +2 nm +100 +35 +This work +500 nm +1.6 ns +200 m/s +320 nm +64 +26 +Note: v is calculated based on magneto-static spin wave using Eq. 74. +E +Other data +E.1 +Nv and Np dependence of performance +Fig. 11 shows Nv and Np dependencies of MC, IPC and NRMSE for NARMA10 task. As Nv +and Np are increased, MC and IPC increase. Then, NARMA10 prediction task becomes better +with increasing Nv and Np. MC and NRMSE for NARMA10 with different Np with fixed Nv += 8 are compared with other reservoirs shown in Fig. 8 in the main text. +E.2 +exchange interaction +In the main text, we use the dipole interaction to compute the response function as (38) and +(39). 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Mochizuki, Reservoir Computing with Spin Waves in a Skyrmion Crystal. +Physical Review Applied 18, 014074 (2022). +48 + +(a) (mx), MC and IPC +MNO +Pm +x), NARMA10 +MC +IPC +0 +20 +40 +60 +80 +5 +2.5 +α = 5×10-4 +Frequency, 1/θ (GHz) +0 +20 +40 +60 +80 +α = 5×10-3 +Linear and non-linear memory capacity +0.2 +0.4 +0 +20 +40 +60 +80 +α = 5×10-2 +Distance of virtual nodes, θ (ns) +0 +0.5 +1 +5 +2.5 +α = 5×10-4 + Training, + Test +Frequency, 1/θ (GHz) +0 +0.5 +1 +α = 5×10-3 +Normalized root mean square error, NRMSE +for NARMA10 task +0.2 +0.4 +0 +0.5 +1 +α = 5×10-2 +Distance of virtual nodes, θ (ns) +(c) (m +Q, mz) MC and IPC +(d) (m +R, mz), NARMA10 +MC +IPC +0 +20 +40 +60 +80 +5 +2.5 +α = 5×10-4 +Frequency, 1/θ (GHz) +0 +20 +40 +60 +80 +α = 5×10-3 +Linear and non-linear memory capacity +0.2 +0.4 +0 +20 +40 +60 +80 +α = 5×10-2 +Distance of virtual nodes, θ (ns) +0 +0.5 +1 +5 +2.5 +α = 5×10-4 + Training, + Test +Frequency, 1/θ (GHz) +0 +0.5 +1 +α = 5×10-3 +Normalized root mean square error, NRMSE +for NARMA10 task +0.2 +0.4 +0 +0.5 +1 +α = 5×10-2 +Distance of virtual nodes, θ (ns) +Figure 10: +Reservoir computing with various parameter combinations obtained using micro- +magnetic Mumax3 simulation. Linear memory capacity, MC and nonlinear memory capacity, +IPC plotted as a function of θ obtained using linear mx output only (a) and using mx, mz (c). +Normalized root mean square error, NRMSE for NARMA10 task plotted as a function of θ +obtained using linear mx output only (b) and using mx, mz (d). +49 + +2 +4 +6 +8 +2 +4 +6 +8 +Number of physical nodes, Np +Number of virtual nodes, Nv +0 +5 +10 +15 +20 +25 +30 +Memory capacity, MC +2 +4 +6 +8 +2 +4 +6 +8 +Number of physical nodes, Np +Number of virtual nodes, Nv +0 +10 +20 +30 +40 +50 +60 +S T +Nonlinear memory capacity, IPC +2 +4 +6 +8 +2 +4 +6 +8 +Number of physical nodes, Np +Number of virtual nodes, Nv +0.00 +0.20 +0.40 +0.60 +0.80 +1.00 +Normalized mean square error, NRMSE +for NARMA10 task +(a) +U VW +(c) +Figure 11: (a) Memory capacity, MC (b) Nonlinear memory capacity, IPC and (c) Normalized +root mean square error, NRMSE for NARMA10 task plotted as a function of the number of +virtual and physical nodes. The parameters used in the simulation are α = 5 × 10−4, θ = 0.2 +ns. +(a) +(b) +(c) +wave speed (log m/s) +characteristic size (log nm) +3.0 +2.0 +2.0 +3.0 +4.0 +MC +20 +30 +40 +50 +damping time +1.0 +5.0 +4.0 +wave speed (log m/s) +characteristic size (log nm) +2.0 +4.0 +3.0 +2.0 +3.0 +4.0 +IPC +20 +30 +40 +50 +damping time +1.0 +5.0 +0 +20 +40 +60 +80 +5 +2.5 +� = 5×10-4 +Frequency, 1/� (GHz) +0 +20 +40 +60 +80 +� = 5×10-3 +Linear and non-linear memory capacity +0.2 +0.4 +0 +20 +40 +60 +80 +� = 5×10-2 +Distance of virtual nodes, � (ns) +Figure 12: (a) Memory capacity, MC (solid symbols) and nonlinear memory capacity, IPC +(open symbols) obtained using the response function method for exchange interaction plotted +as a function of θ with different damping parameters α. (b) MC and (c) IPC plotted as a function +of characteristic size and wave speed. +50 + diff --git a/DtE0T4oBgHgl3EQfQgBB/content/tmp_files/load_file.txt b/DtE0T4oBgHgl3EQfQgBB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5ad87465e419e3e85be7e2738e18833b83f1c42c --- /dev/null +++ b/DtE0T4oBgHgl3EQfQgBB/content/tmp_files/load_file.txt @@ -0,0 +1,1551 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf,len=1550 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='02193v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='app-ph] 5 Jan 2023 Universal scaling between wave speed and size enables nanoscale high-performance reservoir computing based on propagating spin-waves Satoshi Iihama,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 Yuya Koike,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 Shigemi Mizukami,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 Natsuhiko Yoshinaga2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5∗ 1Frontier Research Institute for Interdisciplinary Sciences (FRIS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Tohoku University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Sendai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 980-8578,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Japan 2WPI Advanced Institute for Materials Research (AIMR),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Tohoku University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Katahira 2-1-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Sendai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 980-8577,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Japan 3Department of Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Tohoku University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Sendai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 980-8579,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Japan 4Center for Science and Innovation in Spintronics (CSIS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Tohoku University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Sendai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 980-8577,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Japan 5MathAM-OIL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' AIST,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Sendai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 980-8577,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Japan ∗To whom correspondence should be addressed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' E-mail: yoshinaga@tohoku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='jp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Neuromorphic computing using spin waves is promising for high-speed nanoscale devices, but the realization of high performance has not yet been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Here we show, using micromagnetic simulations and simplified theory with response functions, that spin-wave physical reservoir computing can achieve miniaturization down to nanoscales keeping high computational power com- parable with other state-of-art systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We also show the scaling of system sizes with the propagation speed of spin waves plays a key role to achieve high performance at nanoscales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 1 Introduction Non-local magnetization dynamics in a nanomagnet, spin-waves, can be used for processing in- formation in an energy-efficient manner since spin-waves carry information in a magnetic ma- terial without Ohmic losses (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The wavelength of the spin-wave can be down to the nanometer scale, and the spin-wave frequency becomes several GHz to THz frequency, which are promis- ing properties for nanoscale and high-speed operation devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Recently, neuromorphic com- puting using spintronics technology has attracted great attention for the development of future low-power consumption artificial intelligence (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Spin-waves can be created by various means such as magnetic field, spin-transfer torque, spin-orbit torque, voltage induced change in mag- netic anisotropy and can be detected by the magnetoresistance effect (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Therefore, neuromor- phic computing using spin waves may have a potential of realisable devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Reservoir computing (RC) is a promising neuromorphic computation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' RC is a variant of recurrent neural networks (RNNs) and has a single layer, referred to as a reservoir, to transform an input signal into an output (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In contrast with the conventional RNNs, RC does not update the weights in the reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Therefore, by replacing the reservoir of an artificial neural network with a physical system, for example, magnetization dynamics, we may realize a neural network device to perform various tasks, such as time-series prediction (4,5), short-term memory (6, 7), pattern recognition, and pattern generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Several physical RC has been pro- posed: spintronic oscillators (8,9), optics (10), photonics (11,12), fluids, soft robots, and others (see reviews (13–15)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Among these systems, spintronic RC has the advantage in its potential realization of nanoscale devices at high speed of GHz frequency with low power consumption, which may outperform conventional electric computers in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' So far, spintronic RC has been considered using spin-torque oscillators (8, 9), magnetic skyrmion (16), and spin waves in garnet thin films (17–19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' However, the current performance of spintronic RC still remains 2 poor compared with the Echo State Network (ESN) (6, 7), idealized RC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The biggest issue is a lack of our understanding of how to achieve high performance in the RC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' To achieve high performance, the reservoir has to have a large degree of freedom, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' How- ever, in practice, it is difficult to increase the number of physical nodes, Np, because it requires more wiring of multiple inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In this respect, wave-based computation in continuum media has attracting features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The dynamics in the continuum media have large, possibly infinite, de- grees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In fact, several wave-based computations have been proposed (20, 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The challenge is to use the advantages of both wave-based computation and RC to achieve high- performance computing of time-series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' For spin wave-based RC, so far, the large degrees of freedom are extracted only by using a large number of input and/or output nodes (19, 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Here, to propose a realisable spin wave RC, we use an alternative route;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' we extract the informa- tion from the continuum media using a small number of physical nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Along this direction, using Nv virtual nodes for the dynamics with delay was proposed to increase N in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This idea was applied in optical fibres with a long delay line (11) and a net- work of oscillators with delay (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Nevertheless, the mechanism of high performance remains elusive, and no unified understanding has been made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The increase of N = NpNv with Nv does not necessarily improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In fact, RC based-on STO struggles with insufficient per- formance both in experiments (9) and simulations (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The photonic RC requires a large size of devices due to the long delay line (11,12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In this work, we show nanoscale and high-speed RC based on spin wave propagation with a small number of inputs can achieve performance comparable with the ESN and other state-of-art RC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' More importantly, by using a simple theoretical model, we clarify the mechanism of the high performance of spin wave RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We show the scaling between wave speed and system size to make virtual nodes effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 3 Results Reservoir computing using wave propagation The basic task of RC is to transform an input signal Un to an output Yn for the discrete step n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , T at the time tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' For example, for speech recognition, the input is an acoustic wave, and the output is a word corresponding to the sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Each word is determined not only by the instantaneous input but also by the past history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Therefore, the output is, in general, a function of all the past input, Yn = g ({Um}n m=1) as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The RC can also be used for time-series prediction by setting the output as Yn = Un+1 (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In this case, the state at the next time step is predicted from all the past data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' namely, the effect of delay is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The performance of the input-output transformation g can be characterized by how much past information does g have, and how much nonlinear transformation does g perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We will discuss that the former is expressed by memory capacity (MC) (26), whereas the latter is measured by information processing capacity(IPC) (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We propose physical computing based on a propagating wave (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 1(b,c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Time series of an input signal Un can be transformed into an output signal Yn (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' As we will discuss below, this transformation requires large linear and nonlinear memories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' for example, to predict Yn, we need to memorize the information of Un−2 and Un−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The input signal is injected in the first input node and propagates in the device to the output node spending a time τ1 as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Then, the output may have past information at tn − τ1 corresponding to the step n − m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The output may receive the information from another input at different time tn − τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The sum of the two peices of information is mixed and transformed as Un−m1Un−m2 either by nonlinear readout or by nonlinear dynamics of the reservoir (see also Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' B in Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We will demonstrate the wave propagation can indeed enhances memory capacity and learning performance of the input-output relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 4 Figure 1: Illustration of physical reservoir computing and reservoir based on propa- gating spin-wave network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (a) Schematic illustration of output function prediction by using time-series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Output signal Y is transformed by past information of input signal U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (b) Schematic illustration of reservoir computing with multiple physical nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The output signal at physical node A contains past input signals in other physical nodes, which are memorized by the reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (c) Schematic illustration of reservoir computing based on propagating spin-wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Propagating spin-wave in ferromagnetic thin film (m ∥ ez) is excited by spin-transfer torque at multiple physical nodes with reference magnetic layer (m ∥ ex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' x-component of magnetization is detected by the magnetoresistance effect at each physical node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 5 a (b) Y(t) α U(t - T1) · U(t - t2) output input g((U(t))) U(t - T1) X(t) α U(t - T1) + U(t - T2) input reservoir U(t - T2) Spin injector and detector (c) Ferromagnetic thin filmBefore explaining our learning strategy, we discuss how to achieve accurate learning of the input-output relationship Yn = g ({Um}n m=1) from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Here, the output may be dependent on a whole sequence of the input {Um}n m=1 = (U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , Un).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Even when both Un and Yn are one-variable time-series data, the input-output relationship g(·) may be T-variable polynomials, where T is the length of the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Formally, g(·) can be expanded in a polynomial series (Volterra series) such that g ({Um}n m=1) = � k1,k2,··· ,kt βk1,k2,··· ,ktUk1 1 Uk2 2 · · · Ukn n with the coefficients βk1,k2,··· ,kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Therefore, even for the linear input-output relationship, we need T coefficients in g(·), and as the degree of powers in the polynomials increases, the number of the coefficients increases exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This observation implies that a large number of data is required to estimate the input-output relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Nevertheless, we may expect a dimensional reduction of g(·) due to its possible dependence on the time close to t and on the lower powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Still, our physical computers should have degrees of freedom N ≫ 1, if not exponentially large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The reservoir computing framework is used to handle time-series data of the input U and the output Y (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In this framework, the input-output relationship is learned through the reservoir dynamics X(t), which in our case, is magnetization at the detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The reservoir state at a time tn is driven by the input at the nth step corresponding to tn as X(tn+1) = f (X(tn), Un) (1) with nonlinear (or possibly linear) function f(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The output is approximated by the readout operator ψ(·) as ˆYn = ψ (X(tn)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (2) Our study uses the nonlinear readout ψ (X(t)) = W1X(t) + W2X2(t) (5, 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The weight matrices W1 and W2 are estimated from the data of the reservoir dynamics X(t) and the true output Yn, where X(t) is obtained by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' With the nonlinear readout, the RC with linear 6 dynamics can achieve nonlinear transformation, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We stress that the system also works with linear readout when the RC has nonlinear dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We discuss this case in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Spin wave reservoir computing We consider a magnetic device of a thin rectangular system with cylindrical injectors (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The size of the device is L × L × D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Under the uniform external magnetic field, the magnetization is along the z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Electric current is injected at the Np injectors with the radius a and the same height with the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The spin-torque by the current drives mag- netization m(x, t) and propagating spin-waves as schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The actual demonstration of the spin-wave reservoir computing is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We demonstrate the spin-wave RC using two methods: the micromagnetic simulations and the theoretical model using a response function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In the micromagnetic simulations, we analyze the Landau-Lifshitz-Gilbert (LLG) equation with the effective magnetic field Heff = Hext+Hdemag+Hexch consists of the external field, de- magnetization, and the exchange interaction (see Theoretical analysis using response function in Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The spin waves are driven by Slonczewski spin-transfer torque (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The driving term is proportional to the DC current j(t) at the nanocontact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We inject the DC current propor- tional to the input time series U with a pre-processing filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' From the resulting spatially inho- mogeneous magnetization m(x, t), we measure the averaged magnetization at ith nanocontact mi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We use the method of time multiplexing with Nv virtual nodes (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We choose the x- component of magnetization mx,i as a reservoir state, namely, Xn = {mx,i(tn,k)}i∈[1,Np],k∈[1,Nv] (see (14) in Methods for its concrete form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' For the output transformation, we use ψ(mi,x) = W1,imi,x + W2,im2 i,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Therefore, the dimension of our reservoir is 2NpNv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The nonlinear output transformation can enhance the nonlinear transformation in reservoir (5), and it was shown that even under the linear reservoir dynamics, RC can learn any nonlinearity (28, 30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' B in 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 ・ ・ ・ �i n n+1 n+2 n+3 n+4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 Time step higher damping region cylindrical region to apply spin-transfer torque (a) (b) Training � Input, � Time Binary mask, �i �n,� �n �� �n �� �n�� � �� Time (ns) ・ ・ ・ Masked input, Time step Input, � Time step Output, � Time step � � Time step Time � 0 � �n �n+1 �n,3 �n,5 �n,7 (t) (t) Figure 2: Dimension of spin-wave reservoir and prediction of NARMA10 task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (a) Input signals U are multiplied by binary mask Bi(t) and transformed into injected current j(t) = 2jc ˜Ui(t) for the ith physical node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Current is injected into each physical node with the cylindrical region to apply spin-transfer torque and to excite spin-wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Higher damping regions in the edges of the rectangle are set to avoid reflection of spin-waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (b) Prediction of NARMA10 task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' x-component of magnetization at each physical and virtual node are collected and output weights are trained by linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='01 Node (1, 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='01 (2,1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='01 (Ny,Np 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='02 1000 2000 3000 4000 5000 6000- OQutput, Predicted 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 Error 0 6000 7000 8000 9000 10000 110000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 0 1000 2000 3000 4000 5000 60000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 1000 2000 3000 4000 5000 60001000nm 1000nm Q 500nm 4nmSupplementary Information, we also discuss the linear readout, but including the z-component of magnetization X = (mx, mz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In this case, mz plays a similar role to m2 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The performance of the RC is measured by three tasks: MC, IPC, and NARMA10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The weights in the readout are trained by reservoir variable X and the output Y (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2(b), see also Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' To understand the mechanism of high performance of learning by spin wave propagation, we also consider a simplified model using the response function of the spin wave dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' By linearizing the magnetization around m = (0, 0, 1) without inputs, we may express the linear response of the magnetization at the ith readout mi = mx,i + imy,i to the input as(see Methods) mi(t) = Np � j=1 � dt′Gij(t, t′)U(j)(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (3) Here, U(j)(t) is the input time series at jth nanocontact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The response function has a self part Gii, that is, input and readout nanocontacts are the same, and the propagation part Gij, where the distance between the input and readout nanocontacts is |Ri − Rj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We use the quadratic nonlinear readout, which has a structure m2 i (t) = Np � j1=1 Np � j2=1 � dt1 � dt2G(2) ij1j2(t, t1, t2)U(j1)(t1)U(j2)(t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (4) The response function of the nonlinear readout is G(2) ij1j2(t, t1, t2) ∝ Gij1(t, t1)Gij2(t, t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The same structure as (4) appears when we use a second-order perturbation for the input (see Meth- ods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In general, we may include the cubic and higher-order terms of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This expansion leads to the Volterra series of the output in terms of the input time series, and suggests how the spin wave RC works (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='1 in Supplementary Information for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Once the magnetization at each nanocontact is computed, we may estimate MC and IPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Figure 3 shows the results of the three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' When the time scale of the virtual node θ is small and the damping is small, the performance of spin wave RC is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 3(a) shows, we achieve MC ≈ 60 and IPC ≈ 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Accordingly, we achieve a small error in the NARMA10 9 task, NRMSE ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 3(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Theses performances are comparable with state-of-the-art ESN with the number of nodes ∼ 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' When the damping is stronger, both MC and IPC become smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Because the NARMA10 task requires the memory with the delay steps ≈ 10 and the second order nonlinearity with the delay steps ≈ 10 (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='A in Supplementary Information), the NRMSE becomes larger when MC ≲ 10 and IPC ≲ 102/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The results of the micromagnetic simulations are semi-quantitatively reproduced by the the- oretical model using the response function, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This result suggests that the linear response function G(t, t′) captures the essential feature of delay t − t′ due to wave propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (a) (b) Non-linear, IPC Linear, MC 0 20 40 60 80 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 α = 5×10-4 Frequency, 1/θ (GHz) 0 20 40 60 80 α = 5×10-3 Linear and non-linear memory capacity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 0 20 40 60 80 α = 5×10-2 Distance of virtual nodes, θ (ns) (c) Normalized root mean square error, NRMSE for NARMA10 task 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 1 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 α = 5×10-4 Training, Test Frequency, 1/θ (GHz) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 1 α = 5×10-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 1 α = 5×10-2 Distance of virtual nodes, θ (ns) 0 20 40 60 80 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 α = 5×10-4 Frequency, 1/θ (GHz) 0 20 40 60 80 α = 5×10-3 Linear and non-linear memory capacity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 0 20 40 60 80 α = 5×10-2 Distance of virtual nodes, θ (ns) Figure 3: Effect of virtual node distance on performance of spin-wave reservoir computing obtained with 8 physical nodes and 8 virtual nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Memory capacity MC and information processing capacity IPC obtained by (a) micromagnetics simulation and (b) response function method plotted as a function of virtual node distance θ with different damping parameters α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (c) Normalized root mean square error, NRMSE for NARMA10 task is plotted as a function of θ with different α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' To confirm the high MC and IPC are due to spin-wave propagation, we perform micromag- netic simulations with damping layers between nodes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 4(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The damping layers inhibit spin wave propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The result of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 4(b) shows that the memory capacity is substantially lower than that without damping, particularly when θ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The NARMA10 task shows a 10 larger error (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 4(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' When θ is small, the suppression is less effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This may be due to incomplete suppression of wave propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We also analyze the theoretical model with the response function by neglecting the inter- action between two physical nodes, namely, Gij = 0 for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In this case, information transmission between two physical nodes is not allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We obtain smaller MC and IPC than the system with wave propagation, supporting our claim (see (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 4(c))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Our spin wave RC also works for the prediction of time-series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In the study of (5), the functional relationship between the state at t + ∆t and the states before t is learned by the ESN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The trained ESN can estimate the state at t + ∆t from the past states, and therefore, it can predict the dynamics without the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In (5), the prediction for the chaotic time-series data was demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Figure 5 shows the prediction using our spin wave RC for the Lorenz model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We can demonstrate that the RC shows short-time prediction and, more importantly, reconstruct the chaotic attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Scaling of system size and wave speed To clarify the mechanism of the high performance of our spin wave RC, we investigate MC and IPC of the system with different characteristic length scales L and different wave propagat- ing speed v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The characteristic length scale is controlled by the radius of the circle on which inputs are located (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We use our theoretical model with the response function to compute MC and IPC in the parameter space (v, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This calculation can be done because the computational cost of our model is much cheaper than numerical micromagnetic simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Figure 6(a,b) shows that both MC and IPC have maximum when L ∝ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' To obtain a deeper understanding of the result, we perform the same analyzes for the further simplified model, in 11 (� � (c) (d) Linear, MC Non-linear, IPC 0 20 40 60 80 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 α = 5 × 10-4 C � \x0e\x0f\x10 \x11\x12 \x13\x14 \x15 \x16\x17 Frequency, 1/θ (GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 0 20 40 60 80 No connection Linear and non-linear memory capacity Distance of virtual nodes, θ (ns) Normalized root mean square error, NRMSE for NARMA10 task 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 1 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 \x18 \x19\x1a\x1b !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' "#$ %&\' Training, Test Frequency, 1/θ (GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 1 No-connection Distance of virtual nodes, θ (ns) (a) )*+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='/ 123 4 56789 : No-connection 0 20 40 60 80 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 α = 5 × 10-4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' <= >?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' @ ABD EFG H I JK Frequency, 1/θ (GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 0 20 40 60 80 No connection (G iL = 0 ) Linear and non-linear memory capacity Distance of virtual nodes, θ (ns) Figure 4: Effect of the network connection on the performance of reservoir computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (a) Schematic illustration of the network of physical nodes connected through propagating spin- wave [left] and physical nodes with no connection [right].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Memory capacity MC and informa- tion processing capacity IPC obtained using a connected network with 8 physical nodes [top] and physical nodes with no connection [bottom] calculated by (a) micromagnetics simulation and (b) response function method plotted as a function of virtual node distance θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 8 virtual nodes are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (c) Normalized root mean square error, NRMSE for NARMA10 task obtained by micromagnetics simulation is plotted as a function of θ with a connected network [top] and physical nodes with no connection [bottom].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 12 ground truth prediction time 10 0 10 20 30 40 5 15 25 35 45 20 10 0 10 20 20 10 10 20 0 training prediction A1 A1 A3 A3 A1 A2 A3 Figure 5: Prediction of time-series data for the Lorenz system using the RC with micro- magnetic simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The parameters are θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4ns and α = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0×10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (a) The ground truth (A1(t), A2(t), A3(t)) and the estimated time series ( ˆ A1(t), ˆ A1(t), ˆ A3(t)) are shown in blue and red, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The training steps are during t < 0, whereas the prediction steps are during t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (b) The attractor in the A1A3 plane for the ground truth and during the prediction steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 13 45 40 35 30 25 20 15 10 5 20 10 0 10 2045 40 35 30 25 20 15 10 5 20 10 0 10 20which the response function is replaced by the Gaussian function Gij(t) = exp � − 1 2w2 � t − Rij v �2� (5) where Rij is the distance between ith and jth physical nodes, and w is the width of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Even in this simplified model, we obtain MC≈ 40 and IPC≈ 60, and also the maximum when L ∝ v (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 6(c,d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' From this result, the origin of the optimal ratio between the length and speed becomes clearer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' when L ≪ v, the response functions under different Rij overlap so that different physical nodes cannot carry the information of different delay times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' On the other hand, when L ≫ v, the characteristic delay time L/v exceeds the maximum delay time to compute MC and IPC, or exceeds the total length of the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Note that we set the maximum delay time as 100, which is much longer than the value necessary for the NARMA10 task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The result suggests the universal scaling between the size of the system and the speed of the RC based on wave propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Our system of the spin wave has a characteristic length L ∼ 500 nm and a speed of v ∼ 200 m s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In fact, the reported photonic RC has characteristic length scale of optical fibres close to the scaling in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Discussion Figure 7 shows reports of reservoir computing in literature with multiple nodes plotted as a function of the length of nodes L and products of wave speed and delay time vτ0 for both photonic and spintronic RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' For the spintronic RC, the dipole interaction is considered for wave propagation in which speed is proportional to both saturation magnetization and thickness of the film (31)(See supplementary information sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' For the photonic RC, the characteristic speed is the speed of light, v ∼ 108 m s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Symbol size corresponds to MC taken from the literature [See details of plots in supplementary information sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Plots are roughly on a broad oblique 14 (a) (b) (c) (d) wave speed (log m/s) characteristic size (log nm) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 MC 10 20 30 40 50 60 damping time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 wave speed (log m/s) characteristic size (log nm) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 IPC 10 20 30 40 50 60 damping time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 wave speed (log m/s) characteristic size (log nm) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 MC 10 20 30 40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 wave speed (log m/s) characteristic size (log nm) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 IPC 10 20 30 40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 60 50 time response function 11( ) G t 12( ) G t 13( ) G t 14( ) G t 15( ) G t time dense sparse memorise (e) Figure 6: Scaling between characteristic size and propagating wave speed obtained by response function method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' MC (a,c) and IPC (b,d) as a function of the characteristic length scale between physical nodes R and the speed of wave propagation v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The results with the response function for the dipole interaction (a,b) and for the Gaussian function (5) (c,d) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (e) Schematic illustration of the response function and its relation to wave propagation between physical nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' When the speed of the wave is too fast, all the response functions are overlapped (dense regime), while the response functions cannot cover the time windows when the speed of the wave is too slow (sparse regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 15 line with a ratio L/(vτ0) ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Therefore, the photonic RC requires a larger system size, as long as the delay time of the input τ0 = Nvθ is the same order (τ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='3 − 3 ns in our spin wave RC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 6, if one wants to reduce the length of physical nodes, one must reduce wave speed or delay time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' otherwise the information is dense, and the reservoir cannot memorize many degrees of freedom (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 6(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Reducing delay time is challenging since the experimental demonstration of the photonic reservoirs has already used the short delay close to the instrumental limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Also, reducing wave speed in photonics systems is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' On the other hand, the wave speed of propagating spin-wave is much lower than the speed of light and can be tuned by configuration, thickness and material parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' If one reduces wave speed or delay time over the broad line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 7, information becomes sparse and cannot be used efficiently(See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 6(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Therefore, there is an optimal condition for high-performance RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The performance is comparable with other state of the art techniques, which are summa- rized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' For example, for the spintronic RC, MC ≈ 30 (19) and NRMSE ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 (22) in the NARMA10 task are obtained using Np ≈ 100 physical nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The spintronic RC with one physical node but with 101 − 102 virtual nodes do not show high performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' MC is less than 10 (the bottom left points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This fact suggests that the spintronic RC so far cannot use virtual nodes effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' On the other hand, for the photonic RC, comparable performances are achieved using Nv ≈ 50 virtual nodes, but only one physical node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' As we discussed, however, the photonic RC requires mm system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Our system achieves comparable performances using ≲ 10 physical nodes, and the size is down to nanoscales keeping the 2 − 50 GHz compu- tational speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We also demonstrate that the spin wave RC can perform time-series prediction and reconstruction of an attractor for the chaotic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' To our knowledge, this has not been done in nanoscale systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Our results of micromagnetic simulations suggest that our system can be physically im- 16 0 10 20 30 40 50 60 MC This work ( t 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='6 ns) This work ( t 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='16 ns) Spintronic RC ( 19 ) ( 22 ) Photonic RC ( 46 ) ( 47 ) (48 ) (12 ) ( 50 ) vt 0 (m) Length, L (m) Dense Sparse Figure 7: Reports of reservoir computing using multiple nodes are plotted as a function of the length between nodes and characteristic wave speed (v) times delay time (τ0) for photonics system (open symbols) and spintronics system (solid symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The size of sym- bols corresponds to memory capacity, which is taken from literature (12,19,22,32–35) and this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The gray scale represents memory capacity evaluated by using the response function method [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (5)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 17 VJKhJK-Jh-J1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='1 1 This work (Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=', Nv = 8, θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 ns) This work (Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=', Nv = 8, θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='02 ns) Spintronic RC (Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=') (22) Photonic RC (45) (Nv = 50) (48) (Nv = 50) (49) (Nv = 50) Normalized root mean square error, NRMSE for NARMA10 task Number of physical nodes, Np 1 10 100 10 100 This work (Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=', Nv = 8, θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 ns) This work (Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=', Nv = 8, θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='02 ns) Spintronic RC (Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=') (44) (19) (22) Spintronic RC (Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=') (9) (Nv = 250) (51) (Nv = 40) Photonic RC (46) (Nv = 50) (47) (Nv = 50) (48) (Nv = 50) Memory capacity, MC Number of physical nodes, Np (a) (b) Figure 8: Reservoir computing performance compared with different systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (a) Memory capacity, MC reported plotted as a function of physical nodes Np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (b) Normalized root mean square error, NRMSE for NARMA10 task is plotted as a function of Np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Open blue symbols are values reported using photonic RC while solid red symbols are values reported using spintronic RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' MC and NRMSE for NARMA10 task are taken from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (9,19,22,36,37) for spintronic RC and Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (32–34,38,39) for photonic RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' plemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' All the parameters in this study are feasible using realistic materials (40–43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Nanoscale propagating spin waves in a ferromagnetic thin film excited by spin-transfer torque using nanometer electrical contacts have been observed (44–46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Patterning of multiple elec- trical nanocontacts into magnetic thin films was demonstrated in mutually synchronized spin- torque oscillators (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In addition to the excitation of propagating spin-wave in a magnetic thin film, its non-local magnetization dynamics can be detected by tunnel magnetoresistance effect at each electrical contact, as schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 1(c), which are widely used for the development of spintronics memory and spin-torque oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In addition, virtual nodes are effectively used in our system by considering the speed of propagating spin-wave and distance of physical nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' thus, high-performance reservoir computing can be achieved with the small number of physical nodes, contrary to many physical nodes used in previous reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This work provides a way to realize nanoscale high-performance reservoir computing based on propagat- ing spin-wave in a ferromagnetic thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' There is an interesting connection between our study to the recently proposed next-generation 18 RC (28, 47), in which the linear ESN is identified with the NVAR (nonlinear vectorial autore- gression) method to estimate a dynamical equation from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Our formula of the response func- tion (3) results in the linear input-output relationship with a delay Yn+1 = anUn+an−1Un−1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' A in Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' More generally, with the nonlinear readout or with higher-order response functions, we have the input-output relationship with delay and non- linearity Yn+1 = anUn + an−1Un−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' + an,nUnYn + an,n−1UnUn−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' B in Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' These input-output relations are nothing but Volterra series of the output as a function of the input with delay and nonlinearity (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The coefficients of the ex- pansion are associated with the response function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Therefore, the performance of RC falls into the independent components of the matrix of the response function, which can be evaluated by how much delay the response functions between two nodes cover without overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The results would be helpful to a potential design of the network of the physical nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We should note that the polynomial basis of the input-output relation in this study originates from spin wave excitation around the stationary state mz = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' When the input data has a hier- archical structure, another basis may be more efficient than the polynomial expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Another setup of magnetic systems may lead to a different basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We believe that our study shows simple but clear intuition of the mechanism of high-performance RC, that can lead to the exploration of another setup for more practical application of the physical RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 19 Materials and Methods Micromagnetic simulations We analyze the LLG equation using the micromagnetic simulator mumax3 (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The LLG equation for the magnetization M(x, t) yields ∂tM(x, t) = − γµ0 1 + α2M × Heff − αγµ0 Ms(1 + α2)M × (M × Heff) + ℏPγ 4M2s eDJ(x, t)M × (M × mf) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (6) We consider the effective magnetic field as Heff = Hext + Hdemag + Hexch, (7) Hext = H0ez (8) Hms = − 1 4π � ∇∇ 1 |r − r′|dr′ (9) Hexch = 2Aex µ0Ms ∆M, (10) where Hext is the external magnetic field, Hms is the magnetostatic interaction, and Hexch is the exchange interaction with the exchange parameter Aex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The size of our system is L = 1000 nm and D = 4 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The number of mesh points is 200 in the x and y directions, and 1 in the z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We consider Co2MnSi Heusler alloy ferromagnet, which has a low Gilbert damping and high spin polarization with the parameter Aex = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 pJ/m, Ms = 1000 kA/m, and α = 5 × 10−4 (40,41,41–43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Out-of-plane magnetic field µ0H0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 T is applied so that magnetization is pointing out-of-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The spin-polarized current field is included by the Slonczewski model (29) with polarization parameter P = 1 and spin torque asymmetry parameter λ = 1 with the reduced Planck constant ℏ and the charge of an electron e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The uniform fixed layer magnetization is mf = ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We use absorbing boundary layers for spin waves to ensure the magnetization vanishes at the boundary of the system (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We set the initial magnetization as m = ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 20 The reference time scale in this system is τ0 = 1/γµ0Ms ≈ 5 ps, where γ is the gyromag- netic ratio, µ0 is permeability, and Ms is saturation magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The reference length scale is the exchange length l0 ≈ 5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The relevant parameters are Gilbert damping α, the time scale of the input time series θ, and the characteristic length between the input nodes R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The injectors and detectors of spin are placed as cylindrical nanocontacts embedded in the region with their radius a and height D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We set a = 20nm unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The input time series is uniform random noise Un ∈ U(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The injected density current is set as j(tn) = 2jcUn with jc = 2 × 10−4/(πa2)A/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Under a given input time series of the length T, we apply the current during the time θ, and then update the current at the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The same input current with different filters is injected for different virtual nodes (see Learning with reservoir computing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The total simulation time is, therefore, TθNv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Learning with reservoir computing Our RC architecture consists of reservoir state variables X(t + ∆t) = f (X(t), U(t)) (11) and the readout Yn = W · ˜˜X(tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (12) In our spin wave RC, the reservoir state is chosen as x-component of the magnetization X = � mx,1(tn), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , mx,i(tn), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , mx,Np(tn) �T , (13) for the indices for the physical nodes i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , Np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Here, Np is the number of physical nodes, and each mx,i(tn) is a T-dimensional row vector with n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We use a time- multiplex network of virtual nodes in RC (23), and use Nv virtual nodes with time interval θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 21 The expanded reservoir state is expressed by NpNv × T matrix ˜X as (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2(b)) ˜X = (mx,1(tn,1), mx,1(tn,2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , mx,1(tn,k), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , mx,1(tn,Nv), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , mx,i(tn,1), mx,i(tn,2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , mx,i(tn,k), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , mx,i(tn,Nv), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , mx,Np(tn,1), mx,Np(tn,2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , mx,Np(tn,k), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , mx,Np(tn,Nv) �T , (14) where tn,k = ((n − 1)Nv − (k − 1))θ for the indices of the virtual nodes k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , Nv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The total number of rows is N = NpNv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We use the nonlinear readout by augmenting the reservoir state as ˜˜X = � ˜X ˜X ◦ ˜X � , (15) where ˜X(t) ◦ ˜X(t) is the Hadamard product of ˜X(t), that is, component-wise product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The readout weight W is trained by the data of the output Y (t) W = Y · ˜˜X† (16) where X† is pseudo-inverse of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In the time-multiplexing approach, the input time-series U = (U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , UT) ∈ RT is translated into piece-wise constant time-series ˜U(t) = Un with t = (n − 1)Nvθ + s under k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , T and s = [0, Nvθ) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This means that the same input remains during the time period τ0 = Nvθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' To use the advantage of physical and virtual nodes, the actual input Ji(t) at the ith physical node is ˜U(t) multiplied by τ0-periodic random binary filter Bi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Here, Bi(t) ∈ {0, 1} is piece-wise constant during the time θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' At each physical node, we use different realizations of the binary filter as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Unless otherwise stated, We use 1000 steps of the input time-series as burn-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' After these steps, we use 5000 steps for training and 5000 steps for test for the MC, IPC, and NARMA10 tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 22 NARMA task The NARMA10 task is based on the discrete differential equation, Yn+1 = αYn + βYn 9 � p=0 Yn−p + γUnUn−9 + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (17) Here, Un is an input taken from the uniform random distribution U(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5), and yk is an output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We choose the parameter as α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='3, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='05, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5, and δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In RC, the input is U = (U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , UT) and the output Y = (Y1, Y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , YT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The goal of the NARMA10 task is to estimate the output time-series Y from the given input U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The training of RC is done by tuning the weights W so that the estimated output ˆY (tn) is close to the true output Yn in terms of squared norm | ˆY (tn) − Yn|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The performance of the NARMA10 task is measured by the deviation of the estimated time series ˆY = W · ˜˜X from the true output Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The normalized root-mean-square error (NRMSE) is NRMSE ≡ �� n( ˆY (tn) − Yn)2 � n Y 2 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (18) Performance of the task is high when NRMSE ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In the ESN, it was reported that NRMSE ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 for N = 50 and NRMSE ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 for N = 200 (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The number of node N = 200 was used for the speech recognition with ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='02 word error rate (51), and time-series prediction of sptio- temporal chaos (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Therefore, NRMSE ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 is considered as reasonably high performance in practical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We also stress that we use the same order of nodes (virtual and physical nodes) N = 128 to achieve NRMSE ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Memory capacity and information processing capacity Memory capacity (MC) is a measure of the short-term memory of RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This was introduced in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' For the input Un of random time series taken from the uniform distribution, the network 23 is trained for the output Yn = Un−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The MC is computed from MCk = ⟨Un−k, W · X(tn)⟩2 ⟨U2 n⟩⟨(W · X(tn))2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (19) This quantity is decaying as the delay k increases, and MC is defined as MC = kmax � k=1 MCk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (20) Here, kmax is a maximum delay, and in this study we set it as kmax = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The advantage of MC is that when the input is independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' ), and the output function is linear, then MC is bounded by N, the number of internal nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Information processing capacity (IPC) is a nonlinear version of MC (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In this task, the output is set as Yn = � k Pdk(Un−k) (21) where dk is non-negative integer, and Pdk(x) is the Legendre polynomials of x order dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We may define IPCd0,d1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=',dT −1 = ⟨Yn, W · X(tn)⟩2 ⟨Y 2 n ⟩⟨(W · X(tn))2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (22) and then compute jth order IPC as We may define IPCj = � dks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='j=� k dk IPCd1,d2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=',dT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (23) When j = 1, the IPC is, in fact, equivalent to MC, because P0(x) = 1 and P1(x) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In this case, Yn = Un−k for di = 1 when i = k and di = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (23) takes the sum over all possible delay k, which is nothing but MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' When j > 1, IPC captures all the nonlinear transformation and delays up to the jth polynomial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' For example, when j = 2, the output can be Yn = Un−k1Un−k2 or Yn = U2 n−k + const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In this study, we focus on j = 2 because 24 the second-order nonlinearity is essential for the NARMA10 task (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' A in Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The relevance of MC and IPC is clear by considering the Volterra series of the input-output relation, Yn = � k1,k2,··· ,kt βk1,k2,··· ,knUk1 1 Uk2 2 · · · Ukn n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (24) Instead of polynomial basis, we may use orthonormal basis such as the Legendre polynomials Yn = � k1,k2,··· ,kn βk1,k2,··· ,knPk1(U1)Pk2(U2) · · ·Pkn(Un).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (25) Each term in (25) is characterized by the non-negative indices (k1, k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , kn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Therefore, the terms corresponding to j = � i ki = 1 in Yn have information on linear terms with time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Similarly, the terms corresponding to j = � i ki = 2 have information of second-order nonlinearity with time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In this view, the estimation of the output Y (t) is nothing but the estimation of the coefficients βk1,k2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=',kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In RC, the readout of the reservoir state at ith node (either physical or virtual node) can also be expanded as the Volterra series ˜˜X(i)(tn) = � k1,k2,··· ,kn ˜˜β(i) k1,k2,··· ,knUk1 1 Uk2 2 · · · Ukn n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (26) Therefore, MC and IPC are essentially a reconstruction of βk1,k2,··· ,kn from ˜˜β(i) k1,k2,··· ,kn with i ∈ [1, N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This can be done by regarding βk1,k2,··· ,kn as a T + T(T − 1)/2 + · · · -dimensional vector, and using the matrix M associated with the readout weights as βk1,k2,··· ,kn = M · \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed ˜˜β(1) k1,k2,··· ,kn ˜˜β(2) k1,k2,··· ,kn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' ˜˜β(N) k1,k2,··· ,kn \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (27) MC corresponds to the reconstruction of βk1,k2,··· ,kn for � i ki = 1, whereas the second-order IPC is the reconstruction of βk1,k2,··· ,kn for � i ki = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' If all of the reservoir states are indepen- 25 dent, we may reconstruct N components in βk1,k2,··· ,kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In realistic cases, the reservoir states are not independent, and therefore, we can estimate only < N components in βk1,k2,··· ,kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Prediction of chaotic time-series data Following (5), we perform the prediction of time-series data from the Lorenz model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The model is a three-variable system of (A1(t), A2(t), A3(t)) yielding the following equation dA1 dt = 10(A2 − A1) (28) dA2 dt = A1(28 − A3) − A2 (29) dA3 dt = A1A2 − 8 3A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (30) The parameters are chosen such that the model exhibits chaotic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Similar to the other tasks, we apply the different masks of binary noise for different physical nodes, B(l) i (t) ∈ {−1, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Because the input time series is three-dimensional, we use three independent masks for A1, A2, and A3, therefore, l ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The input for the ith physical node after the mask is given as Bi(t) ˜Ui(t) = B(1) i (t)A1(t)+B(2) i (t)A2(t)+B(3) i (t)A3(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Then, the input is normalized so that its range becomes [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5], and applied as an input current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Once the input is prepared, we may compute magnetization dynamics for each physical and virtual node, as in the case of the NARMA10 task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We note that here we use the binary mask of {−1, 1} instead of {0, 1} used for other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We found that the {0, 1} does not work for the prediction of the Lorenz model, possibly because of the symmetry of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The ground-truth data of the Lorenz time-series is prepared using the Runge-Kutta method with the time step ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The time series is t ∈ [−60, 75], and t ∈ [−60, −50] is used for relaxation, t ∈ (−50, 0] for training, and t ∈ (0, 75] for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' During the training steps, we compute the output weight by taking the output as Y = (A1(t + ∆t), A2(t + ∆t), A3(t + ∆t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' After training, the RC learns the mapping (A1(t), A2(t), A3(t)) → (A1(t + ∆t), A2(t + 26 ∆t), A3(t + ∆t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' For the prediction steps, we no longer use the ground-truth input but the estimated data ( ˆ A1(t), ˆ A2(t), ˆ A3(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Using the fixed output weights computed in the training steps, the time evolution of the estimated time-series ( ˆ A1(t), ˆ A2(t), ˆ A3(t)) is computed by the RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Theoretical analysis using response function We consider the Landau-Lifshitz-Gilbert equation for the magnetization field m(x, t), ∂tm(x, t) = −m × heff − m × (m × heff) + σ(x, t)m × (m × mf) (31) We normalize both the magnetic and effective fields by saturation magnetization as m = M/Ms and heff = Heff/Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This normalization applies to all the fields including external and anisotropic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We also normalize the current density as σ(x, t) = J(x, t)/j0 for the current density J(x) and the unit of current density j0 = 4M2 s eπa2Dµ0 ℏP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We apply the current density at the nanocontact as J(x, t) = 2jc ˜U(t) Np � i=1 χa(|x − Ri|) (32) Here χa(x) is a characteristic function χa(x) = 1 when x ≤ a and χa(x) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We expand the solution of (31) around the uniform magnetization m(x, t) = (0, 0, 1) with- out current injection as m(x, t) = m0(x, t) + ǫm(1)(x, t) + O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (33) Here, m0(x, t) = (0, 0, 1) and ǫ ≪ 1 is a small parameter corresponding to the magnitude of the input σ(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The first-order term corresponds to a linear response of the magnetization to the input σ, whereas the higher-order terms describe nonlinear responses, for example, m(2)(x, t) ∼ σ(x1, t1)σ(x2, t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Because our input is driven by the spin torque with fixed layer magnetization in the x-direction, mf = ex, only mx and my appear in the first-order term O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Deviation of 27 mz from mz = 1 appears in O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Therefore, for the first-order term m(1), we may define the complex magnetization m = mx + imy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (34) Here, we will show the magnetization is expressed by the response function Gij(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The input at the jth physical node affects the magnetization at the ith physical node as mi(t) = � dτGii(t − τ)σi(τ) + � i̸=j � dτGij(t − τ)σj(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (35) The input for the jth physical node is expressed by σj(t) = 2jcBj(t) ˜Uj(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Because different physical nodes have different masks discussed in Learning with reservoir computing in Meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' When the wave propagation is dominated by the exchange interaction, the response func- tion for the same node is Gii(t − τ) = 1 2πe−˜h(α+i)(t−τ) � 1 − e− a2 4(α+i)(t−τ) � (36) and for different nodes, it becomes Gij(t − τ) = a2 2πe−˜h(α+i)(t−τ)e− |Ri−Rj|2 4(α+i)(t−τ) 1 2(α + i)(t − τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (37) When the wave propagation is dominated by the dipole interaction, the response function for the same node is Gii(t − τ) = 1 2πe−˜h(α+i)(t−τ) −1 + � 1 + a2 (d/4)2(α+i)2(t−τ)2 � 1 + a2 (d/4)2(α+i)2(t−τ)2 (38) and for different nodes it becomes Gij(t − τ) = a2 2πe−˜h(α+i)(t−τ) × 1 (d/4)2(α + i)2(t − τ)2 � 1 + |Ri−Rj|2 (d/4)2(α+i)2(t−τ)2 �3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (39) 28 Clearly, Gii(0) → 1 and Gij(0) → 0, while Gii(∞) → 0 and Gij(∞) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Once the magnetization is expressed in the form of (35), we may compute the reservoir state X under the input U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Then, we may use the same method as in Learning with reservoir computing, and estimate the output ˆY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Similar to the micromagnetic simulations, we evaluate the perfor- mance by MC, IPC, and NARMA10 tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We may extend the analyzes for the higher-order terms in the expansion of (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='B in Supplementary Materials, we show the second-order term m(2)(x, t) has only the z-component, and moreover, it is dependent only on the first-order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' As a result, the second-order term is expressed as m(2) z (x, t) = −1 2 � (m(1) x )2 + (m(1) y )2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (40) To compute the response functions, we linearize (31) for the complex magnetization m(x, t) as ∂tm(x, t) = Lm + σ(x, t), (41) where the linear operator is expressed as L = � −˜h + ∆ � (α + i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (42) In the Fourier space, the linearized equation becomes ∂tmk(t) = Lkmk + σk(t), (43) with Lk = − � ˜h + k2� (α + i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (44) The solution of ((43)) is obtained as mk(t) = � dτeLk(t−τ)σk(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (45) 29 We have Np cylindrical shape inputs with radius a and the ith input is located at Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The input function is expressed as σ(x) = Np � i=1 χa (|x − Ri|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (46) We are interested in the magnetization at the input mi(t) = m(x = Ri, t), which is mi = 1 (2π)2 � j � dτe−˜h(α+i)(t−τ) � dke−k2(α+i)(t−τ)eik·(Ri−Rj)2πaJ1(ka)σj(t) = a 2π � j � dτe−˜h(α+i)(t−τ) � dke−k2(α+i)(t−τ)J0 (k|Ri − Rj|) J1(ka)σj(t) (47) For the same node, |Ri − Rj| = 0, and we may compute the integral explicitly as (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' When ka ≪ 1, we may assume J1(ka) ≈ ka/2, and finally, come up with (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' When the thickness d of the material is thin, the dispersion relation becomes Lk = −˜h(α + i) �� 1 + k2 ˜h � � 1 + k2 ˜h + βk ˜h � (48) where β = d 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (49) We assume for k ≪ β � ˜h, then the linearized operator becomes Lk = −(α + i) � ˜h + kd 4 � (50) leading to (38) and (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Acknowledgements: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' thanks to CSRN at Tohoku University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Numerical simulations in this work were carried out in part by AI Bridging Cloud Infrastructure (ABCI) at National Institute of Advanced In- dustrial Science and Technology (AIST), and by the supercomputer system at the information 30 initiative center, Hokkaido University, Sapporo, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Funding: This work is support by JSPS KAKENHI grant numbers 21H04648, 21H05000 to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=', by JST, PRESTO Grant Number JPMJPR22B2 to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=', X-NICS, MEXT Grant Number JPJ011438 to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=', and by JST FOREST Program Grant Number JPMJFR2140 to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Author Contributions S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' conceived the research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' carried out simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' an- alyzed the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' wrote the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' All the authors discussed the results and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Competing Interests The authors declare that they have no competing financial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Data and materials availability: All data are available in the main text or the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' A Connection between the NARMA10 task and MC/IPC In this section, we discuss the necessary properties of reservoir computing to achieve high per- formance of the NARMA10 task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In short, the NARMA10 task is dominated by the memory of 31 nine step previous data and second-order nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We discuss these properties in two meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The first method is based on the extended Dynamic Mode Decomposition (DMD) (52) and the higher-order DMD (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The second method is a regression of the input-output relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We will discuss the details of the two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Our results are consistent with previous studies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' the requirement of memory was discussed in (54), and the second-order nonlinear terms with a time delay in (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The NARMA10 task is based on the discrete differential equation, Yn+1 = αYn + βYn 9 � i=0 Yn−i + γUnUn−9 + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (51) Here, Un is an input at the time step n taken from the uniform random distribution U(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5), and Yn is an output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We choose the parameter as α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='3, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='05, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5, and δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In the first method, we estimate the transition matrix A from the state variable Yn = (Y1, Y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , Yn) to Yn+1 = (Y2, Y3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , Yn+1) yielding Yn+1 = A · Yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (52) We may extend the notion of the state variable to contain delayed data and polynomials of the output with time delay as Yn = (Yn, Yn−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , Y1, YnYn, YnYn−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , Y1Y1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (53) Including the delay terms following from the higher-order DMD (53), while the polynomial nonlinear terms are used as a polynomial dictionary in the extended DMD (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Here, (53) contains all the combination of the second-order terms with time delay, Yn−i1Yn−i2 with the integers i1 and i2 in 0 ≤ i1 ≤ l2 ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We may straightforwardly include higher-order terms in powers in (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In the NARMA10 task, the output Yn+1 is also affected by the input Un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Therefore, the extended DMD is generalized to include the control as (56) Yn+1 = (A B) · �Yn Un � , (54) 32 where the state variable corresponding to the input includes time delay and nonlinearity, and is described as Un = (Un, Un−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , U1, UnUn, UnUn−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , U1U1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (55) We denote the generalized transition matrix as Ξ = (A B) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (56) The idea of DMD is to estimate the transition matrix from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This is done by taking pseudo inverse of the state variables as ˆΞ = Yk+1 · � Yk Uk �† .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (57) Here, M† is the pseudoinverse of the matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This is nothing but a least-square estimation for the cost function of l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='s minus r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='s of (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We may include the Tikhonov regularization term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Note that for the extended DMD (52) and the higher-order DMD (53), the transition matrix Ξ is further decomposed into characteristic modes associated with its eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The decom- position gives us a dimensional reduction of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The estimation of the transition matrix is also called nonlinear system identification, particularly, nonlinear autoregression with exoge- nous inputs (NARX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In this work, we focus on the estimation of the input-output relationship, and do not discuss the dimensional reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' For time-series prediction, we estimate the func- tion Yn+1 = f(Yn, Yn−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , Y1), and we do not need the input Un in (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Even in this case, we may consider a similar estimation of Ξ (in fact, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This estimation is the method used in the next-generation RC (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The second method is based on the Volterra series of the state variable Yn by the input Un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In this method, we assume that the state variable is independent of its initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Then, 33 we may express the state variable as Yn = G · Un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (58) Note that Un includes the input and its polynomials with a time delay as in (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Similar to the first method, we estimate G by ˆG = Yt · U† t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (59) The estimated ˆG gives us information on which time delay and nonlinearity dominate the state variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 0 5 10 15 20 25 30 delay NRMSE 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 test training linear (A) (B) (C) (D) (E) (F) 0 5 10 15 20 25 30 delay NRMSE 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 second-order nonlinearity 0 5 10 15 20 25 30 delay NRMSE 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 third-order nonlinearity 0 5 10 15 20 25 30 delay NRMSE 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 linear 0 5 10 15 20 25 30 delay NRMSE 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 second-order nonlinearity 0 5 10 15 20 25 30 delay NRMSE 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 third-order nonlinearity Figure 9: (A-C) the estimation based on the extended DMD, (D-F) the estimation based on the Volterra series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The dictionary of each case is (A,D) first-order (linear) delay terms, (B,E) up to second-order delay terms, and (C,F) up to third-order delay terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The results of the two estimation methods are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Both approaches suggest that memory of ≈ 10 steps is enough to get high performance, and further memory does not improve the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The second-order nonlinear term shows a reasonably small NRMSE of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Including the third-order nonlinearity improves the error, but there is a sign of overfitting 34 at a longer delay because the number of the state variables is too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' It should also be noted that even with the linear terms, the NRMSE becomes ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This result implies that although NRMSE ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='35 is often considered good performance, nonlinearity of the data is not learned at the error of this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='1 The MC and IPC tasks as Volterra series for linear and nonlinear readout In (3) and (4) in the main text, we show that the magnetization at the input region is expressed by the response function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The magnetization at the time tn corresponding to the input Un at the n step is expressed as m(tn) = anUn + an−1Un−1 + · · · , (60) where the coefficients an can be computed from the response function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We first consider the linear case, but we will generalize the expression for the nonlinear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Because we use virtual nodes, the input Un at the step n continues during the time period t ∈ [tn, tn+1) discretized by Nv steps as (tn,1, tn,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' , tn,Nv), and is multiplied by the filter of the binary noise (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 and Methods in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Therefore, the magnetization is expressed by the response functions G(t − t′) is formally expressed as m(tn) = Np � i [(G(0) + G(θ) + · · · G(θ(Nv − 1))) σi(tn) + (G(θNv) + G(θ(Nv + 1)) + · · · G(θ(2Nv − 1))) σi(tn−1) + · · ·] , (61) where σi(tn) ∝ Un is the non-dimensionalized current injection at the time tn at the ith physical node, which is proportional to Un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Therefore, (61) results in the expression of (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Our input is taken from a uniform random distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Therefore, the inner product of the reservoir state, 35 which is nothing but magnetization, and (delayed) input to learn MC is ⟨m(tn), Un⟩ = T � n=1 m(tn)Un = an⟨U2 n⟩ + O(1/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (62) Similarly, the variance of the magnetization is equal to the variance of the input with the coef- ficient associated with m(tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We may express the MC and IPC tasks in a matrix form as ˜S ≈ W · G · (S ◦ Win) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (63) Here, S is the matrix associated with the original input, and ˜S is the delayed one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The output weight is denoted by W, and Win is the matrix associated with the mask of binary noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The goal of MC and IPC tasks is to approximate the delayed input ˜S by the reservoir states G · S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Here, the reservoir states are expressed by the response function G and input denoted by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We define delayed input ˜S ∈ RK×T ˜S = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed Un Un+1 Un+2 · · Un−1 Un Un+1 · · Un−2 Un−1 Un · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (64) Here, T is the number of the time series, and K is the total length of the delay that we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The ith row shows the i−1 delayed time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The input S ∈ RTNv×T to compute the reservoir states are expressed as S = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed σ(tn) σ(tn+1) σ(tn+2) · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' σ(tn) σ(tn+1) σ(tn+2) · · σ(tn−1) σ(tn) σ(tn+1) · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' σ(tn−2) σ(tn−1) σ(tn) · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (65) Note that σ(tn) ∝ Un upto constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Due to time multiplexing, each row is repeated Nv times, and then the time series is delayed in the next row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' After multiplying the input filter Win, the 36 input is fed into the response function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The input filter Win ∈ RTNv×T is a stack of constant row vectors with the length T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The Nv different realizations of row vectors are taken from binary noise, and then the resulting Nv × T matrix is repeated T times in the row direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This input is multiplied by the coefficients of the Volterra series G ∈ RN×TNv G = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed G(1)(0) · · G(1)(θ(Nv − 1)) G(1)(θNv) · · G(1)(θ(2Nv − 1)) · · G(2)(0) · · G(2)(θ(Nv − 1)) G(2)(θNv) · · G(2)(θ(2Nv − 1)) · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' G(N)(0) · · G(N)(θ(Nv − 1)) G(N)(θNv) · · G(N)(θ(2Nv − 1)) · · \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (66) (63) implies that by choosing the appropriate W, we can get a canonical form of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' If the canonical form has N × N identity matrix in the left part of W · G, then the reservoir reproduces the time series up to N − 1 delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This means that the rank of the matrix G, or the number of independent rows, is the maximum number of steps of the delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This is consistent with the known fact that MC is bounded by the number of independent components of reservoir variables (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Next we extend the Volterra series of the magnetization, including nonlinear terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The magnetization is expressed as m(tn) = anσ(tn) + an−1σ(tn−1) + · · · + an,nσ(tn)σ(tn) + an,n−1σ(tn)σ(tn−1) + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (67) The delayed input ˜S is rewritten as ˜S = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed Un Un+1 Un+2 · · Un−1 Un Un+1 · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' UnUn Un+1Un+1 Un+2Un+2 · · UnUn−1 Un+1Un Un+2Un+1 · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (68) The matrix ˜S contains all the nonlinear combinations of the input series (Un, Un+1, · · ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Ac- cordingly, we should modify S and also G to include the nonlinear response functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Note that to guarantee the orthogonality, Legendre polynomials (or other orthogonal polynomials) 37 should be used instead of polynomials in powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Nevertheless, up to the second order of nonlinearity, which is relevant to consider the performance of NARMA10 (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' A), the dif- ference is only in the constant terms (P2(x) = x2 − 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Because we subtract the mean value of the time series of all the input, output, and reservoir states, these constant terms do not change our conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' With nonlinear terms, (66) is extended as G = (Glin, Gnonl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Still, the rank of the matrix remains N at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This is the reason why the total sum of IPC, including all the lin- ear and nonlinear delays, is bounded by the number of independent reservoir variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' When Gnonl = 0, the reservoir can memorize only the linear delay terms, but MC can be maximized to be N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' On the other hand, when Gnonl ̸= 0, it is possible that MC is less than N, but the reservoir may have finite IPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' When the readout is nonlinear, we use the reservoir state variable as X = � M M ◦ M � , (69) where ◦ is the Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' If M is linear in the input, G has a structure of G = � Glin 0 0 Gnonlin � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (70) In this case, rank(G) = rank(Glin) + rank(Gnonlin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' B Learning with multiple variables In the main text, we use only mx for the readout as in (13)-(15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The readout is nonlinear and has both the information of mx and m2 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In this section, we consider the linear readout, but use both mx and mz for the output in micromagnetic simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We begin with the linear readout only with mx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The results of the MC and IPC tasks are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 10(a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We obtain a similar performance for the MC task with the result in the main text (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' On the other hand, the performance for the IPC task in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 10(a) is significantly poorer than the result 38 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This result demonstrates that the linear readout only with mx does not learn the nonlinearity effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Note that in the theoretical model with the response function, the IPC is exactly zero when we use the linear readout only with mx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The discrepancy arises from the expansion (33) around m0 = (0, 0, 1) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Strictly speaking, the expansion should be made around m0 under the constant input ⟨σ⟩ averaged over time at the input nanocontact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' This reference state is inhomogeneous in space, and is hard to compute analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Due to this effect, mx in the micromagnetic simulations contain small nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Next, we consider the linear readout with mx and mz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 10(c,d), mz carries nonlinear information, and enhances the IPC and learning performance of NARMA10 com- pared with linear readout only with mx (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 10 (a,b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The performance is IPC ≈ 60 under α = 5 × 10−4, which is comparable value with the results in the main text (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 3(a,c)) where the readout is (mx, m2 x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Also, high performance for NARMA10 task, NRMSE ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2, can be obtained using variables (mx, mz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' These results show that adding mz into the readout has a similar effect to adding m2 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Similarity between m2 x and mz can be understood by using the theoretical formula with the response function in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' We continue the expansion (33) at the second order, and obtain ∂tm(2)(x, t) = − m(1) × ∆m(1) − αm(1) × �� ˜hm(1) − ∆m(1)� × ez � + σ(x, t)m(1) × ey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (71) This result suggests that m(2) contains only the z component, and is slaved by m(1), which does not have z component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Therefore, m(2) z can be computed as m(2) z (x, t) = −1 2 � (m(1) x )2 + (m(1) y )2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (72) Because mx and my carry similar information, mz in the readout has a similar effect with m2 x in the readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 39 C Speed of propagating spin wave using dipole interaction Propagating spin wave when magnetization is pointing along film normal is called magneto- static forward volume mode, and its dispersion relation can be described by the following equa- tion (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' ω(k) = γµ0 � (H0 − Ms) � H0 − Ms 1 − e−kd kd � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (73) Then, one can obtain the group velocity at k ∼ 0 as, vg = dω dk (k = 0) = γµ0Msd 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (74) In the magneto-static spin wave driven by dipole interaction, group velocity is proportional to both Ms and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' vg ∼ 200 m/s is obtained when the following parameters are used: µ0H = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 T, Ms = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 × 106 A/m, d = 4 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The same estimation is used for calculating the speed of information propagation for spin reservoirs in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (19) and (22), which are used to plot Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 7 in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' D Details of reservoir computing scaling compared with lit- erature In this section, details of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 7 shown in the main text are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' MC and NRMSE for NARMA10 tasks using photonic and spintronic RC are reported in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (12,32–35,38,39) for photonic RC and (9,19,22,25,36,37,57,58) for spintronic RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Table 1 and 2 shows reports of MC for photonic and spintronic RC with different length scales, which are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 7 in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 40 Table 1: Report of photonic RC with different length scales used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 7 in the main text Reports Length, L Time interval, τ0 vτ0 N MC Duport et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (32) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='6 km 8 µs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 km 50 21 Dejonckheere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (33) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='6 km 8 µs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 km 50 37 Vincker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (34) 230 m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='1 µs 340 m 50 21 Takano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (12) 11 mm 200 ps 60 mm 31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 Sugano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (35) 10 mm 240 ps 72 mm 240 10 Note: speed of light, v = 3 × 108 m/s is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Table 2: Report of spin reservoirs with different length scales used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 7 in the main text Reports L τ0 v vτ0 N MC Nakane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (19) 5 µm 2 ns 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 km/s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='8 µm 72 21 Dale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (22) 50 nm 10 ps 200 m/s 2 nm 100 35 This work 500 nm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='6 ns 200 m/s 320 nm 64 26 Note: v is calculated based on magneto-static spin wave using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' E Other data E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='1 Nv and Np dependence of performance Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 11 shows Nv and Np dependencies of MC, IPC and NRMSE for NARMA10 task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' As Nv and Np are increased, MC and IPC increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Then, NARMA10 prediction task becomes better with increasing Nv and Np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' MC and NRMSE for NARMA10 with different Np with fixed Nv = 8 are compared with other reservoirs shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 8 in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 exchange interaction In the main text, we use the dipole interaction to compute the response function as (38) and (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' In this section, we show the result using the exchange interaction shown in (36) and (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Figure 12 shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 41 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} 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1/θ (GHz) 0 20 40 60 80 α = 5×10-3 Linear and non-linear memory capacity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 0 20 40 60 80 α = 5×10-2 Distance of virtual nodes, θ (ns) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 1 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 α = 5×10-4 Training, Test Frequency, 1/θ (GHz) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 1 α = 5×10-3 Normalized root mean square error, NRMSE for NARMA10 task 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 1 α = 5×10-2 Distance of virtual nodes, θ (ns) (c) (m Q, mz) MC and IPC (d) (m R, mz), NARMA10 MC IPC 0 20 40 60 80 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 α = 5×10-4 Frequency, 1/θ (GHz) 0 20 40 60 80 α = 5×10-3 Linear and non-linear memory capacity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 0 20 40 60 80 α = 5×10-2 Distance of virtual nodes, θ (ns) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 1 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 α = 5×10-4 Training, Test Frequency, 1/θ (GHz) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 1 α = 5×10-3 Normalized root mean square error, NRMSE for NARMA10 task 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 1 α = 5×10-2 Distance of virtual nodes, θ (ns) Figure 10: Reservoir computing with various parameter combinations obtained using micro- magnetic Mumax3 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Linear memory capacity, MC and nonlinear memory capacity, IPC plotted as a function of θ obtained using linear mx output only (a) and using mx, mz (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' Normalized root mean square error, NRMSE for NARMA10 task plotted as a function of θ obtained using linear mx output only (b) and using mx, mz (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 49 2 4 6 8 2 4 6 8 Number of physical nodes, Np Number of virtual nodes, Nv 0 5 10 15 20 25 30 Memory capacity, MC 2 4 6 8 2 4 6 8 Number of physical nodes, Np Number of virtual nodes, Nv 0 10 20 30 40 50 60 S T Nonlinear memory capacity, IPC 2 4 6 8 2 4 6 8 Number of physical nodes, Np Number of virtual nodes, Nv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='00 Normalized mean square error, NRMSE for NARMA10 task (a) U VW (c) Figure 11: (a) Memory capacity, MC (b) Nonlinear memory capacity, IPC and (c) Normalized root mean square error, NRMSE for NARMA10 task plotted as a function of the number of virtual and physical nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' The parameters used in the simulation are α = 5 × 10−4, θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (a) (b) (c) wave speed (log m/s) characteristic size (log nm) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 MC 20 30 40 50 damping time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 wave speed (log m/s) characteristic size (log nm) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 IPC 20 30 40 50 damping time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='0 0 20 40 60 80 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='5 � = 5×10-4 Frequency, 1/� (GHz) 0 20 40 60 80 � = 5×10-3 Linear and non-linear memory capacity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content='4 0 20 40 60 80 � = 5×10-2 Distance of virtual nodes, � (ns) Figure 12: (a) Memory capacity, MC (solid symbols) and nonlinear memory capacity, IPC (open symbols) obtained using the response function method for exchange interaction plotted as a function of θ with different damping parameters α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' (b) MC and (c) IPC plotted as a function of characteristic size and wave speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} +page_content=' 50' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfQgBB/content/2301.02193v1.pdf'} diff --git a/DtE1T4oBgHgl3EQfWQTV/vector_store/index.faiss b/DtE1T4oBgHgl3EQfWQTV/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..5f56fb8611be7a9117e3481b3a6fb7cb8fd590ef --- /dev/null +++ b/DtE1T4oBgHgl3EQfWQTV/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:368b98a89d5fa2006a6172b0c52dad1c573023570afe04cf497f268ac010fa1a +size 3342381 diff --git a/FdAyT4oBgHgl3EQfrPl7/content/tmp_files/2301.00557v1.pdf.txt b/FdAyT4oBgHgl3EQfrPl7/content/tmp_files/2301.00557v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..77406309c76c35df06f21bce1bdcf009c1dbcd44 --- /dev/null +++ b/FdAyT4oBgHgl3EQfrPl7/content/tmp_files/2301.00557v1.pdf.txt @@ -0,0 +1,2010 @@ +Learning to Maximize Mutual Information for Dynamic Feature Selection +Ian Covert 1 Wei Qiu 1 Mingyu Lu 1 Nayoon Kim 1 Nathan White 2 Su-In Lee 1 +Abstract +Feature selection helps reduce data acquisition +costs in ML, but the standard approach is to train +models with static feature subsets. Here, we con- +sider the dynamic feature selection (DFS) prob- +lem where a model sequentially queries features +based on the presently available information. DFS +is often addressed with reinforcement learning +(RL), but we explore a simpler approach of greed- +ily selecting features based on their conditional +mutual information. This method is theoretically +appealing but requires oracle access to the data +distribution, so we develop a learning approach +based on amortized optimization. The proposed +method is shown to recover the greedy policy +when trained to optimality and outperforms nu- +merous existing feature selection methods in our +experiments, thus validating it as a simple but +powerful approach for this problem. +1. Introduction +A machine learning model’s inputs can be costly to obtain, +and feature selection is often used to reduce data acquisition +costs. In applications where information is gathered sequen- +tially, a natural option is to select features adaptively based +on the currently available information rather than using a +fixed feature set. This setup is known as dynamic feature +selection (DFS)1, and the problem has been considered by +several works in the last decade (Saar-Tsechansky et al., +2009; Dulac-Arnold et al., 2011; Chen et al., 2015b; Early +et al., 2016a; He et al., 2016a; Kachuee et al., 2018). +Compared to static feature selection with a fixed feature +set (Li et al., 2017; Cai et al., 2018), DFS can offer better +performance given a fixed budget. This is easy to see, be- +cause selecting the same features for all instances (e.g., all +1Paul G. Allen School of Computer Science & Engineering, +University of Washington 2Department of Emergency Medicine, +University of Washington. +Correspondence to: +Ian Covert +. +1The problem is also sometimes referred to as sequential fea- +ture selection or active feature acquisition. +patients visiting a hospital’s emergency room) is suboptimal +when the most informative features vary across individuals. +Although it should in theory offer better performance, DFS +also presents a more challenging learning problem, because +it requires learning both a feature selection policy and how +to make predictions given variable feature sets. +Prior work has approached DFS in several ways, though of- +ten using reinforcement learning (RL) (Dulac-Arnold et al., +2011; Shim et al., 2018; Kachuee et al., 2018; Janisch et al., +2019; Li & Oliva, 2021). RL is a natural approach for se- +quential decision-making problems, but current methods are +difficult to train and do not reliably outperform static fea- +ture selection methods (Henderson et al., 2018; Erion et al., +2021). Our work therefore explores a simpler approach: +greedily selecting features based on their conditional mutual +information (CMI) with the response variable. +The greedy approach is known from prior work (Fleuret, +2004; Chen et al., 2015b; Ma et al., 2019) but is difficult +to use in practice, because calculating CMI requires oracle +access to the data distribution (Cover & Thomas, 2012). +Our focus is therefore developing a practical approximation. +Whereas previous work makes strong assumptions about the +data (e.g., binary features in Fleuret 2004) or approximates +the data distribution with generative modeling (Ma et al., +2019), we develop a flexible approach that directly predicts +the optimal selection at each step. Our method is based on a +variational perspective on the greedy CMI policy, and it uses +a technique known as amortized optimization (Amos, 2022) +to enable training using only a standard labeled dataset. +Notably, the model is trained with an objective that recovers +the greedy policy when it is trained to optimality. +Our contributions in this work are the following: +1. We derive a variational, or optimization-based perspec- +tive on the greedy CMI policy, which shows it to be +equivalent to minimizing the one-step-ahead prediction +loss given an optimal classifier. +2. We develop a learning approach based on amortized op- +timization, where a policy network is trained to directly +predict the greedy selection at each step. Rather than re- +quiring a dataset that indicates the correct selections, our +training approach is based on a standard labeled dataset +and an objective function whose global optimizer is the +arXiv:2301.00557v1 [cs.LG] 2 Jan 2023 + +Learning to Maximize Mutual Information for Dynamic Feature Selection +greedy CMI policy. +3. We propose a continuous relaxation for the inherently +discrete learning objective, which enables efficient and +architecture-agnostic training with stochastic gradient +descent. +Our experiments evaluate the proposed method on numer- +ous datasets, and the results show that it outperforms many +recent dynamic and static feature selection methods. Over- +all, our work shows that when learned properly, the greedy +CMI policy is a simple and powerful method for DFS. +2. Problem formulation +In this section, we describe the DFS problem and introduce +notation used throughout the paper. +2.1. Notation +Let x denote a vector of input features and y a response +variable for a supervised learning task. The input consists +of d distinct features, or x = (x1, . . . , xd). We use the nota- +tion s ⊆ [d] ≡ {1, . . . , d} to denote a subset of indices and +xs = {xi : i ∈ s} a subset of features. Bold symbols x, y +represent random variables, the symbols x, y are possible +values, and p(x, y) denotes the data distribution. +Our goal is to design a policy that controls which features +are selected given the currently available information. The +selection policy can be viewed as a function π(xs) ∈ [d], +meaning that it receives a subset of features as its input +and outputs the next feature index to query. The policy is +accompanied by a predictor f(xs) that can make predic- +tions given the set of available features; for example, if y +is discrete then predictions lie in the probability simplex, +or f(xs) ∈ ∆K−1 for K classes. The notation f(xs ∪ xi) +represents the prediction given the combined features. We +initially consider policy and predictor functions that operate +on feature subsets, and Section 4 proposes an implementa- +tion using a mask variable m ∈ [0, 1]d where the functions +operate on x ⊙ m. +2.2. Dynamic feature selection +The goal of DFS is to select features with minimal budget +that achieve maximum predictive accuracy. Having access +to more features generally makes prediction easier, so the +challenge is selecting a small number of informative features. +There are multiple formulations for this problem, including +non-uniform feature costs and different budgets for each +sample (Kachuee et al., 2018), but we focus on the setting +with a fixed budget and uniform costs. Our goal is to handle +data samples at test-time by beginning with no features, +sequentially selecting features xs such that |s| = k for a +fixed budget k < d, and finally making accurate predictions +for the response variable y. +Given a loss function that measures the discrepancy between +predictions and labels ℓ(ˆy, y), a natural scoring criterion is +the expected loss after selecting k features. The scoring is +applied to a policy-predictor pair (π, f), and we define the +score for a fixed budget k as follows, +vk(π, f) = Ep(x,y) +� +ℓ +� +f +� +{xit}k +t=1 +� +, y +�� +, +(1) +where feature indices are chosen sequentially for each (x, y) +according to in = π({xit}n−1 +t=1 ). The goal is to minimize +vk(π, f), or equivalently, to maximize our final predictive +accuracy. +One approach is to frame this as a Markov decision process +(MDP) and solve it using standard RL techniques, so that +π and f are trained to optimize a reward function based on +eq. (1). Several recent works have designed such formula- +tions (Shim et al., 2018; Kachuee et al., 2018; Janisch et al., +2019; Li & Oliva, 2021). However, these approaches are +difficult to train effectively, so our work focuses on a greedy +approach that is easier to learn and simpler to interpret. +3. Greedy information maximization +This section first defines the greedy CMI policy, and then +describes an existing approximation strategy based on gen- +erative modeling. +3.1. The greedy selection policy +As an idealized approach to DFS, we are interested in the +greedy algorithm that selects the most informative feature +at each step. This feature can be defined in multiple ways, +but we focus on the information-theoretic perspective that +the most useful feature has maximum CMI with the re- +sponse variable (Cover & Thomas, 2012). CMI, denoted as +I(xi; y | xs), quantifies how much information an unknown +feature xi provides about the response y when accounting +for the current features xs, and it is defined as the KL diver- +gence between the joint and factorized distributions: +I(xi; y | xs) = DKL +� +p(xi, y | xs) || p(xi | xs)p(y | xs) +� +. +Based on this, we define the greedy CMI policy as π∗(xs) ≡ +arg maxi I(xi; y | xs), so that features are sequentially se- +lected to maximize our information about the response vari- +able. We can alternatively understand the policy as perform- +ing greedy uncertainty minimization, because this is equiva- +lent to minimizing y’s conditional entropy at each step, or +π∗(xs) = arg mini H(y | xi, xs) (Cover & Thomas, 2012). +For a complete characterization of this idealized approach +to DFS, we also consider that the policy is paired with the +Bayes classifier as a predictor, or f ∗(xs) = p(y | xs). + +Learning to Maximize Mutual Information for Dynamic Feature Selection +Maximizing the information about y at each step is intuitive +and should be effective in many problems. Prior work has +considered the same idea, but from two perspectives that +differ from ours. First, Chen et al. (2015b) take a theoretical +perspective and prove that the greedy algorithm has bounded +suboptimality relative to the optimal policy-predictor pair; +the proof requires specific distributional assumptions, but +we find that the greedy algorithm performs well with many +real datasets (Section 6). Second, from an implementation +perspective, two works aim to provide practical approxi- +mations; however, these suffer from several limitations, so +our work aims to develop a simple and flexible alternative +(Section 4). In these works, Fleuret (2004) requires binary +features, and Ma et al. (2019) requires a conditional gen- +erative model of the data distribution, which we discuss +next. +3.2. Estimating conditional mutual information +The greedy policy is trivial to implement if we can directly +calculate CMI, but this is rarely the case in practice. Instead, +one option to to estimate it. We now describe a procedure to +do so iteratively for each feature, assuming for now that we +have oracle access to the response distributions p(y | xs) +for all s ⊆ [d] and the feature distributions p(xi | xs) for +all s ⊆ [d] and i ∈ [d]. +At any point in the selection procedure, given the current +features xs, we can estimate the CMI for a feature xi where +i /∈ s as follows. First, we can sample multiple values for +xi from its conditional distribution, or xj +i ∼ p(xi | xs) for +j ∈ [n]. Next, we can generate Bayes optimal predictions +for each sampled value, or p(y | xs, xj +i). Finally, we can +calculate the mean prediction and the mean KL divergence +relative to the mean prediction, which yields the following +CMI estimator: +In +i = 1 +n +n +� +j=1 +DKL +� +p(y | xs, xj +i) || 1 +n +n +� +l=1 +p(y | xs, xl +i) +� +. +(2) +This score measures the variability among predictions and +captures whether different xi values significantly affect y’s +conditional distribution. The estimator can be used to select +features, or we can set π(xs) = arg maxi In +i , due to the +following limiting result (see Appendix A): +lim +n→∞ In +i = I(y; xi | xs). +(3) +This procedure thus provides a way to identify the correct +greedy selections by estimating the CMI. Prior work has +explored similar ideas for scoring features based on sam- +pled predictions (Saar-Tsechansky et al., 2009; Chen et al., +2015a; Early et al., 2016a;b), but the implementation choices +in these works prevent them from performing greedy infor- +mation maximization. In eq. (2), is it important that our +estimator uses the Bayes classifier, that we sample features +from the conditional distribution p(xi | xs), and that we +use the KL divergence as a measure of prediction variability. +However, this estimator is impractical because we typically +lack access to both p(y | xs) and p(xi | xs). +In practice, we would instead require learned substitutes +for each distribution. For example, we can use a a classi- +fier that approximates f(xs) ≈ p(y | xs) and a generative +model that approximates samples from p(xi | xs). Simi- +larly, Ma et al. (2019) propose jointly modeling (x, y) with +a conditional generative model, which is implemented via +a modified VAE (Kingma et al., 2015). This approach is +limited for several reasons, including (i) the difficulty of +training an accurate conditional generative model, (ii) the +challenge of modeling mixed continuous/categorical fea- +tures (Ma et al., 2020; Nazabal et al., 2020), and (iii) the +slow CMI estimation process. In our approach, which we +discuss next, we bypass all three of these challenges by +directly predicting the best selection at each step. +4. Proposed method +We now introduce our approach, a practical approximation +the greedy policy trained using amortized optimization. Un- +like prior work that estimates CMI as an intermediate step, +we develop a variational perspective on the greedy policy, +which we then leverage to train a policy network that directly +predicts the optimal selection given the current features. +4.1. A variational perspective on CMI +For our purpose, it is helpful to recognize that the greedy +policy can be viewed as the solution to an optimization +problem. Section 3 provides a conventional definition of +CMI as a KL divergence, but this is difficult to integrate into +an end-to-end learning approach. Instead, we now consider +the one-step-ahead prediction achieved by a policy π and +predictor f, and we determine the behavior that minimizes +their loss. Given the current features xs and a selection +i = π(xs), the expected one-step-ahead loss is: +Ey,xi|xs +� +ℓ +� +f(xs ∪ xi), y +�� +. +(4) +The variational perspective we develop here consists of +two main results regarding this expected loss. The first +result concerns the predictor, and we show that the loss- +minimizing predictor can be defined independently of the +policy π. We formalize this in the following proposition for +classification tasks, and our results can also be generalized +to regression tasks (see proofs in Appendix A). +Proposition 1. When y is discrete and ℓ is cross-entropy +loss, eq. (4) is minimized for any policy π by the Bayes +classifier, or f ∗(xs) = p(y | xs). + +Learning to Maximize Mutual Information for Dynamic Feature Selection +The above property requires that features are selected with- +out knowledge of the remaining features or response vari- +able, which is a valid assumption for DFS, but not in scenar- +ios where selections are based on the full feature set (Chen +et al., 2018; Yoon et al., 2018; Jethani et al., 2021). Now, +assuming that we use the Bayes classifier f ∗ as a predictor, +our second result concerns the selection policy. As we show +next, the loss-minimizing policy is equivalent to making +selections based on CMI. +Proposition 2. When y is discrete, ℓ is cross-entropy +loss and the predictor is the Bayes classifier f ∗, eq. (4) +is minimized by the greedy CMI policy, or π∗(xs) = +arg maxi I(y; xi | xs). +With this, we can see that the greedy CMI policy defined +in Section 3 is equivalent to minimizing the one-step-ahead +prediction loss. Next, we exploit this variational perspec- +tive to develop a joint learning procedure for a policy and +predictor network. +4.2. An amortized optimization approach +Instead of estimating each feature’s CMI to identify the next +selection, we now develop an approach that directly pre- +dicts the best selection at step. The greedy policy implicitly +requires solving an optimization problem at each step, or +arg maxi I(y, xi; xs), but since we lack access to this ob- +jective, we now formulate an approach that directly predicts +the solution. Following a technique known as amortized +optimization (Amos, 2022), we do so by casting our varia- +tional perspective on CMI from Section 4.1 as an objective +function to be optimized by a learnable network. +First, because it facilitates gradient-based optimization, we +now consider that the policy outputs a distribution over fea- +ture indices. With slight abuse of notation, this section lets +the policy be a function π(xs) ∈ ∆d−1, which generalizes +the previous definition π(xs) ∈ [d]. Using this stochas- +tic policy, we can now formulate our objective function as +follows. +Let the selection policy be parameterized by a neural +network π(xs; φ) and the predictor by a neural network +f(xs; θ). Let p(s) represent a distribution over subsets with +p(s) > 0 for all |s| < d. Then, our objective function +L(θ, φ) is defined as +L(θ, φ) = Ep(x,y)Ep(s) +� +Ei∼π(xs;φ) +� +ℓ +� +f(xs ∪ xi; θ), y +��� +. +(5) +Intuitively, eq. (5) represents generating a random feature +set xs, sampling a feature index according to i ∼ π(xs; φ), +and then measuring the loss of the prediction f(xs ∪ xi; θ). +Our objective thus optimizes for individual selections and +predictions rather than the entire trajectory, which lets us +build on Proposition 1-2. We describe this as an implemen- +tation of the greedy approach because it recovers the greedy +CMI selections when it is trained to optimality. In the clas- +sification case, we show the following result under a mild +assumption that there is a unique optimal selection. +Theorem 1. When y is discrete and ℓ is cross-entropy loss, +the global optimum of eq. (5) is a predictor that satisfies +f(xs; θ∗) = p(y | xs) and a policy π(xs; φ∗) that puts all +probability mass on i∗ = arg maxi I(y; xi | xs). +If we relax the assumption of a unique optimal selection, +the optimal policy π(xs; φ∗) will simply split probability +mass among the best indices. A similar result holds in the +regression case, where we can interpret the greedy policy as +performing conditional variance minimization. +Theorem 2. When y is continuous and ℓ is squared error +loss, the global optimum of eq. (5) is a predictor that satisfies +f(xs; θ∗) = E[y | xs] and a policy π(xs; φ∗) that puts all +probability mass on i∗ = arg mini Exi|xs[Var(y | xi, xs)]. +Proofs for these results are in Appendix A. This approach +has two key advantages over the CMI estimation procedure +from Section 3.2. First, we avoid modeling the feature +conditional distributions p(xi | xs) for all (s, i). Modeling +these distributions is a difficult intermediate step, and our +approach instead aims to directly output the optimal index. +Second, our approach is faster because each selection is +made in a single forward pass: selecting k features using +the Ma et al. (2019) procedure requires O(dk) scoring steps, +but our approach requires only k forward passes through the +policy π(xs; φ). +Furthermore, compared to a policy trained by RL, the +greedy approach is easier to learn. Our training proce- +dure can be viewed as a form of reward shaping (Sutton +et al., 1998; Randløv & Alstrøm, 1998), where the reward +accounts for the loss after each step and provides a strong +signal about whether each selection is helpful. In compar- +ison, observing the reward only after selecting k features +provides a comparably weak signal to the policy network +(see eq. (1)). RL methods generally face a challenging +exploration-exploitation trade-off, but learning the greedy +policy is simpler because it only requires finding the locally +optimal choice at each step. +4.3. Training with a continuous relaxation +Our objective in eq. (5) yields the correct greedy policy +when it is perfectly optimized, but L(θ, φ) is difficult to +optimize by gradient descent. In particular, gradients are +difficult to propagate through the policy network given a +sampled index i ∼ π(xs; φ). The REINFORCE trick +(Williams, 1992) is one way to get stochastic gradients, +but high gradient variance can make it ineffective in many +problems. There is a robust literature on reducing gradient + +Learning to Maximize Mutual Information for Dynamic Feature Selection +𝜋 ⋅ ; 𝜙 +𝑓 ⋅ ; 𝜃 +Policy +Predictor +#𝑦 +Repeat for 𝑘 selection steps +Concrete 𝛼, 𝜏 +Update masked +input +0 +𝑥# +0 +𝑥% +𝑥" +𝑥# +0 +𝑥% +≈ +Figure 1. Diagram of our training approach. Left: features are selected by making repeated calls to the policy network using masked +inputs. Right: predictions are made after each selection using the predictor network. Only solid lines are backpropagated through when +performing gradient descent. +variance in this setting (Tucker et al., 2017; Grathwohl et al., +2018), but we propose a simple alternative: the Concrete +distribution (Maddison et al., 2016). +An index sampled according to i ∼ π(xs; φ) can be rep- +resented by a one-hot vector m ∈ {0, 1}d indicating the +chosen index, and with the Concrete distribution we instead +sample an approximately one-hot vector in the probability +simplex, or m ∈ ∆d−1. This continuous relaxation lets us +calculate gradients using the reparameterization trick (Mad- +dison et al., 2016; Jang et al., 2016). Relaxing the subset +s ⊆ [d] to a continuous vector also requires relaxing the +policy and predictor functions, so we let these operate on +a masked input x, or the element-wise product x ⊙ m. To +avoid ambiguity about whether features are zero or masked, +we can also pass the mask as an input. +Training with the Concrete distribution requires specifying +a temperature parameter τ > 0 to control how discrete the +samples are. Previous works have typically trained with a +fixed temperature or annealed it over a pre-determined num- +ber of epochs (Chang et al., 2017; Chen et al., 2018; Balın +et al., 2019), but we instead train with a sequence of τ values +and perform early stopping at each step. This removes the +temperature and number of epochs as important hyperpa- +rameters to tune. Our training procedure is summarized in +Figure 1, and in more detail by Algorithm 1. +There are also several optional steps that we found can +improve optimization: +• Parameters can be shared between the predictor and pol- +icy networks f(x; θ), π(x, φ). This does not complicate +their joint optimization, and learning a shared represen- +tation in the early layers can in some cases help the +networks optimize faster. +• Rather than training with a random subset distribution +p(s), we generate subsets using features selected by +the policy π(x; φ). This allows the models to focus on +subsets likely to be encountered at inference time, and +it does not affect the globally optimal policy/predictor: +gradients are not propagated between selections, so both +eq. (5) and this sampling approach treat each feature +set as an independent optimization problem, only with +different weights (see Appendix D). +• We pre-train the predictor f(x; θ) using random subsets +before jointly training the policy-predictor pair. This +works better than optimizing L(θ, φ) from a random ini- +tialization, because a random predictor f(x; θ) provides +no signal to π(x; φ) about which features are useful. +5. Related work +Prior work has frequently addressed DFS using RL. For +example, Dulac-Arnold et al. (2011); Shim et al. (2018); +Janisch et al. (2019); Li & Oliva (2021) optimize a reward +based on the final prediction accuracy, and Kachuee et al. +(2018) use a reward that accounts for prediction uncertainty. +RL is a natural approach for sequential decision-making +problems, but it can be difficult to optimize in practice: +RL requires complex architectures and training routines, is +slow to converge, and is highly sensitive to its initialization +(Henderson et al., 2018). As a result, RL-based DFS does +not reliably outperform static feature selection, as shown by +Erion et al. (2021) and confirmed in our experiments. +Several other approaches include imitation learning (He +et al., 2012; 2016a) and iterative feature scoring methods +(Melville et al., 2004; Saar-Tsechansky et al., 2009; Chen +et al., 2015a; Early et al., 2016b;a). Imitation learning casts +DFS as supervised classification, whereas our training ap- +proach bypasses the need for an oracle policy. Most existing +feature scoring techniques are greedy methods, like ours, +but they use scoring heuristics unrelated to maximizing +CMI (see Section 3.2). Two feature scoring methods are +specifically designed to calculate CMI, but they suffer from +practical limitations: Fleuret (2004) requires binary features, +and Ma et al. (2019) relies on difficult-to-train generative +models. Our approach is simpler, faster and more flexi- +ble, because the selection logic is contained within a policy +network that avoids the need for generative modeling. + +Learning to Maximize Mutual Information for Dynamic Feature Selection +0 +5 +10 +15 +20 +25 +# Selected Features +0.55 +0.60 +0.65 +0.70 +0.75 +AUROC +Bleeding AUROC Comparison +0 +5 +10 +15 +20 +25 +# Selected Features +0.65 +0.70 +0.75 +0.80 +0.85 +AUROC +Respiratory AUROC Comparison +2 +4 +6 +8 +10 +# Selected Features +0.70 +0.75 +0.80 +0.85 +AUROC +Fluid AUROC Comparison +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +IntGrad +DeepLift +SAGE +Perm Test +CAE +Opportunistic (OL) +CMI (Marginal) +CMI (PVAE) +Greedy (Ours) +0 +5 +10 +15 +20 +25 +# Selected Features +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +AUROC +Spam AUROC Comparison +0 +5 +10 +15 +20 +25 +# Selected Features +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +AUROC +MiniBooNE AUROC Comparison +2 +4 +6 +8 +10 +# Selected Features +0.75 +0.80 +0.85 +0.90 +0.95 +AUROC +Diabetes AUROC Comparison +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +IntGrad +DeepLift +SAGE +Perm Test +CAE +Opportunistic (OL) +CMI (Marginal) +CMI (PVAE) +Greedy (Ours) +Figure 2. Evaluating the greedy approach on six tabular datasets. The results for each method are the average across five runs. +Static feature selection is a long-standing problem (Guyon +& Elisseeff, 2003; Cai et al., 2018). There are no default ap- +proaches for neural networks, but one option is ranking fea- +tures by local or global importance scores (Breiman, 2001; +Shrikumar et al., 2017; Sundararajan et al., 2017; Covert +et al., 2020). In addition, several prior works have leveraged +continuous relaxations to learn feature selection strategies +by gradient descent: for example, Chang et al. (2017); Balın +et al. (2019); Yamada et al. (2020); Lee et al. (2021) perform +static feature selection, and Chen et al. (2018); Jethani et al. +(2021) perform instance-wise feature selection given all the +features. Our work uses a similar continuous relaxation +for optimization but in the DFS context, where our method +learns a selection policy rather than a static selection layer. +Finally, several works have examined greedy feature selec- +tion algorithms from a theoretical perspective. For example, +Das & Kempe (2011); Elenberg et al. (2018) show that +weak submodularity implies near-optimal performance in +the static feature selection setting. Chen et al. (2015b) find +that the related notion of adaptive submodularity (Golovin +& Krause, 2011) does not not apply to DFS when evaluated +via mutual information, but manage to provide performance +guarantees under specific distributional assumptions. +6. Experiments +We now demonstrate the use of our greedy approach on +several datasets. We first explore tabular datasets of vari- +ous sizes, including four medical diagnosis tasks, and we +then consider two image classification datasets. Several +of the tasks are natural candidates for DFS, and the re- +maining ones serve as useful tasks to test the effectiveness +of our approach. Code for reproducing our experiments +is available online: https://github.com/iancovert/ +dynamic-selection. +We evaluate our method by comparing to both dynamic and +static feature selection methods. We also ensure consistent +comparisons by only using methods applicable to neural +networks. As static baselines, we use permutation tests +(Breiman, 2001) and SAGE (Covert et al., 2020) to rank +features by their importance to model accuracy, as well as +per-prediction DeepLift (Shrikumar et al., 2017) and Int- +Grad (Sundararajan et al., 2017) scores aggregated across +the dataset. We then use a supervised version of the Con- +crete Autoencoder (CAE, Balın et al. 2019), a state-of-the- +art static feature selection method. As dynamic baselines, +we use two versions of the CMI estimation procedure de- +scribed in Section 3.2. First, we use the PVAE generative +model from Ma et al. (2019) to sample unknown features, +and second, we instead sample unknown features from their +marginal distribution; in both cases, we use a classifier +trained with random feature subsets to make predictions. +Finally, we also use the RL-based Opportunistic Learning +(OL) approach (Kachuee et al., 2018). Appendix C provides +more information about each of the baselines. +6.1. Tabular datasets +We first applied our method to three medical diagnosis tasks +derived from an emergency medicine setting. The tasks +involve predicting a patient’s bleeding risk via a low fibrino- +gen concentration (bleeding), whether the patient requires +endotracheal intubation for respiratory support (respiratory), +and whether the patient will be responsive to fluid resusci- + +Learning to Maximize Mutual Information for Dynamic Feature Selection +Table 1. AUROC averaged across budgets of 1-10 features (with 95% confidence intervals). +Spam +MiniBooNE +Diabetes +Bleeding +Respiratory +Fluid +Static +IntGrad +82.84 ± 0.68 +89.10 ± 0.33 +88.91 ± 0.24 +66.70 ± 0.27 +81.10 ± 0.04 +79.94 ± 0.94 +DeepLift +90.16 ± 1.24 +88.62 ± 0.30 +95.42 ± 0.13 +67.75 ± 0.49 +76.05 ± 0.35 +76.96 ± 0.56 +SAGE +89.70 ± 1.10 +92.64 ± 0.03 +95.43 ± 0.01 +71.34 ± 0.19 +82.92 ± 0.26 +83.27 ± 0.53 +Perm Test +85.64 ± 3.58 +92.19 ± 0.15 +95.46 ± 0.02 +68.89 ± 1.06 +81.56 ± 0.28 +81.35 ± 1.04 +CAE +92.28 ± 0.27 +92.76 ± 0.41 +95.91 ± 0.07 +70.69 ± 0.57 +83.10 ± 0.45 +79.40 ± 0.86 +Dynamic +Opportunistic (OL) +85.94 ± 0.20 +69.23 ± 0.64 +83.07 ± 0.82 +60.63 ± 0.55 +74.44 ± 0.42 +78.13 ± 0.31 +CMI (Marginal) +86.57 ± 1.54 +92.21 ± 0.40 +95.48 ± 0.05 +70.57 ± 0.46 +79.62 ± 0.62 +81.97 ± 0.93 +CMI (PVAE) +89.01 ± 1.40 +88.94 ± 1.25 +90.50 ± 5.16 +70.17 ± 0.74 +74.12 ± 3.50 +80.27 ± 1.02 +Greedy (Ours) +93.91 ± 0.17 +94.46 ± 0.12 +96.03 ± 0.02 +72.64 ± 0.31 +84.48 ± 0.08 +86.59 ± 0.25 +tation (fluid). See Appendix B for more details about the +datasets. In each scenario, gathering all possible inputs at +test-time is challenging due to time and resource constraints, +thus making DFS a natural solution. +We use fully connected networks for all methods, and we +use dropout to reduce overfitting (Srivastava et al., 2014). +Figure 2 (top) shows the results of applying each method +with various feature budgets. The classification accuracy is +measured via AUROC, and the greedy method achieves the +best results for nearly all feature budgets on all three tasks. +Among the baselines, several static methods are sometimes +close, but the CMI estimation method is rarely competitive. +Additionally, OL provides unstable and weak results. The +greedy method’s advantage is often largest when selecting a +small number of features, and it usually becomes narrower +once the accuracy saturates. +Next, we conducted experiments using three publicly avail- +able tabular datasets: spam classification (Dua & Graff, +2017), particle identification (MiniBooNE) (Roe et al., +2005) and diabetes diagnosis (Miller, 1973). The diabetes +task is a natural application for DFS and was used in prior +work (Kachuee et al., 2018). We again tested various num- +bers of features, and Figure 2 (bottom) shows plots of the +AUROC for each feature budget. On these tasks, the greedy +method is once again most accurate for nearly all numbers +of features. Table 1 summarizes the results via the mean +AUROC across k = 1, . . . , 10 features, further emphasizing +the benefits of the greedy method across all six datasets. +Appendix E shows larger versions of the AUROC curves +(Figure 4 and Figure 5), as well as plots demonstrating the +variability of selections within each dataset. +The results with these datasets reveal that, perhaps surpris- +ingly, dynamic methods can be outperformed by static meth- +ods. Interestingly, this point was not highlighted in prior +work where strong static baselines were not used (Kachuee +et al., 2018; Janisch et al., 2019). For example, OL is never +competitive on these datasets, and the two versions of the +CMI estimation method are not consistently among the top +baselines. Dynamic methods are in principle capable of +performing better, so the sub-par results from these methods +underscore the difficulty of learning both a selection policy +and a prediction function that works for multiple feature +sets. In these experiments, our approach is the only dynamic +method to do both successfully. +6.2. Image classification datasets +Next, we considered two standard image classification +datasets: MNIST (LeCun et al., 1998) and CIFAR-10 +(Krizhevsky et al., 2009). Our goal is to begin with a blank +image, sequentially reveal multiple pixels or patches, and +ultimately make a classification using a small portion of +the image. Although this is not an obvious use case for +DFS, it represents a challenging problem for our method, +and similar tasks were considered in several earlier works +(Karayev et al., 2012; Mnih et al., 2014; Early et al., 2016a; +Janisch et al., 2019). +For MNIST, we use fully connected architectures for both +the policy and predictor, and we treat pixels as individual +features, where d = 784. For CIFAR-10, we use a shared +ResNet backbone (He et al., 2016b) for the policy and pre- +dictor networks, and each network uses its own output head. +The 32 × 32 images are coarsened into d = 64 patches +of size 4 × 4, so the selector head generates logits corre- +sponding to each patch, and the predictor head generates +probabilities for each class. +Figure 3 shows our method’s accuracy for different feature +budgets. For MNIST, we use the previous baselines but ex- +clude the CMI estimation method due to its computational +cost. We observe a large benefit for our method, particu- +larly when making a small number of selections: our greedy +method reaches nearly 90% accuracy with just 10 pixels, +which is roughly 10% higher than the best baseline and con- +siderably higher than prior work (Balın et al., 2019; Yamada +et al., 2020; Covert et al., 2020). OL yields the worst results, +and it also trains slowly due to the large number of states. +For CIFAR-10, we use two simple baselines: center crops +and random masks of various sizes. For each method, we +plot the mean and 95% confidence intervals determined +from five trials. Our greedy approach is slightly less ac- +curate with 1-2 patches, but it reaches significantly higher + +Learning to Maximize Mutual Information for Dynamic Feature Selection +10 +20 +30 +40 +50 +# Selected Pixels +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Top-1 Accuracy +MNIST Accuracy Comparison +IntGrad +DeepLift +SAGE +Perm Test +CAE +Opportunistic (OL) +Greedy (Ours) +0 +5 +10 +15 +20 +25 +30 +# Selected Patches +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Top-1 Accuracy +CIFAR-10 Accuracy Comparison +Center Crop +Random Mask +Greedy (Ours) +Horse +Truck +Airplane +Ship +Frog +Horse +Ship +Deer +Dog +Bird +Prob = 55.34% +Prob = 98.69% +Prob = 99.98% +Prob = 80.30% +Prob = 97.80% +Prob = 20.27% +Prob = 99.01% +Prob = 51.61% +Prob = 52.17% +Prob = 99.86% +Figure 3. Greedy feature selection for image classification. Top left: accuracy comparison on MNIST with results averaged across five +runs. Top right: accuracy comparison on CIFAR-10 with 95% confidence intervals. Bottom: example selections and predictions for the +greedy method with 10 out of 64 patches for CIFAR-10 images. +accuracy when using 5-20 patches. Figure 3 (bottom) also +shows qualitative examples of our method’s predictions af- +ter selecting 10 out of 64 patches, and Appendix E shows +similar plots with different numbers of patches. +7. Conclusion +In this work, we explored a greedy algorithm for DFS that se- +lects features based on their CMI with the response variable. +We proposed an approach to approximate this policy by di- +rectly predicting the optimal selection at each step, and we +conducted experiments that show our method outperforms +a variety of existing feature selection methods, including +both dynamic and static baselines. Future work on this +topic may include incorporating non-uniform features costs, +determining the feature budget on a per-sample basis, and +further characterizing the greedy suboptimality gap: some +progress has been made in analyzing the greedy algorithm’s +suboptimality in the dynamic setting (Chen et al., 2015b), +but more general characterizations remain an open topic for +future work. +Acknowledgements +We thank Samuel Ainsworth, Kevin Jamieson, Mukund +Sudarshan and the Lee Lab for helpful discussions. This +work was funded by NSF DBI-1552309 and DBI-1759487, +NIH R35-GM-128638 and R01-NIA-AG-061132. +References +National health and nutrition examination survey, 2018. +URL https://www.cdc.gov/nchs/nhanes. +Amos, B. Tutorial on amortized optimization for learning +to optimize over continuous domains. arXiv preprint +arXiv:2202.00665, 2022. +Balın, M. F., Abid, A., and Zou, J. Concrete autoencoders: +Differentiable feature selection and reconstruction. 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Machine +Learning, 8(3):229–256, 1992. +Yamada, Y., Lindenbaum, O., Negahban, S., and Kluger, Y. +Feature selection using stochastic gates. In International +Conference on Machine Learning. PMLR, 2020. +Yoon, J., Jordon, J., and van der Schaar, M. +INVASE: +Instance-wise variable selection using neural networks. +In International Conference on Learning Representations, +2018. + +Learning to Maximize Mutual Information for Dynamic Feature Selection +A. Proofs +In this section, we re-state and prove our main theoretical results. We begin with our proposition regarding the optimal +predictor for an arbitrary policy π. +Proposition 1. When y is discrete and ℓ is cross-entropy loss, eq. (4) is minimized for any policy π by the Bayes classifier, +or f ∗(xs) = p(y | xs). +Proof. Given the predictor inputs xs, our goal is to determine the prediction that minimizes the expected loss. Because +features are selected sequentially by π with no knowledge of the non-selected values, there is no other information to +condition on; for the predictor, we do not even need to distinguish the order in which features were selected. We can +therefore derive the optimal prediction ˆy ∈ ∆K−1 for a discrete response y ∈ [K] as follows: +f ∗(xs) = arg min +ˆy +Ey|xs +� +ℓ(ˆy, y) +� += arg min +ˆy +� +i∈Y +p(y = i | xs) log ˆyi += arg min +ˆy +DKL +� +p(y | xs) || ˆy +� ++ H(y | xs) += p(y | xs). +In the case of a continuous response y ∈ R with squared error loss, we have a similar result involving the response’s +conditional expectation: +f ∗(xs) = arg min +ˆy +Ey|xs +� +(ˆy − y)2� += arg min +ˆy +Ey|xs +� +(ˆy − E[y | xs])2� ++ Var(y | xs) += E[y | xs]. +Proposition 2. When y is discrete, ℓ is cross-entropy loss and the predictor is the Bayes classifier f ∗, eq. (4) is minimized +by the greedy CMI policy, or π∗(xs) = arg maxi I(y; xi | xs). +Proof. Following eq. (4), the policy network’s selection i = π(xs) incurs the following expected loss with the distribution +p(y, xi | xs): +Ey,xi|xs +� +ℓ(f ∗(xs ∪ xi), y) +� += Ey,xi|xs +� +ℓ(p(y | xi, xs), y) +� += Exi|xs +� +Ey|xi,xs[ℓ(p(y | xi, xs), y)] +� += Exi|xs +� +H(y | xi, xs) +� += H(y | xs) − I(y; xi | xs). +Note that H(y | xs) is a constant that does not depend on i. When identifying the index that minimizes the expected loss, +we thus have the following result: +arg min +i +Ey,xi|xs +� +ℓ(f ∗(xs ∪ xi), y) +� += arg max +i +I(y; xi | xs). + +Learning to Maximize Mutual Information for Dynamic Feature Selection +In the case of a continuous response with squared error loss and an optimal predictor given by f ∗(xs) = E[y | xs], we have +a similar result: +Ey,xi|xs +� +(f ∗(xs ∪ xi) − y)2� += Ey,xi|xs +� +(E[y | xi, xs] − y)2� += Exi|xs +� +Ey|xi,xs[(E[y | xi, xs] − y)2] +� += Exi|xs[Var(y | xi, xs)]. +When we aim to minimize the expected loss, our selection is thus the index that yields the lowest expected conditional +variance: +arg min +i +Exi|xs[Var(y | xi, xs)]. +We also prove the limiting result presented in eq. (3), which states that In +i → I(y; xi | xs). +Proof. Conditional mutual information I(y; xi | xs) is defined as follows (Cover & Thomas, 2012): +I(y; xi | xs) = DKL +� +p(xi, y | xs) || p(xi | xs)p(y | xs) +� += Ey,xi|xs +� +log +p(y, xi | xs) +p(xi | xs)p(y | xs) +� +. +Rearranging terms, we can write this as an expected KL divergence with respect to xi: +I(y; xi | xs) = Exi|xsEy|xs,xi +� +log +p(y, xi | xs) +p(xi | xs)p(y | xs) +� += Exi|xsEy|xs,xi +� +log p(y | xi, xs) +p(y | xs) +� += Exi|xs +� +DKL +� +p(y | xi, xs) || p(y | xs) +�� +Now, when we sample multiple values x1 +i , . . . , xn +i ∼ p(xi | xs) and make predictions using the Bayes classifier, we have +the following mean prediction as n becomes large: +lim +n→∞ +1 +n +n +� +j=1 +p(y | xs, xj +i) = Exi|xs +� +p(y | xi, xs) +� += p(y | xs). +Calculating the mean KL divergence across the predictions, we arrive at the following result: +lim +n→∞ In +i = Exi|xs +� +DKL +� +p(y | xi, xs) || p(y | xs) +�� += I(y; xi | xs). +Theorem 1. When y is discrete and ℓ is cross-entropy loss, the global optimum of eq. (5) is a predictor that satisfies +f(xs; θ∗) = p(y | xs) and a policy π(xs; φ∗) that puts all probability mass on i∗ = arg maxi I(y; xi | xs). + +Learning to Maximize Mutual Information for Dynamic Feature Selection +Proof. We first consider the predictor network f(xs; θ). When the predictor is given the feature values xs, it means that +one index i ∈ s was chosen by the policy according to π(xs\i; φ) and the remaining indices s \ i were sampled from p(s). +Because s is sampled independently from (x, y), and because π(xs\i; φ) is not given access to (x[d]\s, xi, y), the predictor’s +expected loss must be considered with respect to the distribution y | xs. The globally optimal predictor f(xs; θ∗) is thus +defined as follows, regardless of the selection policy π(xs; φ) and which index i was selected last: +f(xs; θ∗) = arg min +ˆy +Ey|xs +� +ℓ(ˆy, y) +� += p(y | xs). +The above result follows from our proof for Proposition 1. Now, given the optimal predictor f(xs; θ∗), we can define the +globally optimal policy by minimizing the expected loss for a fixed input xs. Denoting the probability mass placed on each +index i ∈ [d] as πi(xs; φ), where π(xs; φ) ∈ ∆d−1, the expected loss is the following: +Ei∼π(xs;φ)Ey,xi|xs +� +ℓ(f(xs ∪ xi; θ∗), y) +� += +� +i∈[d] +πi(xs; φ)Ey,xi|xs +� +ℓ +� +f(xs ∪ xi; θ∗), y +�� += +� +i∈[d] +πi(xs; φ)Exi|xs[H(y | xi, xs)]. +The above result follows from our proof for Proposition 2. If there exists a single index i∗ ∈ [d] that yields the lowest +expected conditional entropy, or +Exi∗|xs[H(y | xi∗, xs)] < Exi|xs[H(y | xi, xs)] +∀i ̸= i∗, +then the optimal predictor must put all its probability mass on i∗, or πi∗(xs; φ∗) = 1. Note that the corresponding feature +xi∗ has maximum conditional mutual information with y, because we have +I(y; xi∗ | xs) = H(y | xs) +� +�� +� +Constant +−Exi∗|xs[H(y | xi∗, xs)]. +To summarize, we derived the global optimum to our objective L(θ, φ) by first considering the optimal predictor f(xs; θ∗), +and then considering the optimal policy π(xs; φ∗) when we assume that we use the optimal predictor. +Theorem 2. When y is continuous and ℓ is squared error loss, the global optimum of eq. (5) is a predictor that satisfies +f(xs; θ∗) = E[y | xs] and a policy π(xs; φ∗) that puts all probability mass on i∗ = arg mini Exi|xs[Var(y | xi, xs)]. +Proof. Our proof follows the same logic as our proof for Theorem 1. For the optimal predictor given an arbitrary policy, we +have: +f(xs; θ∗) = arg min +ˆy +Ey|xs +� +(ˆy − y)2� += E[y | xs]. +Then, for the policy’s expected loss, we have: +Ei∼π(xs;φ)Ey,xi|xs +�� +f(xs ∪ xi; θ∗) − y +�2� += +� +i∈[d] +πi(xs; φ)Exi|xs[Var(y | xi, xs)]. +If there exists an index i∗ ∈ [d] that yields the lowest expected conditional variance, then the optimal policy must put all its +probability mass on i∗, or πi∗(xs; φ∗) = 1. + +Learning to Maximize Mutual Information for Dynamic Feature Selection +B. Datasets +The datasets used in our experiments are summarized in Table 2. Three of the tabular datasets and the two image classification +datasets are publicly available, and the three emergency medicine tasks were privately curated from the Harborview Medical +Center Trauma Registry. +Table 2. Summary of datasets used in our experiments. +Dataset +# Features +# Feature Groups +# Classes +# Samples +Fluid +224 +162 +2 +2,770 +Respiratory +112 +35 +2 +65,515 +Bleeding +121 +44 +2 +6,496 +Spam +58 +– +2 +4,601 +MiniBooNE +51 +– +2 +130,064 +Diabetes +45 +– +3 +92,062 +MNIST +784 +– +10 +60,000 +CIFAR-10 +1,024 +64 +10 +60,000 +B.1. MiniBooNE and spam classification +The spam dataset includes features extracted from e-mail messages to predict whether or not a message is spam. Three +features describes the usage of capital letters in the e-mail, and the remaining 54 features describe the frequency with which +certain key words or characters are used. The MiniBooNE particle identification dataset involves distinguishing electron +neutrinos from muon neutrinos based on various continuous features (Roe et al., 2005). Both datasets were obtained from +the UCI repository (Dua & Graff, 2017). +B.2. Diabetes classification +The diabetes dataset was obtained from from the National Health and Nutrition Examination Survey (NHANES) (NHA, +2018), an ongoing survey designed to assess the well-being of adults and children in the United States. We used a version +of the data pre-processed by Kachuee et al. (2018; 2019) that includes data collected from 1999 through 2016. The input +features include demographic information (age, gender, ethnicity, etc.), lab results (total cholesterol, triglyceride, etc.), +examination data (weight, height, etc.), and questionnaire answers (smoking, alcohol, sleep habits, etc.). An expert was also +asked to suggest costs for each feature based on the financial burden, patient privacy, and patient inconvenience, but we +assume uniform feature costs in our experiments. Finally, the fasting glucose values were used to define three classes based +on standard threshold values: normal, pre-diabetes, and diabetes. +B.3. Image classification datasets +The MNIST and CIFAR-10 datasets were downloaded using PyTorch (Paszke et al., 2017). We used the standard train-test +splits, and we split the train set to obtain a validation set with the same size as the test set (10,000 examples). +B.4. Emergency medicine datasets +The emergency medicine datasets used in this study were gathered over a 13-year period (2007-2020) and encompass 14,463 +emergency department admissions. We excluded patients under the age of 18, and we curated 3 clinical cohorts commonly +seen in pre-hospitalization settings. These include 1) pre-hospital fluid resuscitation, 2) emergency department respiratory +support, and 3) bleeding after injury. These datasets are not publicly available due to patient privacy concerns. +Pre-hospital fluid resuscitation +We selected 224 variables that were available in the pre-hospital setting, including +dispatch information (injury date, time, cause, and location), demographic information (age, sex), and pre-hospital vital +signs (blood pressure, heart rate, respiratory rate). The outcome was each patient’s response to fluid resuscitation, following +the Advanced Trauma Life Support (ATLS) definition (Subcommittee et al., 2013). + +Learning to Maximize Mutual Information for Dynamic Feature Selection +Emergency department respiratory support +In this cohort, our goal is to predict which patients require respiratory +support upon arrival in the emergency department. Similar to the previous dataset, we selected 112 pre-hospital clinical +features including dispatch information (injury date, time, cause, and location), demographic information (age, sex), and +pre-hospital vital signs (blood pressure, heart rate, respiratory rate). The outcome is defined based on whether a patient +received respiratory support, including both invasive (intubation) and non-invasive (BiPap) approaches. +Bleeding +In this cohort, we only included patients whose fibrinogen levels were measured, as this provides an indicator for +bleeding or fibrinolysis (Mosesson, 2005). As with the previous datasets, demographic information, dispatch information, +and pre-hospital observations were used as input features. The outcome, based on experts’ opinion, was defined by whether +an individual’s fibrinogen level is below 200 mg/dL, which represents higher risk of bleeding after injury. +C. Baselines +This section provides more details on the baseline methods used in our experiments (Section 6). +C.1. Global feature importance methods +Two of our static feature selection baselines, permutation tests and SAGE, are global feature importance methods that rank +features based on their role in improving model accuracy (Covert et al., 2021). In our experiments, we ran each method +using a single classifier trained on the entire dataset, and we then selected the top k features depending on the budget. +When running the permutation test, we calculated the validation AUROC while replacing values in the corresponding feature +column with random draws from the training set. When running SAGE, we used the authors’ implementation with automatic +convergence detection (Covert et al., 2020). To handle held-out features, we averaged across 128 sampled values for the six +tabular datasets, and for MNIST we used a zeros baseline to achieve faster convergence. +C.2. Local feature importance methods +Two of our static feature selection baselines, DeepLift and Integrated Gradients, are local feature importance methods that +rank features based on their importance to a single prediction. In our experiments, we generated feature importance scores +for the true class using all examples in the validation set. We then selected the top k features based on their mean absolute +importance. We used a mean baseline for Integrated Gradients (Sundararajan et al., 2017), and both methods were run using +the Captum package (Kokhlikyan et al., 2020). +C.3. CMI estimation +Our experiments use two versions of the CMI estimation approach described in Section 3.2. Both are inspired by the +EDDI method introduced by Ma et al. (2019), but a key difference is that we do not jointly model (x, y) within the same +conditional generative model: we instead separately model the response with a classifier f(xs) ≈ p(y | xs) and the features +with a generative model of p(xi | xs). This partially mitigates one challenge with this approach, which is working with +mixed continuous/categorical data (i.e., we do not need to jointly model categorical response variables). +For the first version of this approach, we train a PVAE as a generative model (Ma et al., 2019). The encoder and decoder both +have two hidden layers, the latent dimension is set to 16, and we use 128 samples from the latent posterior to approximate +p(xi | xs) = +� +p(xi | z)p(z | xs). We use Gaussian distributions for both the latent and decoder spaces, and we generate +samples using the decoder mean, similar to the original approach (Ma et al., 2019). In the second version, we bypass the +need for a generative model with a simple approximation: we sample features from their marginal distribution, which is +equivalent to assuming feature independence. +C.4. Opportunistic learning +Kachuee et al. (2018) proposed Opportunistic Learning (OL), an approach to solve DFS using RL. The model consists +of two networks analogous to our policy and predictor: a Q-network that estimates the value associated with each action, +where actions correspond to features, and a P-network responsible for making predictions. When using OL, we use the same +architectures as our approach, and OL shares network parameters between the P- and Q-networks. +The authors introduce a utility function for their reward, shown in eq. (6), which calculates the difference in prediction + +Learning to Maximize Mutual Information for Dynamic Feature Selection +uncertainty as approximated by MC dropout (Gal & Ghahramani, 2016). The reward also accounts for feature costs, but we +set all feature costs to ci = 1: +ri = ||Cert(xs) − Cert(xs ∪ xi)|| +ci +(6) +To provide a fair comparison with the remaining methods, we made several modifications to the authors’ implementation. +These include 1) preventing the prediction action until the pre-specified budget is met, 2) setting all feature costs to be +identical, and 3) supporting pre-defined feature groups as described in Appendix D.3. When training, we update the P-, +Q-, and target Q-networks every 1 + +d +100 experiences, where d is the number of features in a dataset. In addition, the +replay buffer is set to store the 1000d most recent experiences, and the random exploration probability is decayed so that it +eventually reaches a value of 0.1. +D. Training approach and hyperparameters +This section provides more details on our training approach and hyperparameter choices. +D.1. Training pseudocode +Algorithm 1 summarizes our training approach. Briefly, we select features by drawing a Concrete sample using policy +network’s logits, we calculate the loss based on the subsequent prediction, and we then update the mask for the next step +using a discrete sample from the policy’s distribution. We implemented this approach using PyTorch (Paszke et al., 2017) +and PyTorch Lightning2. +Algorithm 1: Training pseudocode +Input: Data distribution p(x, y), budget k > 0, learning rate γ > 0, temperature τ > 0 +Output: Predictor model f(x; θ), policy model π(x; φ) +initialize f(x; θ), π(x; φ) +while not converged do +sample x, y ∼ p(x, y) +initialize L = 0, m = [0, . . . , 0] +for j = 1 to k do +calculate logits α = π(x ⊙ m; φ), sample Gi ∼ Gumbel for i ∈ [d] +set ˜m = max +� +m, softmax(G + α, τ) +� +// update with Concrete +set m = max +� +m, softmax(G + α, 0) +� +// update with one-hot +update L ← L + ℓ +� +f(x ⊙ ˜m; θ), y +� +end +update θ ← θ − γ∇θL, φ ← φ − γ∇φL +end +return f(x; θ), π(x; φ) +One notable difference between Algorithm 1 and our objective L(θ, φ) in the main text is the use of the policy π(x; φ) for +generating feature subsets. This differs from eq. (5), which generates feature subsets using a subset distribution p(s). The +key shared factor between both approaches is that there are separate optimization problems over each feature set that are +effectively treated independently. For each feature set xs, the problem is the one-step-ahead loss, and it incorporates both +the policy and predictor as follows: +Ei∼π(xs;φ) +� +ℓ +� +f(xs ∪ xi; θ), y +�� +. +(7) +The problems for each subset do not interact: during optimization, the selection given xs is based only on the immediate +change in the loss, and gradients are not propagated through multiple selections as they would be for an RL-based solution. +In solving these multiple problems, the difference is simply that eq. (5) weights them according to p(s), whereas Algorithm 1 +weights them according to the current policy π(x, φ). +2https://www.pytorchlightning.ai + +Learning to Maximize Mutual Information for Dynamic Feature Selection +D.2. Hyperparameters +Our experiments with the six tabular datasets all used fully connected architectures with dropout in all layers (Srivastava +et al., 2014). The dropout probability is set to 0.3, the networks have two hidden layers of width 128, and we performed +early stopping using the validation loss. For our method, the predictor and policy were separate networks with identical +architectures. When training models with the features selected by static methods, we reported results using the best model +from multiple training runs based on the validation loss. We did not perform any additional hyperparameter tuning due to +the large number of models being trained. +For MNIST, we used fully connected architectures with two layers of width 512 and the dropout probability set to 0.3. +Again, our method used separate networks with identical architectures. For CIFAR-10, we used a shared ResNet backbone +(He et al., 2016b) consisting of several residually connected convolutional layers. The classification head consists of global +average pooling and a linear layer, and the selection head consisted of a transposed convolution layer followed by a 1 × 1 +convolution, which output a grid of logits with size 8 × 8. Our CIFAR-10 networks are trained using random crops and +random horizontal flips as augmentations. +D.3. Feature grouping +All of the methods used in our experiments were designed to select individual features, but this is undesirable when using +categorical features with one-hot encodings. Each of our three emergency medicine tasks involve such features, so we +extended each method to support feature grouping. +SAGE and permutation tests are trivial to extend to feature groups: we simply removed groups of features rather than +individual features when calculating importance scores. For DeepLift and Integrated Gradients, we used the summed +importance within each group, which preserves each method’s additivity property. For the method based on Concrete +Autoencoders, we implemented a generalized version of the selection layer that operates on feature groups. We also extended +OL to operate on feature groups by having actions map to groups rather than individual features. +Finally, for our method, we parameterized the policy network π(x; φ) so that the number of outputs is the number of groups +g rather than the total number of features d (where g < d). When applying masking, we first generate a binary mask +m ∈ [0, 1]g, and we then project the mask into [0, 1]d using a binary group matrix G ∈ {0, 1}d×g, where Gij = 1 if feature +i is in group j and Gij = 0 otherwise. Thus, our masked input vector is given by x ⊙ (Gm). +E. Additional results +This section provides several additional experimental results. First, Figure 4 and Figure 5 show the same results as Figure 2 +but larger for improved visibility. Next, Figure 6 though Figure 11 display the feature selection frequency for each of the +tabular datasets when using the greedy method. The heatmaps in each plot show the portion of the time that a feature (or +feature group) is selected under a specific feature budget. These plots reveal that our method is indeed selecting different +features for different samples. +Finally, Figure 12 displays examples of CIFAR-10 predictions given different numbers of revealed patches. The predictions +generally become relatively accurate after revealing only a small number of patches, reflecting a similar result as Figure 3. +Qualitatively, we can see that the policy network learns to select vertical stripes, but the order in which it fills out each stripe +depends on where it predicts important information may be located. + +Learning to Maximize Mutual Information for Dynamic Feature Selection +0 +5 +10 +15 +20 +25 +# Selected Features +0.55 +0.60 +0.65 +0.70 +0.75 +AUROC +Bleeding AUROC Comparison +0 +5 +10 +15 +20 +25 +# Selected Features +0.65 +0.70 +0.75 +0.80 +0.85 +AUROC +Respiratory AUROC Comparison +2 +4 +6 +8 +10 +# Selected Features +0.700 +0.725 +0.750 +0.775 +0.800 +0.825 +0.850 +0.875 +AUROC +Fluid AUROC Comparison +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +IntGrad +DeepLift +SAGE +Perm Test +CAE +Opportunistic (OL) +CMI (Marginal) +CMI (PVAE) +Greedy (Ours) +Figure 4. AUROC comparison on the three emergency medicine diagnosis tasks. + +Learning to Maximize Mutual Information for Dynamic Feature Selection +0 +5 +10 +15 +20 +25 +# Selected Features +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +AUROC +Spam AUROC Comparison +0 +5 +10 +15 +20 +25 +# Selected Features +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +AUROC +MiniBooNE AUROC Comparison +2 +4 +6 +8 +10 +# Selected Features +0.75 +0.80 +0.85 +0.90 +0.95 +AUROC +Diabetes AUROC Comparison +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +IntGrad +DeepLift +SAGE +Perm Test +CAE +Opportunistic (OL) +CMI (Marginal) +CMI (PVAE) +Greedy (Ours) +Figure 5. AUROC comparison on the three public tabular datasets. + +Learning to Maximize Mutual Information for Dynamic Feature Selection +0 +5 +10 +15 +20 +25 +30 +35 +40 +Feature Index +0 +5 +10 +15 +20 +25 +# Selections +Bleeding Feature Selection Frequency +0.0 +0.2 +0.4 +0.6 +0.8 +Figure 6. Feature selection frequency for our greedy approach on the bleeding dataset. +0 +5 +10 +15 +20 +25 +30 +Feature Index +0 +5 +10 +15 +20 +25 +# Selections +Respiratory Feature Selection Frequency +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Figure 7. Feature selection frequency for our greedy approach on the respiratory dataset. +0 +20 +40 +60 +80 +100 +120 +140 +160 +Feature Index +0 +5 +10 +15 +20 +25 +# Selections +Fluid Feature Selection Frequency +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Figure 8. Feature selection frequency for our greedy approach on the fluid dataset. + +Learning to Maximize Mutual Information for Dynamic Feature Selection +0 +10 +20 +30 +40 +50 +Feature Index +0 +5 +10 +15 +20 +25 +# Selections +Spam Feature Selection Frequency +0.0 +0.2 +0.4 +0.6 +0.8 +Figure 9. Feature selection frequency for our greedy approach on the spam dataset. +0 +10 +20 +30 +40 +Feature Index +0 +5 +10 +15 +20 +25 +# Selections +MiniBooNE Feature Selection Frequency +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Figure 10. Feature selection frequency for our greedy approach on the MiniBooNE dataset. +0 +5 +10 +15 +20 +25 +30 +35 +40 +Feature Index +0 +5 +10 +15 +20 +25 +# Selections +Diabetes Feature Selection Frequency +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Figure 11. Feature selection frequency for our greedy approach on the diabetes dataset. + +Learning to Maximize Mutual Information for Dynamic Feature Selection +Full Image +Horse +Automobile +Truck +Cat +Dog +Frog +Ship +1 Patches +Prob = 8.04% +Prob = 27.61% +Prob = 12.60% +Prob = 12.08% +Prob = 69.06% +Prob = 33.50% +Prob = 40.99% +2 Patches +Prob = 19.55% +Prob = 94.60% +Prob = 3.71% +Prob = 19.11% +Prob = 85.20% +Prob = 78.33% +Prob = 52.80% +5 Patches +Prob = 48.14% +Prob = 99.99% +Prob = 16.02% +Prob = 27.03% +Prob = 99.98% +Prob = 94.11% +Prob = 94.02% +10 Patches +Prob = 76.57% +Prob = 99.97% +Prob = 94.65% +Prob = 41.52% +Prob = 99.99% +Prob = 99.75% +Prob = 82.71% +15 Patches +Prob = 92.00% +Prob = 100.00% +Prob = 88.88% +Prob = 72.54% +Prob = 99.97% +Prob = 99.90% +Prob = 98.36% +20 Patches +Prob = 81.35% +Prob = 100.00% +Prob = 96.01% +Prob = 79.03% +Prob = 99.93% +Prob = 99.89% +Prob = 99.90% +25 Patches +Prob = 97.02% +Prob = 100.00% +Prob = 96.34% +Prob = 75.32% +Prob = 99.91% +Prob = 99.56% +Prob = 99.88% +30 Patches +Prob = 91.91% +Prob = 100.00% +Prob = 96.29% +Prob = 66.15% +Prob = 99.78% +Prob = 99.35% +Prob = 99.86% +Figure 12. CIFAR-10 predictions with different numbers of patches revealed by our approach. + +1 \ No newline at end of file diff --git a/FdAyT4oBgHgl3EQfrPl7/content/tmp_files/load_file.txt b/FdAyT4oBgHgl3EQfrPl7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e0af3481557543ba19c00a94caff40a431f917be --- /dev/null +++ b/FdAyT4oBgHgl3EQfrPl7/content/tmp_files/load_file.txt @@ -0,0 +1,1501 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf,len=1500 +page_content='Learning to Maximize Mutual Information for Dynamic Feature Selection Ian Covert 1 Wei Qiu 1 Mingyu Lu 1 Nayoon Kim 1 Nathan White 2 Su-In Lee 1 Abstract Feature selection helps reduce data acquisition costs in ML, but the standard approach is to train models with static feature subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=' Here, we con- sider the dynamic feature selection (DFS) prob- lem where a model sequentially queries features based on the presently available information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=' DFS is often addressed with reinforcement learning (RL), but we explore a simpler approach of greed- ily selecting features based on their conditional mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=' This method is theoretically appealing but requires oracle access to the data distribution, so we develop a learning approach based on amortized optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=' The proposed method is shown to recover the greedy policy when trained to optimality and outperforms nu- merous existing feature selection methods in our experiments, thus validating it as a simple but powerful approach for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=' Introduction A machine learning model’s inputs can be costly to obtain, and feature selection is often used to reduce data acquisition costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=' In applications where information is gathered sequen- tially, a natural option is to select features adaptively based on the currently available information rather than using a fixed feature set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=' This setup is known as dynamic feature selection (DFS)1, and the problem has been considered by several works in the last decade (Saar-Tsechansky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=' Dulac-Arnold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=', 2015b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=' Early et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=', 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=', 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=' Kachuee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=' Compared to static feature selection with a fixed feature set (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=' Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=', 2018), DFS can offer better performance given a fixed budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=' This is easy to see, be- cause selecting the same features for all instances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=', all 1Paul G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=' Allen School of Computer Science & Engineering, University of Washington 2Department of Emergency Medicine, University of Washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfrPl7/content/2301.00557v1.pdf'} +page_content=' Correspondence to: Ian Covert > TN). At temperatures above the magnetic or- +der, we have analyzed the spin-spin correlations result- +ing in a development of slowly fluctuating short-range or- +der. They are much stronger pronounced in MnNiP2S6 +arXiv:2301.04239v1 [cond-mat.str-el] 10 Jan 2023 + +2 +compared to Mn2P2S6, which in our previous study has +shown clear cut signatures of 2D correlated spin dy- +namics [25]. +Therefore, enhanced spin fluctuations in +MnNiP2S6 are attributed to the competition of differ- +ent types of magnetic order. Finally, the analysis of the +temperature dependent critical behavior of the magnon +gaps below the ordering temperature in Mn2P2S6 sug- +gest that the character of the spin wave excitations in +this compound undergoes a field induced crossover from +a 3D-like towards 2D XY regime. +II. +EXPERIMENTAL DETAILS +Crystal growth of Mn2P2S6 and MnNiP2S6 samples +investigated in this work was done using the chemical +vapor transport technique with iodine as the transport +agent. Details of their growth, crystallographic, composi- +tional and static magnetic characterization are described +in Refs. [26, 27]. Note, that the experimental value xexp +in (Mn1−xNix)2P2S6 for the nominal MnNiP2S6 com- +pound is found to be xexp = 0.45, considering an uncer- +tainty of approximately 5% [27]. Both materials exhibit +a monoclinic crystal lattice system with a C2/m space +group [27, 28]. Each unit cell contains a [P2S6]4− cluster +with S atoms occupying the edges of TM octahedra and +P-P dumbbells occupying the void of each metal honey- +comb sublattice. The crystallographic c-axis makes an +angle of 17◦ with the normal to the ab-plane [29], which +is known to be one of the magnetic axes and is hereafter +called as c* [18]. +The ordering temperature of Mn2P2S6 is found to be +TN = 77 K [27]. +In contrast, the transition tempera- +ture of MnNiP2S6 is rather uncertain and might depend +on the direction of the applied magnetic field. Various +studies have reported different values of TN, for instance +it amounts to 12 K in [30], 38 K in [27], 41 K in [31] +and 42 K in [32]. +For the samples used in this study +the ordering temperatures were extracted from the tem- +perature dependence of the susceptibility χ measured at +H = 1000 Oe (see appendix, Fig. 10 (a)). The calcula- +tion of the maximum value of the derivative d(χ · T)/dT +yields TN ∼ 57 K for H ∥ c* and TN ∼ 76 K for H ⊥ c* +(hereafter called TN*). +The antiferromagnetic resonance (AFMR) and ESR +measurements (hereafter called HF-ESR) were performed +on several single crystalline samples of Mn2P2S6 and +MnNiP2S6 using a homemade HF-ESR spectrometer. A +superconducting magnet from Oxford instruments with +a variable temperature insert (VTI) was used to gener- +ate magnetic fields up to 16 T allowing a continuous field +sweep. The sample was mounted on a probe head which +was then inserted into the VTI immersed in a 4He cryo- +stat. A piezoelectric step-motor based sample holder was +used for angular dependent measurements. Continuous +He gas flow was utilized to attain stable temperatures +in the range of 3 to 300 K. Generation and detection of +microwaves was performed using a vector network an- +6 +7 +8 +9 10 11 12 13 +SD (arb. u.) +Magnetic Field (T) + b) MnNiP2S6, � = 326 GHz +300 K +250 K +200 K +175 K +150 K +140 K +120 K +100 K +90 K +80 K +70 K +60 K +50 K +45 K +40 K +35 K +30 K +20 K +10 K +7.5 K +5 K +3 K +2 +4 +6 +B5 +SD (arb. u.) +Magnetic Field (T) +70 K +65 K +60 K +90 K +80 K +75 K +* +** +* +* +* +* +**** +* +50 K +40 K +30 K +20 K +10 K +7.5 K +5 K +3 K +293 K +250 K +194 K +171 K +150 K +130 K +110 K +100 K +B4 +a) Mn2P2S6, � = 147 GHz +FIG. 1. Temperature dependence of HF-ESR spectra of (a) +Mn2P2S6 at fixed excitation frequency ν ≈ 147 GHz and (b) +MnNiP2S6 at ν ≈ 326 GHz in H ∥ c* configuration. Spectra +are normalized and vertically shifted for clarity. The temper- +ature independent peaks from the impurity in the probehead +occurring at low frequencies are marked with asterisks. +alyzer (PNA-X) from Keysight Technologies. Equipped +with the frequency extensions from Virginia Diodes, Inc., +the PNA-X can generate a frequency in the range from +75 to 330 GHz. The measurements are performed in the +transmission mode, where the microwaves are directed +to the sample using oversized waveguides. All measure- +ments were made by sweeping the field from 0 to 16 T +and back to 0 T at constant temperature and frequency. +HF-ESR signals generally have a Lorentzian line pro- +file with an absorption and dispersion components. For +such a case, the resonance field (Hres) and linewidth (full +width at half maxima, ∆H) can be extracted by fitting +the signal using the function: +SD(H) = 2Amp +π +× (L1sinα + L2cosα) ++ Coffset + CslopeH +(1) +where SD(H) is the signal at the detector and Amp is +the amplitude. Coffset represents the offset and CslopeH +is the linear background of the spectra. +L1 is the +Lorentzian absorption which is defined in terms of Hres +and ∆H. L2 is the Lorentzian dispersion which is ob- +tained by applying the Kramers-Kronig transformation +to L1. α is a parameter used to define the degree of in- +strumental mixing of the absorption and dispersion com- +ponents which is unavoidable in the used setup. Some +of the HF-ESR signals of Mn2P2S6 could not be fitted +using the above equation due to the development of the +shoulders or the splitting of peaks [33]. ∆H, Hres and, + +3 +0 +50 +100 +150 +200 +250 +300 +-4 +-2 +0 +2 +4 +6 +� H = Hres(T) - Hres(300 K) (T) +Temperature (K) + 88 GHz, H || c* + 88 GHz, H || c* + 147 GHz, H || c* + 147 GHz, H || c* + 329 GHz, H || c* + 329 GHz, H ^ c* +TN +Mn2P2S6 +B5 +B4 +B3 +0 +100 +200 +300 +0.05 +0.10 +0.15 +0.20 +0.25 +� = 329 GHz + H || c* + H ^ c* +∆H = Linewidth (T) +Temperature (K) +TN +100 +150 +200 +250 +300 +-0.1 +0.0 +0.1 +FIG. 2. Shift of the resonance field position δH (main panel) +and linewidth ∆H (inset) as a function of temperature. The +horizontal dashed line represents zero shift from the room +temperature value and the vertical dashed line (also for inset) +represents the N´eel temperature of the material. +therefore, δH = Hres - Hres(300 K) were then obtained +by picking a position of the peak value and by calculating +the full width at half maximum. +III. +RESULTS +A. +Temperature dependence of HF-ESR response +To study the temperature evolution of the spin dynam- +ics, the HF-ESR spectra were measured at several tem- +peratures in the range of 3 - 300 K and at few selected +microwave excitation frequencies ν. +Such dependences +measured in the H ∥ c* configuration at ν = 147 GHz +for Mn2P2S6 and at ν = 326 GHz for MnNiP2S6 are pre- +sented in Fig. 1. As can be seen, in the case of Mn2P2S6 +upon entering the ordered state with lowering tempera- +ture, the single ESR line transforms into two modes B4 +and B5 (see below) at ν = 147 GHz. The temperature +dependence of the spectra for other frequencies can be +found in Appendix in Fig. 9. +The shift of the obtained values of Hres from the +resonance field position at T = 300 K, δH = Hres - +Hres(300 K) is plotted as a function of temperature for +Mn2P2S6 and MnNiP2S6 in Fig. 2 and Fig. 3, respec- +tively. Hres(300 K) was calculated using the equation +hν = gµBµ0Hres, where the g-factor is obtained from +the frequency dependence of the resonance field at 300 K +(see Sec. III B). In the case of Mn2P2S6, δH stays practi- +cally constant down to T ∼ 130 − 150 K for both config- +urations H ∥ c* and H ⊥ c*. Below this temperature it +starts to slightly deviate (lower inset in Fig. 2), suggest- +ing a development of the static on the ESR time scale +0 +50 +100 +150 +200 +250 +300 +-4 +-3 +-2 +-1 +0 + H || c* + H ⊥ c* +� H = Hres (T) - Hres (300 K) (T) +Temperature (K) +MnNiP2S6, � = 326 GHz +TN TN* +0 +100 +200 +300 +0 +1 +2 +3 +� H = Linewidth (T) +Temperature (K) +TN* +TN +FIG. 3. Temperature dependence of δH (main panel) and ∆H +(inset) measured at ν = 325.67 GHz. The horizontal dashed +line represents zero shift from room temperature value and +the vertical dashed line (also for inset) represents the N´eel +temperature of the material. +internal fields. In contrast, the deviations of δH from +zero value in MnNiP2S6 are larger, and are observed at +a higher temperature T ∼ 200 K. In the vicinity of the +ordering temperature TN* there is a strong shift of the +ESR line, observed for both compounds. In the Mn2P2S6 +case the sign of δH below the ordering temperature de- +pends on the particular AFMR mode, which is probed at +the specific frequency. This is detailed in the following +Sec. III C. +Insets of Fig. 2 and Fig. 3 represent the evolution +of the linewidth ∆H as a function of temperature for +Mn2P2S6 and MnNiP2S6 compounds, respectively. +At +T > TN, ∆H remains practically temperature indepen- +dent for both compounds. +A small broadening of the +line is observed in the vicinity of the phase transition +temperature, and there is a drastic increase of ∆H in +the ordered state. Note, that ∆H of MnNiP2S6 is larger +than that of Mn2P2S6 in the whole temperature range. +Moreover, for MnNiP2S6, ∆H increases at low tempera- +tures by almost one order of magnitude from 0.3 to 3 T +(inset in Fig. 3). Such extensive line broadening at low +temperatures hampers the accurate determination of the +linewidth and resonance field, which is accounted for in +the error bars. +B. +Frequency dependence at 300 K +The frequency dependence of the resonance field +ν(Hres) of Mn2P2S6 and MnNiP2S6 compounds mea- +sured in the paramagnetic state at T = 300 K is shown in +Fig. 4. Both plots have a linear dependence which can be +fitted with the conventional paramagnetic resonance con- + +4 +0 +2 +4 +6 +8 +10 12 +g|| = 2.026 ± 0.002 +g⊥ = 2.047 ± 0.004 +SD (arb. u.) +0 +2 +4 +6 +8 +10 12 +0 +50 +100 +150 +200 +250 +300 +350 +a) Mn2P2S6 +H || c* +H ⊥ c* +Frequency (GHz) +Resonance Field (T) +g|| = 1.992 ± 0.001 +g⊥ = 1.999 ± 0.001 +H || c* +H ⊥ c* +b) MnNiP2S6 +FIG. 4. +ν(Hres) dependence measured at 300 K for (a) +Mn2P2S6 and (b) MnNiP2S6. Blue squares represent H ∥ c* +configuration and the red circles represent H ⊥ c* con- +figuration. +Solid lines show the results of the fit accord- +ing to the resonance condition of a conventional paramagnet +hν = gµBµ0Hres. Right vertical axis: Representative spectra +normalized for clarity. The color of the spectra corresponds +to the color of the data points in the ν(Hres) plot with the +same Hres. +dition for a gapless excitation hν = gµBµ0Hres. Here, +h is the Plank constant, µB is the Bohr magneton, µ0 is +the permeability of free space and g is the g-factor of res- +onating spins. For Mn2P2S6, we obtain almost isotropic +values of the g-factor: g∥ = 1.992 ± 0.001 (H ∥ c*) and +g⊥ = 1.999 ± 0.001 (H ⊥ c*), which is expected for a +Mn2+ ion [34]. +In contrast, MnNiP2S6 shows a small +anisotropy of g-factors with g∥ = 2.026 ± 0.002 and g⊥ += 2.047 ± 0.004. In case of Ni2+ ions (3d8, S = 1), g- +factors are expected to be appreciably greater than free +spin value, as is revealed in HF-ESR studies on Ni2P2S6 +[24]. +C. +Frequency dependence at 3 K +1. +Mn2P2S6 +The low temperature resonance modes of Mn2P2S6 ob- +tained at T = 3 K are plotted in Fig. 5. The measure- +ments along the H ∥ c* configuration (Fig. 5) yield three +branches B3, B4 and B5, two of which (B3 and B4) +are observed below the spin-flop field, HSF = 3.62 T. +Branches B1 and B2 are assigned to the measurements +along a- and b-axis, respectively [35]. +Additionally, at +the spin-flop field, a non-resonance absorption peak (full +circles) was observed at high frequencies. +The exact gap values are calculated by fitting the in- +plane resonance branches B1 and B2 using the analytical +expressions for easy-axis AFMs [36]: +hν = [(g⊥µBµ0Hres)2 + ∆2 +1,2]1/2. +(2) +Here ∆1 corresponds to the magnon excitation gap for +branch B2 (also B3), and ∆2 corresponds to B1 (also +B4). The obtained values are ∆1 = ∆Mn2P2S6 +1 += 101.3 ± +0 +2 +4 +6 +8 +10 +12 +0 +50 +150 +200 +250 +300 +350 +Frequency (GHz) +Resonance Field (T) +Spin Flop +Mn2P2S6, T = 3 K +116 +101 +AFM Branch, H || a*-axis +AFM Branch, H || c*-axis +AFM Branch, H || b*-axis +B1 +B2 +B3 +B4 +B5 +H || a +H || b +H || c* +Paramagnetic branch +SD (arb. u.) +FIG. 5. +ν(Hres) dependence of HF-ESR signals measured +at T = 3 K (symbols). +Solid lines are the fit to the phe- +nomenological equations as explained in the text. The dash +gray lines correspond to the frequencies at which temperature +dependent measurements were performed. The dash line in +magenta represents the paramagnetic branch. Right vertical +scale: Normalized ESR spectra for selected frequencies. For +clarity the spectra are shifted vertically. +Error bars in the +Hres are smaller than the symbol size. +0.6 GHz and ∆2 = ∆Mn2P2S6 +2 += 116 ± 2 GHz. +These +values, which agree well with previous measurements by +Okuda et al. [14] and Kobets et al. [18], are then used +in the theoretical description for a rhombic biaxial two- +lattice AFM [18, 37] to match the field dependence of B3 +and B4 [38]: +ν = gµBµ0 +2h +× +� +∆2 +1 + ∆2 +2 + 2H2 +res± +± +� +8H2res(∆2 +1 + ∆2 +2) + (∆2 +1 + ∆2 +2)2 +�1/2 +. (3) +Above the spin-flop field the above model can not be +used to describe the system. Therefore branch B5 [38] +was simulated by the resonance condition of a conven- +tional easy-axis AFM [36]: +hν = [(g∥µBµ0Hres)2 − ∆2 +1]1/2. +(4) +The presence of the second easy-axis within the ab- +plane is further confirmed by the angular dependence of +Hres(θ) in the H ⊥ c* configuration (Fig. 6). It follows +a A + Bsin2(θ) law, which suggests a 180◦ periodicity of +Hres(θ). θ denotes the angle between the applied field +and a-axis. For a honeycomb spin system with a N´eel +type arrangement, a six-fold periodicity of angular de- +pendence in the layer plane can be expected. However, +this is absent in the case of Mn2P2S6 sample due to dom- +inating effects of a two-fold in-plane anisotropy. + +5 +-120 -90 -60 -30 +0 +30 +60 +90 +120 150 180 210 240 +3.90 +3.95 +4.00 +4.05 +4.10 +4.15 +4.20 +4.25 +4.30 +4.35 +4.40 +4.45 +Resonance Field (T) +Theta (°) +Mn2P2S6 +H ⊥ c*, T = 3 K +� = 159.9 GHz +Model +pi_periodicity (User) +Equation +A + B * sin(x*pi/180 + C)^2 +Plot +Resonance Field +A +3.96068 ± 0.01024 +B +0.43992 ± 0.01675 +C +0.03046 ± 0.01979 +Reduced Chi-Sqr +8.35579E-4 +R-Square (COD) +0.97185 +Adj. R-Square +0.96903 +H || a +H || b +FIG. 6. +Resonance field as a function of angle θ at T = +3 K and ν = 160 GHz for Mn2P2S6. +θ denotes the angle +between the direction of the field applied along the ab-plane +and the a-axis. Red dash line represents the result of the fit, +as explained in the text. +To further analyze the measured ν(Hres) dependence +of the AFMR modes in the magnetically ordered state of +Mn2P2S6, that correspond to the collective excitations +of the spin lattice (spin waves), we employed a linear +spin wave theory (LSWT) with the second quantization +formalism [36, 39]. The details of our model are provided +in Ref. [40]. The phenomenological Hamiltonian for the +two-sublattice spin system, used for calculations of the +spin waves energies, has the following form: +H = A(M1M2) +M 2 +0 ++ Kuniax +M1z +2 + M2z +2 +M 2 +0 ++ Kbiax +2 +(M 2 +1x − M 2 +1y) + (M 2 +2x − M 2 +2y) +M 2 +0 +− (HM1) − (HM2) . +(5) +Here the first term represents the exchange interaction +between the magnetic sublattices with respective magne- +tizations M1 and M2, such that M 2 +1 = M 2 +2 = (M0)2 = +(Ms/2)2, with M 2 +s being the square of the saturation +magnetization. A is the mean-field antiferromagnetic ex- +change constant. The second term in Eq. (5) is the uni- +axial part of the magnetocrystalline anisotropy given by +the anisotropy constant Kuniax. The third term describes +an additional anisotropy in the xy-plane with the respec- +tive constant Kbiax. The fourth and fifth terms are the +Zeeman interactions for both sublattice magnetizations. +The results of the calculation match well the measured +data. In the calculation we assumed a full Mn satura- +tion moment of ∼ 5µB, yielding Ms = 446 erg/(G·cm3) += 446 · 103 J/(T·m3), considering 4 Mn ions in the unit +cell. The average g-factor value of 1.995 was taken from +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 + AFM Branch, H || c* + AFM Branch, H ⊥ c* + Paramagnetic branch + H || c* + H ⊥ c* +Resonance Field (T) +Frequency (GHz) +MnNiP2S6, +T = 3 K +SD (arb. u.) +FIG. 7. ν(Hres) dependence measured at 3 K for both con- +figurations of magnetic field. Right vertical scale: Exemplary +spectra positioned above the resonance points. The horizontal +dash gray line represents the frequency at which the temper- +ature dependence was measured. The dash line in magenta +depicts the paramagnetic resonance branch at 300 K. +the frequency dependence measurements at T = 300 K +(Fig. 4). +As the result we obtain the exchange con- +stant A = 2.53 · 108 erg/cm3 = 2.53 · 107 J/m3, uni- +axial anisotropy constant Kuniax = −7.2 · 104 erg/cm3 += −7.2 · 103 J/m3, and an in-plane anisotropy constant +Kbiax = 1.9 · 104 erg/cm3 = 1.9 · 103 J/m3. Within the +mean-field theory A is related to the Weiss constant +Θ = A ∗ C/M 2 +0 , where C is the Curie constant. +Θ, +that provides an average energy scale for the exchange +interaction in the system, amounts therefore at least +ΘMn2P2S6 ≈ 350 K. +2. +MnNiP2S6 +In the case of MnNiP2S6 we observe one branch for +H ∥ c* and another one for H ⊥ c* configuration, re- +spectively, as shown in Fig. 7. +HF-ESR spectra were +also recorded at various angles for the in-plane orienta- +tion. Within the experimental error bars of ∼ 300 mT, no +signatures for an in-plane anisotropy were observed (see +Fig. 10 in Appendix). Both branches follow the resonance +condition for a hard direction of an AFM given by Eq. (2), +which reveals that neither c*-axis nor the ab-plane are +energetically favorable. The magnitude of the gap was +obtained from the fit as ∆MnNiP2S6 +1 += 115 ± 9 GHz for +H ∥ c* and ∆MnNiP2S6 +2 += 215 ± 1 GHz for H ⊥ c* config- +urations, respectively. +Unfortunately, we could not find a good matching of +the calculated frequency dependence to the one mea- +sured at low temperature (Fig. 7) with the AFM Hamil- +tonian for a two sublattice model. +Inclusion of the + +6 +terms describing cubic, hexagonal and symmetric ex- +change anisotropies in addition to those given in Eq. (5) +did not yield a good result either. +This could be ex- +plained by the complicated type of order of two mag- +netically inequivalent ions Mn2+ (S = +5 +2, g = 1.955) +and Ni2+ (S = 1, g = 2.17), which possibly requires a +more sophisticated model than the one used in this study. +The analysis might be even more complicated by poten- +tial disorder in the system due to the stochastic distribu- +tion of these ions on the 4g Wyckoff sites. Therefore the +full description of this system remains an open question. +However, one could draw some conclusions by analyzing +how the magnetization measured at low-T depends on the +Mn/Ni ratio [27]. The reduction of the magnetization +measured at low-T can be explained by the the reduc- +tion of the total moment per formula unit of MnNiP2S6, +which can be found as an average of the Mn and Ni sat- +uration magnetizations and amounts to ∼ 7.2 µB, com- +pared to Mn2P2S6 which has the saturation moment of +∼ 10 µB. Additionally, an almost isotropic behavior of +the magnetization as a function of magnetic field (inset of +Fig. 10 (a)) suggests that the isotropic exchange energy +is by orders of magnitude the strongest term defining the +static magnetic properties of MnNiP2S6. In this case, the +magnetization value, measured at the magnetic field ap- +plied along some hard direction, should be inversely pro- +portional to the mean-field isotropic exchange constant +M ∼ H/A. +The reduced magnetization in MnNiP2S6 +suggests, therefore, that Θ ∼ A should be at least as large +in MnNiP2S6 (ΘMnNiP2S6 ≥ +∼ 350 K) as in Mn2P2S6 +(ΘMn2P2S6 ≈ 350 K, see Sec. III C 1). +IV. +DISCUSSION +A. +Spin-Spin correlations in (Mn1−xNix)2P2S6 +(T > TN*) +As has been shown in our previous work, both the +resonance field and the linewidth of the HF-ESR signal +in Ni2P2S6 remain temperature independent by cooling +the sample down to temperatures close to TN [24]. Usu- +ally, in the quasi-2D spin systems the ESR line broad- +ening and shift occur at T > TN due to the growth +of the in-plane spin-spin correlations resulting in a de- +velopment of slowly fluctuating short-range order [41]. +Specifically, the slowly fluctuating spins produce a static +on the ESR timescale field causing a shift of the reso- +nance line, and a distribution of these local fields and +shortening of the spin-spin relaxation time due to the +slowing down of the spin fluctuations increase the ESR +linewidth. +In the Mn2P2S6 compound these features +are not very pronounced, only in the resonance field of +the HF-ESR response one can detect within error bars +small deviations starting at T ∼ 130 − 150 K. In the +MnNiP2S6 compound, in turn, the critical broadening +and the shift of the resonance line are observed at tem- +perature T ∼ 200 K, which is much higher than TN. +Even though the critical broadening and the line shift +above TN are much stronger pronounced in MnNiP2S6, +our previous low-frequency ESR study shows that the +clear cut signatures of 2D correlated spin dynamics are +present above TN only in the Mn2P2S6 compound [25]. +Interestingly, these signatures, seen in the characteristic +angular dependence of the ESR linewidth, develop only +at elevated temperatures, where the effect of the strong +isotropic AFM coupling (ΘMn2P2S6 ≈ 350 K) on the +spin fluctuations becomes gradually suppressed. Critical +broadening and the shift of the ESR line in MnNiP2S6 +above TN could therefore be due to the stochastic dis- +tribution of Mn and Ni ions on the 4g Wyckoff sites of +the crystal structure causing a competition of different +order types with contrasting magnetic anisotropies. Our +conclusion on the drastic difference in the ground states +is supported by the strong distinction in the energy gaps +and magnetic field dependences of the low-T spin wave +excitations in Mn2P2S6, MnNiP2S6 and Ni2P2S6, respec- +tively. +The competing types of magnetic order might +enhance spin fluctuations seen in the HF-ESR response +at elevated temperatures. Strong fluctuations suppress, +in turn, the ordering temperature for MnNiP2S6 which is +evident in the recent studies on the (Mn1−xNix)2P2S6 se- +ries [27, 30, 32]. Moreover, in this scenario of the stochas- +tic distribution of Mn and Ni, small deviation of the sto- +ichiometry from sample to sample of the same nominal +composition could vary the ordering temperature, which +explains the broad range of TN measured in MnNiP2S6 +samples [27, 30, 32]. +B. +Ground state and anisotropy of +(Mn1−xNix)2P2S6 (T << TN*) +At the lowest measurement temperature Mn2P2S6 has +an antiferromagnetic ground state with biaxial type of +anisotropy, and the spin wave excitations can be suc- +cessfully modeled using LSWT. As the result we obtain +the estimation of the exchange interaction ΘMn2P2S6 ≈ +350 K and the parameters of the anisotropy Kuniax = +−7.2 · 104 erg/cm3 = −7.2 · 103 J/m3 and Kbiax = 1.9 · +104 erg/cm3 = 1.9 · 103 J/m3. There is only about four +times difference between Kuniax and Kbiax, which sug- +gests that the anisotropy in the ab-plane makes a sig- +nificant contribution to the properties of the ground +state of Mn2P2S6. +Interestingly, the value of Kbiax = +1.9 · 103 J/m3 ≈ 2 · 10−25 J/spin is very close to the es- +timation of the anisotropy within the ab-plane made by +Goossens [42], suggesting a possible dipolar nature of this +anisotropy. In the MnNiP2S6 case we could not find an +appropriate Hamiltonian within a two sublattice model +which would fully describe the system, calling for a more +sophisticated theoretical study. Interestingly, the charac- +teristic feature of the MnNiP2S6 compound is the almost +isotropic dependence of the magnetization as a function +of magnetic field, measured at temperature well below TN +[27]. The isothermal magnetization measurements made + +7 +on the sample used in this study, confirm the presence +of this almost isotropic static magnetic response (see ap- +pendix Fig. 10 (a)). Such an isotropic behavior of the +static magnetization is related to the strong isotropic +AFM exchange interaction (ΘMnNiP2S6 ≥ +∼ 350 K), +which is larger than the applied magnetic field and the +observed magnetic anisotropy in this system. However, +the HF-ESR data reveals a substantial anisotropy in the +magnetic field dependence of the spin waves. This seem- +ing contradiction is actually not surprising. The magne- +tization value at the magnetic field applied along some +hard direction is mostly given by the mean-field exchange +constant M ∼ H/A, whereas the magnon gap measured +in the ESR experiment is roughly proportional to the +square root of the product of exchange and magnetic +anisotropy constants [36]. +Qualitatively, the evolution of the type of magnetic +anisotropy with x in (Mn1−xNix)2P2S6 is also evident +from our study, where, e.g., MnNiP2S6 reveals no easy- +axis within or normal to the ab-plane. In order to quan- +tify the change of magnetic anisotropic properties with +the Mn/Ni content the excitation energy gaps can be +used. +The single gap of about 260 GHz was found in +our previous study on Ni2P2S6 [24]. Both Mn containing +compounds have two gaps ∆MnNiP2S6 +1 += 115±9 GHz and +∆MnNiP2S6 +2 += 215 ± 1 GHz in the case of MnNiP2S6, and +∆Mn2P2S6 +1 += 101.3±0.6 GHz and ∆Mn2P2S6 +2 += 116±2 GHz +in the case of Mn2P2S6. As can be seen, there is a no- +ticeable increase of the zero field AFM gaps in the sam- +ples with higher Ni content, suggesting an increase of the +magnetic anisotropy and exchange interaction. Indeed, +the estimated energy scale of the exchange interaction in +Mn2P2S6 is about ∼ 350 K, in MnNiP2S6 is more than +∼ 350 K, and it is even larger in Ni2P2S6, due to the +observation of the larger TN and as it is suggested by the +previous investigations [25, 43–45]. Mn2+ with the half +filled 3d electronic shell, and a small admixture of the +excited state 4P5/2 into the ground state 6S5/2 is an ion +with rather isotropic magnetic properties. In contrast, +the ground state of the Ni2+ ion in the octahedral envi- +ronment [8] is a spin triplet with the higher lying orbital +multiplets, admixed through the spin-orbit coupling [34], +which makes the Ni spin (S = 1) sensitive to the local +crystal field. +This, first, could increase a contribution +of the local (single ion) magnetic anisotropy term in the +Hamiltonian describing the system in the ordered and +in the paramagnetic state, as discussed for the case of +Ni2P2S6 in [24]. +Second, it could yield a deviation of +the g-factor from the free electron value and also induce +an effective g-factor anisotropy. +The effective g-factor +value and its anisotropy, as found in our study, increase +with Ni content. Deviation of the g-factor from the free +electron value (∆g) and the anisotropy of the exchange +originate from the spin-orbit coupling effect, and there- +fore are interrelated. In the case of symmetric anisotropic +exchange the elements of the anisotropic exchange ten- +sor are A ∝ (∆g/g)2J [46–48], where J is the isotropic +exchange interaction constant. Observation of increased +0.1 +1 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +B5, 329 GHz +B5, 147 GHz +B4, 147 GHz +B4, 88 GHz +∆(T)/∆(3 K) +1 - T/TN +b = 0.55 +b = 0.6 +b = 0.29 +b = 0.26 +collinear phase +spin-flop +phase +H||c* +0 +2 +4 +6 +8 +10 +12 +0 +50 +150 +200 +250 +300 +350 +100 +Frequency (GHz) +Resonance Field (T) +T = 3 K +H||c* +B1 +B2 +B3 +B4 +B5 +µ0Hsf +collinear + phase +spin-flop + phase +FIG. 8. Main panel: Temperature dependence of the normal- +ized energy gap ∆(T)/∆(3 K) = [1−(T/TN)]b for Mn2P2S6 at +different field regimes. Symbol shapes and colors correspond +to that in Fig. 2. Inset: Resonance branches at T = 3 K (solid +lines) as in Fig. 5. Symbols (same as in the main panel) indi- +cate the positions of the resonance modes B4 at 147 GHz and +B5 at 147 and 329 GHz. The position of mode B4 at 88 GHz +is not shown here since it can be detected at T ≥ 50 K only. +The temperature dependence of these modes shown in Fig. 2 +was used to estimate that of ∆(T). (see the text) +∆g at higher Ni content suggests that in the Ni contain- +ing (Mn1−xNix)2P2S6 the exchange anisotropy is likely +an important contributor to the anisotropic properties of +the ground state at low temperatures < TN, such as, e.g., +increased magnon gaps. +C. +Critical behavior of Mn2P2S6 (T ≲ TN*) +In the following we discuss the temperature depen- +dence of the excitation energy gap ∆ at finite magnetic +fields in the collinear and the spin-flop AFM ordered +phases of Mn2P2S6, at H < Hsf and H > Hsf, re- +spectively. This should provide useful insights onto the +type of the critical behavior of the Mn spin lattice at +T < TN. +Such a dependence can be obtained by an- +alyzing the temperature dependence of the shift of the +resonance field positions Hres(T) of the excitation modes +B4 and B5 for H ∥ c* (Fig. 2) with the aid of the sim- +plified relations ∆ ≈ hν − g∥µBµ0Hres for mode B4 and +∆ ≈ [(g∥µBµ0Hres)2 − (hν)2]1/2 for mode B5 derived +from Eqs. (3) and (4), respectively. +The result of this analysis is shown in Fig. 8. +The +∆(T) dependence can be well fitted to the power law +∆(T) ∝ [1 − (T/TN)]b in a broad temperature range be- +low TN with some deviations from it at lower T. The +exponents b indicated in this Figure appear to be very +different for modes B4 and B5. Notably, the resonance + +8 +field of mode B4 is always smaller than the spin-flop +field, HB4 +res |88 GHz< HB4 +res |147 GHz< Hsf, whereas mode +B5 occurs at larger fields with Hsf < HB5 +res |145 GHz< +HB5 +res |329 GHz [Fig. 8(inset)]. This suggests a significant +difference in the temperature dependence of the exci- +tation gap in the collinear and spin-flop AFM ordered +phases of Mn2P2S6. +Usually, the magnetic anisotropy gap ∆(T) observed +in quasi-2D antiferromagnets scales with the sublattice +magnetization Msl(T) [49–51] so that the exponent b of +the temperature dependence of ∆ can be treated as a crit- +ical exponent β of the AFM order parameter Msl. If that +were the case for Mn2P2S6, the value of b in the collinear +phase would indicate the mean-field behavior of Msl(T) +for which β = 0.5 (Fig. 8). In contrast, a strong reduction +of b in the spin-flop phase, as seen in Fig. 8, would cor- +respond to the critical behavior of Msl(T) in the 2D XY +model for which β = 0.231 [52]. However, measurements +of the temperature dependence of Msl by elastic and of +∆ by inelastic neutron scattering in zero magnetic field +reveal a more complex scaling between these two param- +eters with b ≈ 3β/2 and β = 0.32 in the vicinity of TN, +and b ≈ β with β = 0.25 at lower temperatures [53–55]. +This finding was tentatively ascribed to different tem- +perature dependence of the competing single-ion and +dipolar anisotropies which are both responsible for a fi- +nite value of ∆ in the AFM ordered state of Mn2P2S6 +[53]. Theoretical analysis in Ref. [42] shows that besides +the dipolar anisotropy which is responsible for the out- +of-plane order of the Mn spins there is a competing, pre- +sumably single ion anisotropy turning the spins into the +ab plane. As argued in Ref. [54], the presence of the latter +contribution gives rise to the 2D XY critical behavior. +It should also be noted that the scaling b ≈ 3β/2 is +a characteristics of a 3D antiferromagnet, as it follows +from the theories of AFM resonance [56–58] and was con- +firmed experimentally (see, e.g., [59, 60]). Thus, a field- +dependent change of b indicates a kind of field-driven di- +mensional crossover of the spin wave excitations at inter- +mediate temperatures below TN while ramping the mag- +netic field across the spin-flop transition. Magnetic fields +H > Hsf push the spins into the plane, boosting the +effective XY anisotropy, which changes the character of +spin wave excitations observed by ESR towards the 2D +XY scaling regime. +V. +CONCLUSION +In summary, we have performed a detailed ESR spec- +troscopic study of the single-crystalline samples of the +van der Waals compounds Mn2P2S6 and MnNiP2S6. The +measurements were carried out in a broad range of ex- +citation frequencies and temperatures, and at different +orientations of the magnetic field with respect to the sam- +ple. Our study suggests a strong sensitivity of the type +of magnetic order and anisotropy below TN, as well as of +the g-factor and its anisotropy above TN to the Ni con- +centration. Stronger deviation of the g-factor from the +free electron value in the samples containing Ni suggests +that the anisotropy of the exchange can be an impor- +tant contributor to the stabilization of the certain type +of magnetic order with particular anisotropy. Analysis of +the spin excitations at T << TN has shown that both +Mn2P2S6 and MnNiP2S6 are strongly anisotropic. +In +fact, increasing the Ni content yields a larger magnon +gap in the ordered state (T << TN). In the Mn2P2S6 +compound we could fully describe the magnetic excita- +tions using a two sublattice AFM Hamiltonian, which +yielded an estimation of the uniaxial anisotropy energy, +the anisotropy energy within the ab-plane, and the av- +erage exchange interaction ΘMn2P2S6 ≈ 350 K. On the +contrary, in the MnNiP2S6 compound the ground state +and the excitations appear too complex to be described +using two-sublattice AFM model. This could be due to a +stochastic mixing of two magnetically inequivalent ions, +Mn and Ni, on the 4g Wyckoff crystallographic sites. +However, the analysis of the magnetization measured at +low-T suggests that the exchange coupling in this com- +pound should be comparable to or stronger than that in +Mn2P2S6. +We have analyzed the spin-spin correlations resulting +in a development of slowly fluctuating short-range order, +which, in the quasi-2D spin systems, manifest in the ESR +line broadening and shift at T > TN. The line broaden- +ing and shift are much stronger pronounced in MnNiP2S6 +compared to Mn2P2S6, suggesting that the critical broad- +ening and the shift of the ESR line in MnNiP2S6 could +be due to the enhanced spin fluctuations at the elevated +temperatures caused by the competition of different types +of magnetic order. Moreover, these strong spin fluctua- +tions in the mixed Mn/Ni compounds could additionally +lower the ordering temperature. +Finally, the analysis of the temperature dependence of +the spin excitation gap in Mn2P2S6 at different applied +fields suggests a kind of field-driven dimensional crossover +of the spin wave excitations at intermediate temperatures +below TN. +Strong magnetic fields push the spins into +the plane, boosting the effective XY anisotropy, which +changes the character of spin wave excitations observed +by ESR from a 3D-like towards the 2D XY scaling regime. +ACKNOWLEDGMENTS +J.J.A. acknowledges the valuable discussions with +Kranthi Kumar Bestha. This work was supported by the +Deutsche Forschungsgemeinschaft (DFG) through grants +No. KA 1694/12-1, AL 1771/8-1, AS 523/4-1, and within +the Collaborative Research Center SFB 1143 “Correlated +Magnetism – From Frustration to Topology” (project-id +247310070), and the Dresden-W¨urzburg Cluster of Excel- +lence (EXC 2147) “ct.qmat - Complexity and Topology +in Quantum Matter” (project-id 390858490), as well as +by the UKRATOP-project (funded by BMBF with Grant +No. 01DK18002). + +9 +Appendix +0 +1 +2 +3 +300 K +200 K +175 K +150 K +140 K +130 K +120 K +110 K +100 K +b) Mn2P2S6, ν = 88 GHz +10 K +7.5 K +5 K +3 K +30 K +20 K +65 K +60 K +50 K +40 K +90 K +80 K +75 K +70 K +11.8 12.0 12.2 12.4 +50 K +40 K +30 K +20 K +10 K +7.5 K +5 K +3 K +SD (arb. u.) +Magnetic +300 K +250 K +200 K +175 K +150 K +140 K +130 K +120 K +110 K +100 K +90 K +80 K +75 K +70 K +65 K +60 K +a) Mn2P2S6, ν = 326 GHz +**** +* +*** +* +* +* +* +6 +7 +8 +9 10 11 12 13 +Field (T) + d) MnNiP2S6, � = 326 GHz +300 K +250 K +200 K +175 K +150 K +140 K +120 K +100 K +90 K +85 K +80 K +75 K +70 K +65 K +60 K +55 K +50 K +40 K +30 K +20 K +10 K +7.5 K +5 K +3 K +11.0 +11.5 +12.0 +300 K +250 K +200 K +175 K +150 K +140 K +120 K +110 K +100 K +95 K +90 K +85 K +80 K + +75 K +70 K + +60 K +50 K +40 K +30 K +20 K +10 K +7.5 K +5 K +3 K +c) Mn2P2S6, � = 329 GHz +FIG. 9. Temperature dependence of the HF-ESR spectra of (a) Mn2P2S6 at the excitation frequency, ν ≈ 326 GHz for H ∥ c* +configuration, (b) Mn2P2S6 at ν ≈ 88 GHz for H ∥ c*. The temperature independent peaks from the impurity in the probehead +occurring only at low frequencies are marked with asterisks. (c) Mn2P2S6 at ν ≈ 329 GHz for H ⊥ c* and (d) MnNiP2S6 at +ν ≈ 326 GHz for H ⊥ c*. Spectra are normalized and vertically shifted for clarity. +0 +50 +100 +150 +200 +250 +300 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +� m (10-2 emu/ mol Oe) +Temperature (K) + H || c* + H ⊥ c* +H = 100 mT +56.7 K +75.5 K +� � � � � � +0 +2 +4 +6 +� � �� +� � �� +0.0 +0.1 +0.2 +M (µB/f.u.) +H (T) +T = 1.8 K +MnNiP2S6 +a) +0 +30 +60 +90 +120 +150 +180 +2.4 +2.6 +2.8 +3.0 +3.2 +3.4 +MnNiP2S6, H � c* +T = 3 K, � = 226 GHz +Resonance Field (T) +Theta (° ) +b) +FIG. 10. +(a) Molar susceptibility at the applied field of 1000 Oe as a function of temperature measured on the sample +of MnNiP2S6, which was used for the ESR investigations. +The gray broken lines represent the magnetic phase transition +temperature in both configurations. Inset: Isothermal magnetization per formula unit as a function of applied field performed +at 1.8 K for MnNiP2S6, depicting the almost isotropic field dependence of magnetic response. 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Rodr´ıguez-Su´arez, +Introduction to antiferromagnetic magnons, Journal of +Applied Physics 126, 151101 (2019). + diff --git a/GNE2T4oBgHgl3EQf-glT/content/tmp_files/load_file.txt b/GNE2T4oBgHgl3EQf-glT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1f593a35070da4109a273274237a94e929cc643b --- /dev/null +++ b/GNE2T4oBgHgl3EQf-glT/content/tmp_files/load_file.txt @@ -0,0 +1,1031 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf,len=1030 +page_content='Magnetic anisotropy and low-energy spin dynamics in magnetic van der Waals compounds Mn2P2S6 and MnNiP2S6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Abraham,1, 2, ∗ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Senyk,1, 2, ∗ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Shemerliuk,1 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Selter,1, 2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Aswartham,1 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' B¨uchner,1, 3 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Kataev,1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Alfonsov1, 3 1Leibniz IFW Dresden, D-01069 Dresden, Germany 2Institute for Solid State and Materials Physics, TU Dresden, D-01062 Dresden, Germany 3Institute for Solid State and Materials Physics and W¨urzburg-Dresden Cluster of Excellence ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='qmat, TU Dresden, D-01062 Dresden, Germany (Dated: January 12, 2023) We report the detailed high-field and high-frequency electron spin resonance (HF-ESR) spectro- scopic study of the single-crystalline van der Waals compounds Mn2P2S6 and MnNiP2S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Analysis of magnetic excitations shows that in comparison to Mn2P2S6 increasing the Ni content yields a larger magnon gap in the ordered state and a larger g-factor value and its anisotropy in the param- agnetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The studied compounds are found to be strongly anisotropic having each the unique ground state and type of magnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Stronger deviation of the g-factor from the free electron value in the samples containing Ni suggests that the anisotropy of the exchange is an important contributor to the stabilization of a certain type of magnetic order with particular anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' At the temperatures above the magnetic order, we have analyzed the spin-spin correlations resulting in a development of slowly fluctuating short-range order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' They are much stronger pronounced in MnNiP2S6 compared to Mn2P2S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The enhanced spin fluctuations in MnNiP2S6 are attributed to the competition of different types of magnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Finally, the analysis of the temperature de- pendent critical behavior of the magnon gaps below the ordering temperature in Mn2P2S6 suggest that the character of the spin wave excitations in this compound undergoes a field induced crossover from a 3D-like towards 2D XY regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' INTRODUCTION In the past recent years magnetic van der Waals (vdW) materials have become increasingly attractive for the fun- damental investigations since they provide immense pos- sibility to study intrinsic magnetism in low dimensional limit [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The weak vdW forces hold together the atomic monolayers in vdW crystals, which results in a poor interlayer coupling, and therefore renders these ma- terials intrinsically two dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In addition to the fundamental research, these materials are very promis- ing as potential candidates for next-generation spintron- ics devices [4–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Among the variety of magnetic vdW materials a par- ticularly interesting subclass is represented by the anti- ferromagnetic (TM)2P2S6 tiophosphates (TM stands for a transition metal ion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Here the transition metal ions are arranged in a graphene-like layered honeycomb lat- tice [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The high flexibility of the choice of the TM ion enables to control the properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Among the tiophos- phates there are examples of superconductors [9], pho- todetectors and field effect transistors [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' They also can be used for ion-exchange applications [12], catalytic activity [13], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Therefore, a proper choice of TM, or of a mixture of magnetically inequivalent ions on the same crystallographic position, could lead to the possibility of engineering of a material with desired magnetic ground state, excitations and correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' ∗ These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In order to establish the connection between the choice of magnetic ion and the resulting ground state and cor- relations, we performed a detailed high-field and high- frequency electron spin resonance (HF-ESR) spectro- scopic study on single crystals of the van der Waals com- pounds Mn2P2S6 and MnNiP2S6 in a broad range of mi- crowave frequencies and temperatures below and above the magnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' ESR spectroscopy is a powerful tool that can provide insights into spin-spin correlations, mag- netic anisotropy and spin dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' This technique has shown to be very effective for exploration of the mag- netic properties of vdW systems [14–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Albeit reso- nance studies on Mn2P2S6 were made by Okuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' [14], Joy and Vasudevan [15] and Kobets et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' [18], a high-frequency ESR study exploring broad range of tem- peratures below and above magnetic order was not yet performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The MnNiP2S6 compound is barely explored from the point of view of spin excitations from the mag- netic ground state below the ordering temperature, and from the point of view of spin-spin correlations in the high temperature regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Investigating Mn2P2S6 and MnNiP2S6 we have found difference in the types of magnetic order, anisotropies be- low the ordering temperature TN, as well as the g-factors and their anisotropy above TN in these compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In fact, increasing the Ni content yields a larger magnon gap in the ordered state (T << TN) and a larger g- factor value and its anisotropy in the paramagnetic state (T >> TN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' At temperatures above the magnetic or- der, we have analyzed the spin-spin correlations result- ing in a development of slowly fluctuating short-range or- der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' They are much stronger pronounced in MnNiP2S6 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='04239v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='str-el] 10 Jan 2023 2 compared to Mn2P2S6, which in our previous study has shown clear cut signatures of 2D correlated spin dy- namics [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Therefore, enhanced spin fluctuations in MnNiP2S6 are attributed to the competition of differ- ent types of magnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Finally, the analysis of the temperature dependent critical behavior of the magnon gaps below the ordering temperature in Mn2P2S6 sug- gest that the character of the spin wave excitations in this compound undergoes a field induced crossover from a 3D-like towards 2D XY regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' EXPERIMENTAL DETAILS Crystal growth of Mn2P2S6 and MnNiP2S6 samples investigated in this work was done using the chemical vapor transport technique with iodine as the transport agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Details of their growth, crystallographic, composi- tional and static magnetic characterization are described in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Note, that the experimental value xexp in (Mn1−xNix)2P2S6 for the nominal MnNiP2S6 com- pound is found to be xexp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='45, considering an uncer- tainty of approximately 5% [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Both materials exhibit a monoclinic crystal lattice system with a C2/m space group [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Each unit cell contains a [P2S6]4− cluster with S atoms occupying the edges of TM octahedra and P-P dumbbells occupying the void of each metal honey- comb sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The crystallographic c-axis makes an angle of 17◦ with the normal to the ab-plane [29], which is known to be one of the magnetic axes and is hereafter called as c* [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The ordering temperature of Mn2P2S6 is found to be TN = 77 K [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In contrast, the transition tempera- ture of MnNiP2S6 is rather uncertain and might depend on the direction of the applied magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Various studies have reported different values of TN, for instance it amounts to 12 K in [30], 38 K in [27], 41 K in [31] and 42 K in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' For the samples used in this study the ordering temperatures were extracted from the tem- perature dependence of the susceptibility χ measured at H = 1000 Oe (see appendix, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 10 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The calcula- tion of the maximum value of the derivative d(χ · T)/dT yields TN ∼ 57 K for H ∥ c* and TN ∼ 76 K for H ⊥ c* (hereafter called TN*).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The antiferromagnetic resonance (AFMR) and ESR measurements (hereafter called HF-ESR) were performed on several single crystalline samples of Mn2P2S6 and MnNiP2S6 using a homemade HF-ESR spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' A superconducting magnet from Oxford instruments with a variable temperature insert (VTI) was used to gener- ate magnetic fields up to 16 T allowing a continuous field sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The sample was mounted on a probe head which was then inserted into the VTI immersed in a 4He cryo- stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' A piezoelectric step-motor based sample holder was used for angular dependent measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Continuous He gas flow was utilized to attain stable temperatures in the range of 3 to 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Generation and detection of microwaves was performed using a vector network an- 6 7 8 9 10 11 12 13 SD (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=') Magnetic Field (T) b) MnNiP2S6, � = 326 GHz 300 K 250 K 200 K 175 K 150 K 140 K 120 K 100 K 90 K 80 K 70 K 60 K 50 K 45 K 40 K 35 K 30 K 20 K 10 K 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='5 K 5 K 3 K 2 4 6 B5 SD (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=') Magnetic Field (T) 70 K 65 K 60 K 90 K 80 K 75 K ** **** 50 K 40 K 30 K 20 K 10 K 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='5 K 5 K 3 K 293 K 250 K 194 K 171 K 150 K 130 K 110 K 100 K B4 a) Mn2P2S6, � = 147 GHz FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Temperature dependence of HF-ESR spectra of (a) Mn2P2S6 at fixed excitation frequency ν ≈ 147 GHz and (b) MnNiP2S6 at ν ≈ 326 GHz in H ∥ c* configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Spectra are normalized and vertically shifted for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The temper- ature independent peaks from the impurity in the probehead occurring at low frequencies are marked with asterisks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' alyzer (PNA-X) from Keysight Technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Equipped with the frequency extensions from Virginia Diodes, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=', the PNA-X can generate a frequency in the range from 75 to 330 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The measurements are performed in the transmission mode, where the microwaves are directed to the sample using oversized waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' All measure- ments were made by sweeping the field from 0 to 16 T and back to 0 T at constant temperature and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' HF-ESR signals generally have a Lorentzian line pro- file with an absorption and dispersion components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' For such a case, the resonance field (Hres) and linewidth (full width at half maxima, ∆H) can be extracted by fitting the signal using the function: SD(H) = 2Amp π × (L1sinα + L2cosα) + Coffset + CslopeH (1) where SD(H) is the signal at the detector and Amp is the amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Coffset represents the offset and CslopeH is the linear background of the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' L1 is the Lorentzian absorption which is defined in terms of Hres and ∆H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' L2 is the Lorentzian dispersion which is ob- tained by applying the Kramers-Kronig transformation to L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' α is a parameter used to define the degree of in- strumental mixing of the absorption and dispersion com- ponents which is unavoidable in the used setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Some of the HF-ESR signals of Mn2P2S6 could not be fitted using the above equation due to the development of the shoulders or the splitting of peaks [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' ∆H, Hres and, 3 0 50 100 150 200 250 300 4 2 0 2 4 6 � H = Hres(T) - Hres(300 K) (T) Temperature (K) 88 GHz, H || c* 88 GHz, H || c* 147 GHz, H || c* 147 GHz, H || c* 329 GHz, H || c* 329 GHz, H ^ c* TN Mn2P2S6 B5 B4 B3 0 100 200 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='25 � = 329 GHz H || c* H ^ c* ∆H = Linewidth (T) Temperature (K) TN 100 150 200 250 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Shift of the resonance field position δH (main panel) and linewidth ∆H (inset) as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The horizontal dashed line represents zero shift from the room temperature value and the vertical dashed line (also for inset) represents the N´eel temperature of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' therefore, δH = Hres - Hres(300 K) were then obtained by picking a position of the peak value and by calculating the full width at half maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Temperature dependence of HF-ESR response To study the temperature evolution of the spin dynam- ics, the HF-ESR spectra were measured at several tem- peratures in the range of 3 - 300 K and at few selected microwave excitation frequencies ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Such dependences measured in the H ∥ c* configuration at ν = 147 GHz for Mn2P2S6 and at ν = 326 GHz for MnNiP2S6 are pre- sented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' As can be seen, in the case of Mn2P2S6 upon entering the ordered state with lowering tempera- ture, the single ESR line transforms into two modes B4 and B5 (see below) at ν = 147 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The temperature dependence of the spectra for other frequencies can be found in Appendix in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The shift of the obtained values of Hres from the resonance field position at T = 300 K, δH = Hres - Hres(300 K) is plotted as a function of temperature for Mn2P2S6 and MnNiP2S6 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 3, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Hres(300 K) was calculated using the equation hν = gµBµ0Hres, where the g-factor is obtained from the frequency dependence of the resonance field at 300 K (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' III B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In the case of Mn2P2S6, δH stays practi- cally constant down to T ∼ 130 − 150 K for both config- urations H ∥ c* and H ⊥ c*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Below this temperature it starts to slightly deviate (lower inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 2), suggest- ing a development of the static on the ESR time scale 0 50 100 150 200 250 300 4 3 2 1 0 H || c* H ⊥ c* � H = Hres (T) - Hres (300 K) (T) Temperature (K) MnNiP2S6, � = 326 GHz TN TN* 0 100 200 300 0 1 2 3 � H = Linewidth (T) Temperature (K) TN* TN FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Temperature dependence of δH (main panel) and ∆H (inset) measured at ν = 325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='67 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The horizontal dashed line represents zero shift from room temperature value and the vertical dashed line (also for inset) represents the N´eel temperature of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' internal fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In contrast, the deviations of δH from zero value in MnNiP2S6 are larger, and are observed at a higher temperature T ∼ 200 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In the vicinity of the ordering temperature TN* there is a strong shift of the ESR line, observed for both compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In the Mn2P2S6 case the sign of δH below the ordering temperature de- pends on the particular AFMR mode, which is probed at the specific frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' This is detailed in the following Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 3 represent the evolution of the linewidth ∆H as a function of temperature for Mn2P2S6 and MnNiP2S6 compounds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' At T > TN, ∆H remains practically temperature indepen- dent for both compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' A small broadening of the line is observed in the vicinity of the phase transition temperature, and there is a drastic increase of ∆H in the ordered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Note, that ∆H of MnNiP2S6 is larger than that of Mn2P2S6 in the whole temperature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Moreover, for MnNiP2S6, ∆H increases at low tempera- tures by almost one order of magnitude from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='3 to 3 T (inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Such extensive line broadening at low temperatures hampers the accurate determination of the linewidth and resonance field, which is accounted for in the error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Frequency dependence at 300 K The frequency dependence of the resonance field ν(Hres) of Mn2P2S6 and MnNiP2S6 compounds mea- sured in the paramagnetic state at T = 300 K is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Both plots have a linear dependence which can be fitted with the conventional paramagnetic resonance con- 4 0 2 4 6 8 10 12 g|| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='026 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='002 g⊥ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='047 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='004 SD (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=') 0 2 4 6 8 10 12 0 50 100 150 200 250 300 350 a) Mn2P2S6 H || c* H ⊥ c* Frequency (GHz) Resonance Field (T) g|| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='992 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='001 g⊥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='999 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='001 H || c* H ⊥ c* b) MnNiP2S6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' ν(Hres) dependence measured at 300 K for (a) Mn2P2S6 and (b) MnNiP2S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Blue squares represent H ∥ c* configuration and the red circles represent H ⊥ c* con- figuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Solid lines show the results of the fit accord- ing to the resonance condition of a conventional paramagnet hν = gµBµ0Hres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Right vertical axis: Representative spectra normalized for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The color of the spectra corresponds to the color of the data points in the ν(Hres) plot with the same Hres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' dition for a gapless excitation hν = gµBµ0Hres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Here, h is the Plank constant, µB is the Bohr magneton, µ0 is the permeability of free space and g is the g-factor of res- onating spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' For Mn2P2S6, we obtain almost isotropic values of the g-factor: g∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='992 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='001 (H ∥ c*) and g⊥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='999 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='001 (H ⊥ c*), which is expected for a Mn2+ ion [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In contrast, MnNiP2S6 shows a small anisotropy of g-factors with g∥ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='026 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='002 and g⊥ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='047 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In case of Ni2+ ions (3d8, S = 1), g- factors are expected to be appreciably greater than free spin value, as is revealed in HF-ESR studies on Ni2P2S6 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Frequency dependence at 3 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Mn2P2S6 The low temperature resonance modes of Mn2P2S6 ob- tained at T = 3 K are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The measure- ments along the H ∥ c* configuration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 5) yield three branches B3, B4 and B5, two of which (B3 and B4) are observed below the spin-flop field, HSF = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='62 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Branches B1 and B2 are assigned to the measurements along a- and b-axis, respectively [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Additionally, at the spin-flop field, a non-resonance absorption peak (full circles) was observed at high frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The exact gap values are calculated by fitting the in- plane resonance branches B1 and B2 using the analytical expressions for easy-axis AFMs [36]: hν = [(g⊥µBµ0Hres)2 + ∆2 1,2]1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' (2) Here ∆1 corresponds to the magnon excitation gap for branch B2 (also B3), and ∆2 corresponds to B1 (also B4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The obtained values are ∆1 = ∆Mn2P2S6 1 = 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='3 ± 0 2 4 6 8 10 12 0 50 150 200 250 300 350 Frequency (GHz) Resonance Field (T) Spin Flop Mn2P2S6, T = 3 K 116 101 AFM Branch, H || a*-axis AFM Branch, H || c*-axis AFM Branch, H || b*-axis B1 B2 B3 B4 B5 H || a H || b H || c* Paramagnetic branch SD (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=') FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' ν(Hres) dependence of HF-ESR signals measured at T = 3 K (symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Solid lines are the fit to the phe- nomenological equations as explained in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The dash gray lines correspond to the frequencies at which temperature dependent measurements were performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The dash line in magenta represents the paramagnetic branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Right vertical scale: Normalized ESR spectra for selected frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' For clarity the spectra are shifted vertically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Error bars in the Hres are smaller than the symbol size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='6 GHz and ∆2 = ∆Mn2P2S6 2 = 116 ± 2 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' These values, which agree well with previous measurements by Okuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' [14] and Kobets et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' [18], are then used in the theoretical description for a rhombic biaxial two- lattice AFM [18, 37] to match the field dependence of B3 and B4 [38]: ν = gµBµ0 2h × � ∆2 1 + ∆2 2 + 2H2 res± ± � 8H2res(∆2 1 + ∆2 2) + (∆2 1 + ∆2 2)2 �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' (3) Above the spin-flop field the above model can not be used to describe the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Therefore branch B5 [38] was simulated by the resonance condition of a conven- tional easy-axis AFM [36]: hν = [(g∥µBµ0Hres)2 − ∆2 1]1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' (4) The presence of the second easy-axis within the ab- plane is further confirmed by the angular dependence of Hres(θ) in the H ⊥ c* configuration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' It follows a A + Bsin2(θ) law, which suggests a 180◦ periodicity of Hres(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' θ denotes the angle between the applied field and a-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' For a honeycomb spin system with a N´eel type arrangement, a six-fold periodicity of angular de- pendence in the layer plane can be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' However, this is absent in the case of Mn2P2S6 sample due to dom- inating effects of a two-fold in-plane anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 5 120 -90 -60 -30 0 30 60 90 120 150 180 210 240 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='90 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='95 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='40 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='45 Resonance Field (T) Theta (°) Mn2P2S6 H ⊥ c*, T = 3 K � = 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='9 GHz Model pi_periodicity (User) Equation A + B * sin(x*pi/180 + C)^2 Plot Resonance Field A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='96068 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='01024 B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='43992 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='01675 C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='03046 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='01979 Reduced Chi-Sqr 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='35579E-4 R-Square (COD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='97185 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' R-Square 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='96903 H || a H || b FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Resonance field as a function of angle θ at T = 3 K and ν = 160 GHz for Mn2P2S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' θ denotes the angle between the direction of the field applied along the ab-plane and the a-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Red dash line represents the result of the fit, as explained in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' To further analyze the measured ν(Hres) dependence of the AFMR modes in the magnetically ordered state of Mn2P2S6, that correspond to the collective excitations of the spin lattice (spin waves), we employed a linear spin wave theory (LSWT) with the second quantization formalism [36, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The details of our model are provided in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The phenomenological Hamiltonian for the two-sublattice spin system, used for calculations of the spin waves energies, has the following form: H = A(M1M2) M 2 0 + Kuniax M1z 2 + M2z 2 M 2 0 + Kbiax 2 (M 2 1x − M 2 1y) + (M 2 2x − M 2 2y) M 2 0 − (HM1) − (HM2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' (5) Here the first term represents the exchange interaction between the magnetic sublattices with respective magne- tizations M1 and M2, such that M 2 1 = M 2 2 = (M0)2 = (Ms/2)2, with M 2 s being the square of the saturation magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' A is the mean-field antiferromagnetic ex- change constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' (5) is the uni- axial part of the magnetocrystalline anisotropy given by the anisotropy constant Kuniax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The third term describes an additional anisotropy in the xy-plane with the respec- tive constant Kbiax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The fourth and fifth terms are the Zeeman interactions for both sublattice magnetizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The results of the calculation match well the measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In the calculation we assumed a full Mn satura- tion moment of ∼ 5µB, yielding Ms = 446 erg/(G·cm3) = 446 · 103 J/(T·m3), considering 4 Mn ions in the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The average g-factor value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='995 was taken from 0 1 2 3 4 5 6 7 8 9 10 11 12 0 50 100 150 200 250 300 350 400 450 AFM Branch, H || c* AFM Branch, H ⊥ c* Paramagnetic branch H || c* H ⊥ c* Resonance Field (T) Frequency (GHz) MnNiP2S6, T = 3 K SD (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=') FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' ν(Hres) dependence measured at 3 K for both con- figurations of magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Right vertical scale: Exemplary spectra positioned above the resonance points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The horizontal dash gray line represents the frequency at which the temper- ature dependence was measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The dash line in magenta depicts the paramagnetic resonance branch at 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' the frequency dependence measurements at T = 300 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' As the result we obtain the exchange con- stant A = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='53 · 108 erg/cm3 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='53 · 107 J/m3, uni- axial anisotropy constant Kuniax = −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='2 · 104 erg/cm3 = −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='2 · 103 J/m3, and an in-plane anisotropy constant Kbiax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='9 · 104 erg/cm3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='9 · 103 J/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Within the mean-field theory A is related to the Weiss constant Θ = A ∗ C/M 2 0 , where C is the Curie constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Θ, that provides an average energy scale for the exchange interaction in the system, amounts therefore at least ΘMn2P2S6 ≈ 350 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' MnNiP2S6 In the case of MnNiP2S6 we observe one branch for H ∥ c* and another one for H ⊥ c* configuration, re- spectively, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' HF-ESR spectra were also recorded at various angles for the in-plane orienta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Within the experimental error bars of ∼ 300 mT, no signatures for an in-plane anisotropy were observed (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 10 in Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Both branches follow the resonance condition for a hard direction of an AFM given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' (2), which reveals that neither c*-axis nor the ab-plane are energetically favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The magnitude of the gap was obtained from the fit as ∆MnNiP2S6 1 = 115 ± 9 GHz for H ∥ c* and ∆MnNiP2S6 2 = 215 ± 1 GHz for H ⊥ c* config- urations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Unfortunately, we could not find a good matching of the calculated frequency dependence to the one mea- sured at low temperature (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 7) with the AFM Hamil- tonian for a two sublattice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Inclusion of the 6 terms describing cubic, hexagonal and symmetric ex- change anisotropies in addition to those given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' (5) did not yield a good result either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' This could be ex- plained by the complicated type of order of two mag- netically inequivalent ions Mn2+ (S = 5 2, g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='955) and Ni2+ (S = 1, g = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='17), which possibly requires a more sophisticated model than the one used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The analysis might be even more complicated by poten- tial disorder in the system due to the stochastic distribu- tion of these ions on the 4g Wyckoff sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Therefore the full description of this system remains an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' However, one could draw some conclusions by analyzing how the magnetization measured at low-T depends on the Mn/Ni ratio [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The reduction of the magnetization measured at low-T can be explained by the the reduc- tion of the total moment per formula unit of MnNiP2S6, which can be found as an average of the Mn and Ni sat- uration magnetizations and amounts to ∼ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='2 µB, com- pared to Mn2P2S6 which has the saturation moment of ∼ 10 µB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Additionally, an almost isotropic behavior of the magnetization as a function of magnetic field (inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 10 (a)) suggests that the isotropic exchange energy is by orders of magnitude the strongest term defining the static magnetic properties of MnNiP2S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In this case, the magnetization value, measured at the magnetic field ap- plied along some hard direction, should be inversely pro- portional to the mean-field isotropic exchange constant M ∼ H/A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The reduced magnetization in MnNiP2S6 suggests, therefore, that Θ ∼ A should be at least as large in MnNiP2S6 (ΘMnNiP2S6 ≥ ∼ 350 K) as in Mn2P2S6 (ΘMn2P2S6 ≈ 350 K, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' III C 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Spin-Spin correlations in (Mn1−xNix)2P2S6 (T > TN*) As has been shown in our previous work, both the resonance field and the linewidth of the HF-ESR signal in Ni2P2S6 remain temperature independent by cooling the sample down to temperatures close to TN [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Usu- ally, in the quasi-2D spin systems the ESR line broad- ening and shift occur at T > TN due to the growth of the in-plane spin-spin correlations resulting in a de- velopment of slowly fluctuating short-range order [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Specifically, the slowly fluctuating spins produce a static on the ESR timescale field causing a shift of the reso- nance line, and a distribution of these local fields and shortening of the spin-spin relaxation time due to the slowing down of the spin fluctuations increase the ESR linewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In the Mn2P2S6 compound these features are not very pronounced, only in the resonance field of the HF-ESR response one can detect within error bars small deviations starting at T ∼ 130 − 150 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In the MnNiP2S6 compound, in turn, the critical broadening and the shift of the resonance line are observed at tem- perature T ∼ 200 K, which is much higher than TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Even though the critical broadening and the line shift above TN are much stronger pronounced in MnNiP2S6, our previous low-frequency ESR study shows that the clear cut signatures of 2D correlated spin dynamics are present above TN only in the Mn2P2S6 compound [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Interestingly, these signatures, seen in the characteristic angular dependence of the ESR linewidth, develop only at elevated temperatures, where the effect of the strong isotropic AFM coupling (ΘMn2P2S6 ≈ 350 K) on the spin fluctuations becomes gradually suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Critical broadening and the shift of the ESR line in MnNiP2S6 above TN could therefore be due to the stochastic dis- tribution of Mn and Ni ions on the 4g Wyckoff sites of the crystal structure causing a competition of different order types with contrasting magnetic anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Our conclusion on the drastic difference in the ground states is supported by the strong distinction in the energy gaps and magnetic field dependences of the low-T spin wave excitations in Mn2P2S6, MnNiP2S6 and Ni2P2S6, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The competing types of magnetic order might enhance spin fluctuations seen in the HF-ESR response at elevated temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Strong fluctuations suppress, in turn, the ordering temperature for MnNiP2S6 which is evident in the recent studies on the (Mn1−xNix)2P2S6 se- ries [27, 30, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Moreover, in this scenario of the stochas- tic distribution of Mn and Ni, small deviation of the sto- ichiometry from sample to sample of the same nominal composition could vary the ordering temperature, which explains the broad range of TN measured in MnNiP2S6 samples [27, 30, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Ground state and anisotropy of (Mn1−xNix)2P2S6 (T << TN*) At the lowest measurement temperature Mn2P2S6 has an antiferromagnetic ground state with biaxial type of anisotropy, and the spin wave excitations can be suc- cessfully modeled using LSWT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' As the result we obtain the estimation of the exchange interaction ΘMn2P2S6 ≈ 350 K and the parameters of the anisotropy Kuniax = −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='2 · 104 erg/cm3 = −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='2 · 103 J/m3 and Kbiax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='9 · 104 erg/cm3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='9 · 103 J/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' There is only about four times difference between Kuniax and Kbiax, which sug- gests that the anisotropy in the ab-plane makes a sig- nificant contribution to the properties of the ground state of Mn2P2S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Interestingly, the value of Kbiax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='9 · 103 J/m3 ≈ 2 · 10−25 J/spin is very close to the es- timation of the anisotropy within the ab-plane made by Goossens [42], suggesting a possible dipolar nature of this anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In the MnNiP2S6 case we could not find an appropriate Hamiltonian within a two sublattice model which would fully describe the system, calling for a more sophisticated theoretical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Interestingly, the charac- teristic feature of the MnNiP2S6 compound is the almost isotropic dependence of the magnetization as a function of magnetic field, measured at temperature well below TN [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The isothermal magnetization measurements made 7 on the sample used in this study, confirm the presence of this almost isotropic static magnetic response (see ap- pendix Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 10 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Such an isotropic behavior of the static magnetization is related to the strong isotropic AFM exchange interaction (ΘMnNiP2S6 ≥ ∼ 350 K), which is larger than the applied magnetic field and the observed magnetic anisotropy in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' However, the HF-ESR data reveals a substantial anisotropy in the magnetic field dependence of the spin waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' This seem- ing contradiction is actually not surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The magne- tization value at the magnetic field applied along some hard direction is mostly given by the mean-field exchange constant M ∼ H/A, whereas the magnon gap measured in the ESR experiment is roughly proportional to the square root of the product of exchange and magnetic anisotropy constants [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Qualitatively, the evolution of the type of magnetic anisotropy with x in (Mn1−xNix)2P2S6 is also evident from our study, where, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=', MnNiP2S6 reveals no easy- axis within or normal to the ab-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In order to quan- tify the change of magnetic anisotropic properties with the Mn/Ni content the excitation energy gaps can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The single gap of about 260 GHz was found in our previous study on Ni2P2S6 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Both Mn containing compounds have two gaps ∆MnNiP2S6 1 = 115±9 GHz and ∆MnNiP2S6 2 = 215 ± 1 GHz in the case of MnNiP2S6, and ∆Mn2P2S6 1 = 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='6 GHz and ∆Mn2P2S6 2 = 116±2 GHz in the case of Mn2P2S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' As can be seen, there is a no- ticeable increase of the zero field AFM gaps in the sam- ples with higher Ni content, suggesting an increase of the magnetic anisotropy and exchange interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Indeed, the estimated energy scale of the exchange interaction in Mn2P2S6 is about ∼ 350 K, in MnNiP2S6 is more than ∼ 350 K, and it is even larger in Ni2P2S6, due to the observation of the larger TN and as it is suggested by the previous investigations [25, 43–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Mn2+ with the half filled 3d electronic shell, and a small admixture of the excited state 4P5/2 into the ground state 6S5/2 is an ion with rather isotropic magnetic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In contrast, the ground state of the Ni2+ ion in the octahedral envi- ronment [8] is a spin triplet with the higher lying orbital multiplets, admixed through the spin-orbit coupling [34], which makes the Ni spin (S = 1) sensitive to the local crystal field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' This, first, could increase a contribution of the local (single ion) magnetic anisotropy term in the Hamiltonian describing the system in the ordered and in the paramagnetic state, as discussed for the case of Ni2P2S6 in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Second, it could yield a deviation of the g-factor from the free electron value and also induce an effective g-factor anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The effective g-factor value and its anisotropy, as found in our study, increase with Ni content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Deviation of the g-factor from the free electron value (∆g) and the anisotropy of the exchange originate from the spin-orbit coupling effect, and there- fore are interrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In the case of symmetric anisotropic exchange the elements of the anisotropic exchange ten- sor are A ∝ (∆g/g)2J [46–48], where J is the isotropic exchange interaction constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Observation of increased 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='2 B5, 329 GHz B5, 147 GHz B4, 147 GHz B4, 88 GHz ∆(T)/∆(3 K) 1 - T/TN b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='55 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='6 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='29 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='26 collinear phase spin-flop phase H||c* 0 2 4 6 8 10 12 0 50 150 200 250 300 350 100 Frequency (GHz) Resonance Field (T) T = 3 K H||c* B1 B2 B3 B4 B5 µ0Hsf collinear phase spin-flop phase FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Main panel: Temperature dependence of the normal- ized energy gap ∆(T)/∆(3 K) = [1−(T/TN)]b for Mn2P2S6 at different field regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Symbol shapes and colors correspond to that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Inset: Resonance branches at T = 3 K (solid lines) as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Symbols (same as in the main panel) indi- cate the positions of the resonance modes B4 at 147 GHz and B5 at 147 and 329 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The position of mode B4 at 88 GHz is not shown here since it can be detected at T ≥ 50 K only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The temperature dependence of these modes shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 2 was used to estimate that of ∆(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' (see the text) ∆g at higher Ni content suggests that in the Ni contain- ing (Mn1−xNix)2P2S6 the exchange anisotropy is likely an important contributor to the anisotropic properties of the ground state at low temperatures < TN, such as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=', increased magnon gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Critical behavior of Mn2P2S6 (T ≲ TN*) In the following we discuss the temperature depen- dence of the excitation energy gap ∆ at finite magnetic fields in the collinear and the spin-flop AFM ordered phases of Mn2P2S6, at H < Hsf and H > Hsf, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' This should provide useful insights onto the type of the critical behavior of the Mn spin lattice at T < TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Such a dependence can be obtained by an- alyzing the temperature dependence of the shift of the resonance field positions Hres(T) of the excitation modes B4 and B5 for H ∥ c* (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 2) with the aid of the sim- plified relations ∆ ≈ hν − g∥µBµ0Hres for mode B4 and ∆ ≈ [(g∥µBµ0Hres)2 − (hν)2]1/2 for mode B5 derived from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' (3) and (4), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The result of this analysis is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The ∆(T) dependence can be well fitted to the power law ∆(T) ∝ [1 − (T/TN)]b in a broad temperature range be- low TN with some deviations from it at lower T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The exponents b indicated in this Figure appear to be very different for modes B4 and B5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Notably, the resonance 8 field of mode B4 is always smaller than the spin-flop field, HB4 res |88 GHz< HB4 res |147 GHz< Hsf, whereas mode B5 occurs at larger fields with Hsf < HB5 res |145 GHz< HB5 res |329 GHz [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 8(inset)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' This suggests a significant difference in the temperature dependence of the exci- tation gap in the collinear and spin-flop AFM ordered phases of Mn2P2S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Usually, the magnetic anisotropy gap ∆(T) observed in quasi-2D antiferromagnets scales with the sublattice magnetization Msl(T) [49–51] so that the exponent b of the temperature dependence of ∆ can be treated as a crit- ical exponent β of the AFM order parameter Msl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' If that were the case for Mn2P2S6, the value of b in the collinear phase would indicate the mean-field behavior of Msl(T) for which β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='5 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In contrast, a strong reduction of b in the spin-flop phase, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 8, would cor- respond to the critical behavior of Msl(T) in the 2D XY model for which β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='231 [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' However, measurements of the temperature dependence of Msl by elastic and of ∆ by inelastic neutron scattering in zero magnetic field reveal a more complex scaling between these two param- eters with b ≈ 3β/2 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='32 in the vicinity of TN, and b ≈ β with β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='25 at lower temperatures [53–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' This finding was tentatively ascribed to different tem- perature dependence of the competing single-ion and dipolar anisotropies which are both responsible for a fi- nite value of ∆ in the AFM ordered state of Mn2P2S6 [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Theoretical analysis in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' [42] shows that besides the dipolar anisotropy which is responsible for the out- of-plane order of the Mn spins there is a competing, pre- sumably single ion anisotropy turning the spins into the ab plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' As argued in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' [54], the presence of the latter contribution gives rise to the 2D XY critical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' It should also be noted that the scaling b ≈ 3β/2 is a characteristics of a 3D antiferromagnet, as it follows from the theories of AFM resonance [56–58] and was con- firmed experimentally (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=', [59, 60]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Thus, a field- dependent change of b indicates a kind of field-driven di- mensional crossover of the spin wave excitations at inter- mediate temperatures below TN while ramping the mag- netic field across the spin-flop transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Magnetic fields H > Hsf push the spins into the plane, boosting the effective XY anisotropy, which changes the character of spin wave excitations observed by ESR towards the 2D XY scaling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' CONCLUSION In summary, we have performed a detailed ESR spec- troscopic study of the single-crystalline samples of the van der Waals compounds Mn2P2S6 and MnNiP2S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The measurements were carried out in a broad range of ex- citation frequencies and temperatures, and at different orientations of the magnetic field with respect to the sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Our study suggests a strong sensitivity of the type of magnetic order and anisotropy below TN, as well as of the g-factor and its anisotropy above TN to the Ni con- centration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Stronger deviation of the g-factor from the free electron value in the samples containing Ni suggests that the anisotropy of the exchange can be an impor- tant contributor to the stabilization of the certain type of magnetic order with particular anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Analysis of the spin excitations at T << TN has shown that both Mn2P2S6 and MnNiP2S6 are strongly anisotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In fact, increasing the Ni content yields a larger magnon gap in the ordered state (T << TN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In the Mn2P2S6 compound we could fully describe the magnetic excita- tions using a two sublattice AFM Hamiltonian, which yielded an estimation of the uniaxial anisotropy energy, the anisotropy energy within the ab-plane, and the av- erage exchange interaction ΘMn2P2S6 ≈ 350 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' On the contrary, in the MnNiP2S6 compound the ground state and the excitations appear too complex to be described using two-sublattice AFM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' This could be due to a stochastic mixing of two magnetically inequivalent ions, Mn and Ni, on the 4g Wyckoff crystallographic sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' However, the analysis of the magnetization measured at low-T suggests that the exchange coupling in this com- pound should be comparable to or stronger than that in Mn2P2S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' We have analyzed the spin-spin correlations resulting in a development of slowly fluctuating short-range order, which, in the quasi-2D spin systems, manifest in the ESR line broadening and shift at T > TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The line broaden- ing and shift are much stronger pronounced in MnNiP2S6 compared to Mn2P2S6, suggesting that the critical broad- ening and the shift of the ESR line in MnNiP2S6 could be due to the enhanced spin fluctuations at the elevated temperatures caused by the competition of different types of magnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Moreover, these strong spin fluctua- tions in the mixed Mn/Ni compounds could additionally lower the ordering temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Finally, the analysis of the temperature dependence of the spin excitation gap in Mn2P2S6 at different applied fields suggests a kind of field-driven dimensional crossover of the spin wave excitations at intermediate temperatures below TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Strong magnetic fields push the spins into the plane, boosting the effective XY anisotropy, which changes the character of spin wave excitations observed by ESR from a 3D-like towards the 2D XY scaling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' ACKNOWLEDGMENTS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' acknowledges the valuable discussions with Kranthi Kumar Bestha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' KA 1694/12-1, AL 1771/8-1, AS 523/4-1, and within the Collaborative Research Center SFB 1143 “Correlated Magnetism – From Frustration to Topology” (project-id 247310070), and the Dresden-W¨urzburg Cluster of Excel- lence (EXC 2147) “ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='qmat - Complexity and Topology in Quantum Matter” (project-id 390858490), as well as by the UKRATOP-project (funded by BMBF with Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 01DK18002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 9 Appendix 0 1 2 3 300 K 200 K 175 K 150 K 140 K 130 K 120 K 110 K 100 K b) Mn2P2S6, ν = 88 GHz 10 K 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='5 K 5 K 3 K 30 K 20 K 65 K 60 K 50 K 40 K 90 K 80 K 75 K 70 K 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='4 50 K 40 K 30 K 20 K 10 K 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='5 K 5 K 3 K SD (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=') Magnetic 300 K 250 K 200 K 175 K 150 K 140 K 130 K 120 K 110 K 100 K 90 K 80 K 75 K 70 K 65 K 60 K a) Mn2P2S6, ν = 326 GHz **** *** 6 7 8 9 10 11 12 13 Field (T) d) MnNiP2S6, � = 326 GHz 300 K 250 K 200 K 175 K 150 K 140 K 120 K 100 K 90 K 85 K 80 K 75 K 70 K 65 K 60 K 55 K 50 K 40 K 30 K 20 K 10 K 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='5 K 5 K 3 K 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='0 300 K 250 K 200 K 175 K 150 K 140 K 120 K 110 K 100 K 95 K 90 K 85 K 80 K 75 K 70 K 60 K 50 K 40 K 30 K 20 K 10 K 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='5 K 5 K 3 K c) Mn2P2S6, � = 329 GHz FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Temperature dependence of the HF-ESR spectra of (a) Mn2P2S6 at the excitation frequency, ν ≈ 326 GHz for H ∥ c* configuration, (b) Mn2P2S6 at ν ≈ 88 GHz for H ∥ c*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The temperature independent peaks from the impurity in the probehead occurring only at low frequencies are marked with asterisks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' (c) Mn2P2S6 at ν ≈ 329 GHz for H ⊥ c* and (d) MnNiP2S6 at ν ≈ 326 GHz for H ⊥ c*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Spectra are normalized and vertically shifted for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 0 50 100 150 200 250 300 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='0 � m (10-2 emu/ mol Oe) Temperature (K) H || c* H ⊥ c* H = 100 mT 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='7 K 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='5 K � � � � � � 0 2 4 6 � � �� � � �� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='2 M (µB/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=') H (T) T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='8 K MnNiP2S6 a) 0 30 60 90 120 150 180 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='4 MnNiP2S6, H � c* T = 3 K, � = 226 GHz Resonance Field (T) Theta (° ) b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' (a) Molar susceptibility at the applied field of 1000 Oe as a function of temperature measured on the sample of MnNiP2S6, which was used for the ESR investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' The gray broken lines represent the magnetic phase transition temperature in both configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' Inset: Isothermal magnetization per formula unit as a function of applied field performed at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content='8 K for MnNiP2S6, depicting the almost isotropic field dependence of magnetic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' (b) In-plane angular dependence of the resonance field at T = 3 K and ν = 226 GHz for MnNiP2S6, showing no systematic angular dependence within the average error bar of 0.' metadata={'source': 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direction is b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In the case of an antifer- romagnet, a given magnetic field applied along the b-axis yields a lower magnetization value, and, therefore, the in- ternal field in such configuration is smaller compared to H ∥ a*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' This further implies that a larger external field is required to reach the resonance condition for H ∥ b* configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} +page_content=' In the experiment, the magnetic field is applied at various angles by rotating the crystal in the H ⊥ c* configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQf-glT/content/2301.04239v1.pdf'} 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0000000000000000000000000000000000000000..d695bf6797508b2f5de3a23b68129da923b1e4ef --- /dev/null +++ b/GNE3T4oBgHgl3EQfWAqd/content/tmp_files/2301.04465v1.pdf.txt @@ -0,0 +1,992 @@ +arXiv:2301.04465v1 [cs.CV] 11 Jan 2023 +CO-TRAINING WITH HIGH-CONFIDENCE PSEUDO LABELS FOR +SEMI-SUPERVISED MEDICAL IMAGE SEGMENTATION +Zhiqiang Shen1,2 +Peng Cao1,2∗ +Hua Yang3 +Xiaoli Liu4 +Jinzhu Yang1,2 +Osmar R. Zaiane5 +1College of Computer Science and Engineering, Northeastern University, Shenyang, China +2Key Laboratory of Intelligent Computing in Medical Image, Ministry of Education, Shenyang, China +3College of Photonic and Electronic Engineering, Fujian Normal University, Fuzhou, China +4DAMO Academy, Alibaba Group, China +5Alberta Machine Intelligence Institute, University of Alberta, Edmonton, Alberta, Canada +xxszqyy@gmail.com, caopeng@mail.neu.edu.cn +ABSTRACT +High-quality pseudo labels are essential for semi-supervised semantic segmentation. Consistency +regularization and pseudo labeling-based semi-supervised methods perform co-training using the +pseudo labels from multi-view inputs. However, such co-training models tend to converge early to +a consensus during training, so that the models degenerate to the self-training ones. Besides, the +multi-view inputs are generated by perturbing or augmenting the original images, which inevitably +introduces noise into the input leading to low-confidence pseudo labels. To address these issues, +we propose an Uncertainty-guided Collaborative Mean-Teacher (UCMT) for semi-supervised se- +mantic segmentation with the high-confidence pseudo labels. Concretely, UCMT consists of two +main components: 1) collaborative mean-teacher (CMT) for encouraging model disagreement and +performing co-training between the sub-networks, and 2) uncertainty-guided region mix (UMIX) +for manipulating the input images according to the uncertainty maps of CMT and facilitating CMT +to produce high-confidence pseudo labels. Combining the strengths of UMIX with CMT, UCMT +can retain model disagreement and enhance the quality of pseudo labels for the co-training seg- +mentation. Extensive experiments on four public medical image datasets including 2D and 3D +modalities demonstrate the superiority of UCMT over the state-of-the-art. Code is available at: +https://github.com/Senyh/UCMT. +1 +Introduction +Semantic segmentation is critical for medical image analysis. Great progress has been made by deep learning-based +segmentation models relying on a large amount of labeled data [1, 2]. However, labeling such pixel-level annotations +is laborious and requires expert knowledge especially in medical images, resulting in that labeled data are expensive +or simply unavailable. Unlabeled data, on the contrary, are cheap and relatively easy to obtain. Under this condition, +semi-supervised learning (SSL) has been the dominant data-efficient strategy through exploiting information from a +limited amount labeled data and an arbitrary amount of unlabeled data, so as to alleviate the label scarcity problem +[3]. +Consistency regularization [4] and pseudo labeling [6] are the two main methods for semi-supervised semantic seg- +mentation. Currently, combining consistency regularization and pseudo labeling via cross supervision between the +sub-networks, has shown promising performance for semi-supervised segmentation [6, 7, 8, 5, 9]. One critical limita- +tion of these approaches is that the sub-networks tend to converge early to a consensus situation causing the co-training +model degenerating to the self-training [10]. Disagreement between the sub-networks is crucial for co-training, where +the sub-networks initialized with different parameters or trained with different views have different biases (i.e., dis- +agreement) ensuring that the information they provide is complementary to each other. Another key factor affecting +∗corresponding author + +UCMT +(d) Co-training disagreement +(e) Pseudo labels uncertainty +� +� �� +� � +�� +� +(a) MT +� +� �� +� �� +�� +�� +(b) CPS +� � +� +� +� �� +� �� +�� +�� +UMIX +(c) UCMT +(f) Semi-supervised Segmentation +EMA +EMA +EMA +Figure 1: Illustration of the architectures and curves for co-training based semi-supervised semantic segmentation. +(a) Mean-teacher [4], (b) Cross pseudo supervision [5], (c) Uncertainty-guided collaborative mean-teacher, (d) the +disagreement between the pseudo labels in terms of dice loss of two branches (Y 1 and Y in MT; Y 1 and Y 2 in CPS; +Y 1 and Y 2 in UCMT) from the co-training sub-networks (w.r.t. number of iterations), (e) the uncertainty variation of +the pseudo labels in terms of entropy w.r.t. number of iterations, and (f) the performance of MT, CPS, and UCMT on +the semi-supervised skin lesion segmentation under different proportion of labeled data. +the performance of these approaches is the quality of pseudo labels. More importantly, these two factors influence +each other. Intuitively, high quality pseudo labels should have low uncertainty [11]. However, increasing the degree +of the disagreement between the co-training sub-networks by different perturbations or augmentations could result +in their opposite training directions, thus increasing the uncertainty of pseudo labels. To investigate the effect of the +disagreement and the quality of pseudo labels for co-training based semi-supervised segmentation, which has not been +studied in the literature, we conduct a pilot experiment to illustrate these correlations. As shown in Figure 1, com- +pared with mean-teacher (MT) [4] [Figure 1 (a)], cross pseudo supervision (CPS) [5] [Figure 1 (b)] with the higher +model disagreement [(d)] and the lower uncertainty [Figure 1 (e)] produces higher performance [Figure 1 (f)] on semi- +supervised segmentation. Note that the dice loss of two branches are calculated to measure the disagreement. The +question that comes to mind is: how to effectively improve the disagreement between the co-training sub-networks and +the quality of pseudo labels jointly in a unified network for SSL. +In this paper, we focus on two major goals: maintaining model disagreement and the high-confidence pseudo labels at +the same time. To this end, we propose the Uncertainty-guided Collaborative Mean Teacher (UCMT) framework that +is capable of retaining higher disagreement between the co-training segmentation sub-networks [Figure 1 (d)] based on +the higher confidence pseudo labels [Figure 1 (e)], thus achieving better semi-supervised segmentation performance +under the same backbone network and task settings [Figure 1 (f)]. Specifically, UCMT involves two major compo- +nents: 1) collaborative mean-teacher (CMT), and 2) uncertainty-guided region mix (UMIX), where UMIX operates +the input images according to the uncertainty maps of CMT while CMT performs co-training under the supervision +of the pseudo labels derived from the UMIX images. Inspired by the co-teaching [12, 10, 5] for struggling with early +converging to a consensus situation and degrading into self-training, we introduce a third component, the teacher +model, into the co-training framework as a regularizer to construct CMT for more effective SSL. The teacher model +acts as self-ensemble by averaging the student models, serving as a third part to guide the training of the two student +models. Further, we develop UMIX to construct high-confident pseudo labels and perform regional dropout for learn- +ing robust semi-supervised semantic segmentation models. Instead of random region erasing or swapping [13, 14], +UMIX manipulates the original image and its corresponding pseudo labels according to the epistemic uncertainty of +the segmentation models, which not only reduces the uncertainty of the pseudo labels but also enlarges the training +data distribution. Finally, by combining the strengths of UMIX with CMT, the proposed approach UCMT significantly +improves the state-of-the-art (sota) results in semi-supervised segmentation on multiple benchmark datasets. For ex- +ample, UCMT and UCMT(U-Net) achieve 88.22% and 82.14% Dice Similarity Coefficient (DSC) on ISIC dataset +under 5% labeled data, outperforming our baseline model CPS [5] and the state-of-the-art UGCL [15] by 1.41% and +9.47%, respectively. +In a nutshell, our contributions mainly include: +2 + +UCMT +• We pinpoint the problem in existing co-training based semi-supervised segmentation methods: the insufficient +disagreement among the sub-networks and the lower-confidence pseudo labels. To address the problem, we +design an uncertainty-guidedcollaborative mean-teacher to maintain co-training with high-confidence pseudo +labels, where we incorporate CMT and UMIX into a holistic framework for semi-supervised medical image +segmentation. +• To avoid introducing noise into the new samples, we propose an uncertainty-guided regional mix algorithm, +UMIX, which encourages the segmentation model to yield high-confident pseudo labels and enlarge the +training data distribution. +• We conduct extensive experiments on four public medical image segmentation datasets including 2D and 3D +scenarios to investigate the effectiveness of our method. Comprehensive results demonstrate the effectiveness +of each component of our method and the superiority of UCMT over the state-of-the-art. +2 +Related work +2.1 +Semi-supervised learning +Semi-supervised learning aims to improve performance in supervised learning by utilizing information generally asso- +ciated with unsupervised learning, and vice versa [3]. A common form of SSL is introducing a regularization term into +the objective function of supervised learning to leverage unlabeled data. From this perspective, SSL-based methods +can be divided into two main lines, i.e., pseudo labeling and consistency regularization. Pseudo labeling attempts to +generate pseudo labels similar to the ground truth, for which models are trained as in supervised learning [6]. Con- +sistency regularization enforces the model’s outputs to be consistent for the inputs under different perturbations [4]. +Current state-of-the-art approaches have incorporated these two strategies and shown superior performance for semi- +supervised image classification [16, 17]. Based on this line of research, we explore more effective consistency learning +algorithms for semi-supervised semantic segmentation. +2.2 +Semi-supervised semantic segmentation +Compared with image classification, semantic segmentation requires much more intensively and costly labeling for +pixel-level annotations. Semi-supervised semantic segmentation inherits the main ideas of semi-supervised image clas- +sification. The combination of consistency regularization and pseudo labeling, mainly conducting cross supervision +between sub-networks using pseudo labels, has become the mainstream strategy for semi-supervised semantic seg- +mentation in both natural images [7, 5] and medical images [18, 19, 20, 21]. Specifically, these combined approaches +enforce the consistency of the predictions under different perturbations, such as input perturbations [22, 23], feature +perturbations [7], and network perturbations [4, 5, 20, 21]. In addition, adversarial learning-based methods, rendering +the distribution of model predictions from labeled data to be aligned with those from unlabeled data, can also be re- +garded as a special form of consistency regularization [24, 25]. However, such cross supervision models may converge +early to a consensus, thus degenerating to self-training ones. We hypothesize that enlarging the disagreement for the +co-training models based on the high-confidence pseudo labels can improve the performance of SSL. Therefore, we +propose a novel SSL framework, i.e., UCMT, to generate more accurate pseudo labels and maintain co-training for +semi-supervised medical image segmentation. +2.3 +Uncertainty-guided semi-supervised semantic segmentation +Model uncertainty (epistemic uncertainty) can guide the SSL models to capture information from the pseudo labels. +Two critical problems for leveraging model uncertainty are how to obtain and exploit model uncertainty. Recently, +there are mainly two strategies to estimate model uncertainty: 1) using Monte Carlo dropout [26], and 2) calculating +the variance among different predictions [27]. For semi-supervised semantic segmentation, previous works exploit +model uncertainty to re-weight the training loss [18] or selecting the contrastive samples [15]. However, these methods +require manually setting a threshold to neglect the low-confidence pseudo labels, where the fixed threshold is hard to +determine. In this paper, we obtain the epistemic uncertainty by the entropy of the predictions of CMT for the same +input and exploit the uncertainty to guide the region mix for gradually exploring information from the unlabeled data. +3 +Methodology +3 + +UCMT +EMA +EMA +� �; �� +� �; �� +� �; � +Collaborative mean-teacher +UMIX +� +� +� +� +MAA +EMA +EMA +� �; �� +� �; �� +� �; � +Collaborative mean-teacher +� +� +� +� +MAA +First step: uncertainty estimation +Second step: training with UMIX +� �; � +Testing Phase +Loss functions +� +�’ +�� +�� +��� +��� +���� +���� +� +� +��� +���� +����� +����� +����� +����� +����� +����� +����� +����� +�� +�� +Training Phase +�� = ���� + ����� +����� +���� = ����� + ����� +���� = ����� + ����� +�� = ���� + ���� +� = �� + ��� +Figure 2: Overview of the proposed UCMT. CMT includes three sub-networks, i.e., the teacher sub-network (f(·; θ)) +and the two student sub-networks (f(·; θ1) and f(·; θ2)). UMIX constructs each new samples X′ by replacing the top +k most uncertain regions (red grids in V 1 and V 2) with the top k most certain regions (green grids in V 2 and V 1) in +the original image X. Note that three sub-networks are collaboratively learning during the training stage, while only +the teacher model is needed in the testing stage. +3.1 +Problem definition +Before introducing our method, we first define the semi-supervised segmentation problem with some notations used +in this work. The training set D = {DL, DU} contains a labeled set DL = {(Xi, Yi)N +i=1} and a unlabeled set +DU = {(Xj)M +j=N+1}, where Xi/Xj denotes the ith/jth labeled/unlabeled image, Yi is the ground truth of the labeled +image, and N and M − N are the number of labeled and unlabeled samples, respectively. Given the training data D, +the goal of semi-supervised semantic segmentation is to learn a model f(·; θ) performing well on unseen test sets. +3.2 +Overview +To avoid the co-training degrading to the self-training, we propose to encourage model disagreement during train- +ing and ensure pseudo labels with low uncertainty. With this motivation, we propose uncertainty-guided collabo- +rative mean-teacher for semi-supervised image segmentation, which includes 1) collaborative mean-teacher, and 2) +uncertainty-guided region mix. As shown in Figure 1 (d), CMT and UCMT gradually enlarge the disagreement +between the co-training sub-networks. Meanwhile, CMT equipped with UMIX guarantees low-uncertainty for the +pseudo labels. With the help of these conditions, we can safely maintain the co-training status to improve the effec- +tiveness of SSL for exploring unlabeled data. Details of CMT and UMIX are presented on Section 3.3 and Section +3.4, respectively. Figure 2 illustrates the schematic diagram of the proposed UCMT. Generally, there are two steps in +the training phase of UCMT. In the first step, we train CMT using the original labeled and unlabeled data to obtain the +uncertainty maps; Then, we perform UMIX to generate the new samples based on the uncertainty maps. In the second +step, we re-train CMT using the UMIX samples. Details of the training process of UCMT are shown in Algorithm +1. Although UCMT includes three models, i.e., one teacher model and two student models, only the teacher model is +required in the testing stage. +3.3 +Collaborative mean-teacher +Current consistency learning-based SSL algorithms, e.g., Mean-teacher [4] and CPS [5], suggest to perform consis- +tency regularization among the pseudo labels in a multi-model architecture rather than in a single model. However, +during the training process, the two-network SSL framework may converge early to a consensus and the co-training +degenerate to the self-training [10]. To tackle this issue, we design the collaborative mean teacher (CMT) framework +by introducing a "arbitrator", i.e., the teacher model, into the co-training architecture [5] to guide the training of the +4 + +UCMT +Algorithm 1 UCMT algorithm +Input: DL = {{(Xi, Yi)}N +i=1}, DU = {{Xj}M +j=N+1} +Parameter: θ, θ1, θ2 +Output: f(·; θ) +1: for T ∈ [1, numepochs] do +2: +for each minibatch B do +3: +// i/j is the index for labeled/unlabeled data +4: +step 1: uncertainty estimation +5: +ˆY 0 +i ← f(Xi ∈ B; θ), ˆY 0 +j ← f(Xj ∈ B; θ) +6: +ˆY 1 +i ← f(Xi ∈ B; θ1), ˆY 1 +j ← f(Xj ∈ B; θ1) +7: +ˆY 2 +i ← f(Xi ∈ B; θ2), ˆY 2 +j ← f(Xj ∈ B; θ2) +8: +L ← Ls( ˆY 0 +i , ˆY 1 +i , ˆY 2 +i , Yi) + λ(T )Lu( ˆY 0 +j , ˆY 1 +j , ˆY 2 +j ) +9: +Update f(·; θ), f(·; θ1), f(·; θ2) using optimizer +10: +U 1 +i ← Uncertain(f(Xi ∈ B; θ1), f(Xi ∈ B; θ)) +11: +U 2 +i ← Uncertain(f(Xi ∈ B; θ2), f(Xi ∈ B; θ)) +12: +U 1 +j ← Uncertain(f(Xj ∈ B; θ1), f(Xj ∈ B; θ)) +13: +U 2 +j ← Uncertain(f(Xj ∈ B; θ2), f(Xj ∈ B; θ)) +14: +step 2: training with UMIX +15: +X′ +i/Y ′ +i ← UMIX(Xi/Yi, U 1 +i , U 2 +i ; k, 1/r) +16: +X′ +j/ ˆY ′0 +j ← UMIX(Xj/ ˆY 0 +j , U 1 +j , U 2 +j ; k, 1/r) +17: +Repeat 3-7 using X′ +i, X′ +j, Y ′ +i , and ˆ +Y ′0 +j +18: +end for +19: end for +20: return f(·; θ) +two student models. As shown in Figure 2, CMT consists of one teacher model and two student models, where the +teacher model is the self-ensemble of the average of the student models. For labeled data, these models are all opti- +mized by supervised learning. For unlabeled data, there are two critical factors: 1) co-training between the two student +models, and 2) direct supervision from the teacher to the student models. Formally, the data flow diagram of CMT can +be illustrated as 2, +ր f (·; θ1) → ˆY 1 +X → f (·; θ) → ˆ +Y 0 +ց f (·; θ2) → ˆY 2, +(1) +where X is an input image of the labeled or unlabeled data, ˆY 0/ ˆY 1/ ˆY 2 is the predicted segmentation map, and +f(·; θ)/f(·; θ1)/f(·; θ2) with parameters θ, θ1 and θ2 denote the teacher model and student models, respectively. +These models have the same architecture but initialized with different weights for network perturbations. +To explore both the labeled and unlabeled data, the total loss L for training UCMT involves two parts, i.e., the super- +vised loss Ls and the unsupervised loss Lu. +L = Ls + λLu, +(2) +where λ is a regularization parameter to balance the supervised and unsupervised learning losses. We adopt a Gaussian +ramp-up function to gradually increase the coefficient, i.e., λ(t) = λm × exp [−5(1 − +t +tm )2], where λm scales the +maximum value of the weighted function, t denotes the current iteration, and tm is the maximum iteration in training. +Supervised learning path. For the labeled data, the supervised loss is formulated as, +Ls = 1 +N +N +� +i=1 +Lseg (f (Xi; θ) , Yi) ++ Lseg (f (Xi; θ1) , Yi) + Lseg (f (Xi; θ2) , Yi) , +(3) +where Lseg can be any supervised semantic segmentation loss, such as cross entropy loss and dice loss. Note that we +choose dice loss in our experiments as its compelling performance in medical image segmentation. +2We omit the image index i to indicate that X can be the labeled data or unlabeled data. +5 + +UCMT +Unsupervised learning path. The unsupervised loss Lu acts as a regularization term to explore potential knowledge +for the labeled and unlabeled data. Lu includes the cross pseudo supervision Lcps between the two student models +and the mean-teacher supervision Lmts for guiding the student models from the teacher, as follow: +Lu = Lcps + Lmts. +(4) +1) Cross pseudo supervision. The purpose of Lcps is to promote two students to learn from each other and to enforce +the consistency between them. Lcps = Lcps1 + Lcps2 encourages bidirectional interaction for the two student sub- +networks f(·; θ1) and f(·; θ2) as follows, +Lcps1 = +1 +M − N +M−N +� +j=1 +Lseg +� +f (Xj; θ1) , ˆY 2 +j +� +Lcps2 = +1 +M − N +M−N +� +j=1 +Lseg +� +f (Xj; θ2) , ˆY 1 +j +� +, +(5) +where ˆY 1 +j and ˆY 2 +j are the pseudo labels (segmentation maps) for Xj predicted by f (·; θ1) and f (·; θ1) , respectively. +2) Mean-teacher supervision. To avoid the two students cross supervision in the wrong direction, we introduce a +teacher model to guide the optimization of the student models. Specifically, the teacher model is updated by the +exponential moving average (EMA) of the average of the student models: +θt = αθt−1 + (1 − α)θt +1 + θt +2 +2 +, +(6) +where t represents the current training iteration. α is the EMA decay that controls the parameters’ updating rate and +we set α = 0.999 in our experiments. +The loss of mean-teacher supervision Lmts = Lmts1 + Lmts2 is calculated from two branches: +Lmts1 = +1 +M − N +M−N +� +j=1 +Lseg +� +f (Xj; θ1) , ˆY 0 +j +� +Lmts2 = +1 +M − N +M−N +� +j=1 +Lseg +� +f (Xj; θ2) , ˆY 0 +j +� +, +(7) +where ˆY 0 +j refers to the predicted segmentation map derived from f (Xj; θ). +3.4 +Uncertainty-guided Mix +Although CMT can promote model disagreement for co-training, it also slightly increases the uncertainty of the pseudo +labels as depicted in Figure 1. On the other hand, random regional dropout can expand the training distribution and +improve the generalization capability of models [13, 14]. However, such random perturbations to the input images in- +evitably introduce noise into the new samples, thus deteriorating the quality of pseudo labels for SSL. One sub-network +may provide some incorrect pseudo labels to the other sub-networks, degrading their performance. To overcome these +limitations, we propose UMIX to manipulate image patches under the guidance of the uncertainty maps produced by +CMT. The main idea of UMIX is constructing a new sample by replacing the top k most uncertain (low-confidence) re- +gions with the top k most certain (high-confidence) regions in the input image. As illustrated in Figure 2, for example, +we obtain the most uncertain regions (the red grids) and the most certain regions (the green grids) from an uncertainty +map U. Then, we replace the red regions with the green regions in the input image X to construct a new sample X′. +Formally, UMIX constructs a new sample X′ = UMIX(X, U 1, U 2; k, 1/r) by replacing the top k most uncertain +regions (red grids in V 1 and V 2) with the top k most certain regions (green grids in V 2 and V 1) in X, where each +region has size 1/r to the image size. To ensure the reliability of the uncertainty evaluation, we obtain the uncertain +maps by integrating the outputs of the teacher and the student model instead of performing T stochastic forward +passes designed by Monte Carlo Dropout estimate model [26, 18], which is equivalent to sampling predictions from +the previous and current iterations. This process can be formulated as: +U m = Uncertain(f(X; θm), f(X; θ)) = − +� +c +Pc log(Pc), +Pc = 1 +2(Softmax(f(X; θm)) + Softmax(f(X; θ))), +(8) +where m = 1, 2 denotes the index of the student models and c refers to the class index. +6 + +UCMT +Table 1: Comparison with state-of-the-art methods on ISIC dataset. 5% DL and 10% DL of the labeled data are used +for training, respectively. Results are measured by DSC. +Method +5% DL +10% DL +MT [4] +86.67 +87.42 +CCT [7] +83.97 +86.43 +CPS [5] +86.81 +87.70 +UGCL(U-Net) [15] +72.67 +79.48 +UCMT(U-Net) (ours) +82.14 +83.33 +CMT (ours) +87.86 +88.10 +UCMT (ours) +88.22 +88.46 +4 +Experiments and results +4.1 +Experiments Settings +Datasets. We conduct extensive experiments on different medical image segmentation tasks to evaluate the proposed +method, including skin lesion segmentation from dermoscopy images, polyp segmentation from colonoscopy images, +and the 3D left atrium segmentation from cardiac MRI images. +Dermoscopy. We validate our method on the ISIC dataset [28] including 2594 dermoscopy images and corresponding +annotations. Following [15], we adopt 1815 images for training and 779 images for validation. +Colonoscopy. We evaluate the proposed method on the two public colonoscopy datasets, including Kvasir-SEG [29] +and CVC-ClinicDB [30]. Kvasir-SEG and CVC-ClinicDB contain 1000 and 612 colonoscopy images with correspond- +ing annotations, respectively. +Cardiac MRI. We evaluate our method on the 3D left atrial (LA) segmentation challenge dataset, which consists +of 100 3D gadolinium-enhanced magnetic resonance images and LA segmentation masks for training and validation. +Following [18], we split the 100 scans into 80 samples for training and 20 samples for evaluation. +4.1.1 +Implementation details +We use DeepLabv3+ [1] equipped with ResNet50 as the baseline architecture for 2D image segmentation, whereas +adopt VNet [31] as the baseline in the 3D scenario. All images are resized to 256×256 for inference, while the outputs +are recovered to the original size for evaluation, in the 2D scenario. For 3D image segmentation, we randomly crop +80 × 112 × 112(Depth × Height × Width) patches for training and iteratively crop patches using a sliding window +strategy to obtain the final segmentation mask for testing. We implement our method using PyTorch framework on a +NVIDIA Quadro RTX 6000 GPU. We adopt AdamW as an optimizer with the fixed learning rate of le-4. The batchsize +is set to 16, including 8 labeled samples and 8 unlabeled samples. All 2D models are trained for 50 epochs, while the +3D models are trained for 1000 epochs 3. We empirically set k = 2 and r = 16 for our method in the experiment. +4.2 +Comparison with state of the arts +We compare the proposed method with state-of-the art on the four public medical image segmentation datasets. We +re-implement MT [4], CCT [7], and CPS [5] by adopting implementations from [5]. For other approaches, we directly +use the results reported in their original papers. +Results on Dermoscopy. In Table 1, we report the results of our methods on ISIC and compare them with other state- +of-the-art approaches. UCMT substantially outperforms all previous methods and sets new state-of-the-art of 88.22% +DSC and 88.46 DSC under 5% and 10% labeled data. For fair comparison with UGCL [15], replace the backbone +of UCMT with U-Net. The results indicate that our UCMT(U-Net) exceeds UGCL by a large margin. Moreover, our +CMT version also outperforms other approaches under the two labeled data rates. For example, CMT surpasses MT +and CPS by 1.19% and 1.08% on 5% DL labeled data, showing the superiority of collaborative mean-teacher against +the current consistency learning framework. By introducing UMIX, UCMT consistently increases the performance +under different labeled data rates, which implies that promoting model disagreement and guaranteeing high-confident +pseudo labels are beneficial for semi-supervised segmentation. +3Since UCMT performs the two-step training within one iteration, it is trained for half of the epochs. +7 + +UCMT +Table 2: Comparison with state-of-the-art methods on Kvasir-SEG and CVC-ClinicDB datasets. 15% DL and 30% +DL of the labeled data are individually used for training. Results are measured by DSC. +Method +Kvasir-SEG +CVC-ClinicDB +15% DL +30% DL +15% DL +30% DL +AdvSemSeg [24] +56.88 +76.09 +68.39 +75.93 +ColAdv [32] +76.76 +80.95 +82.18 +89.29 +MT [4] +87.44 +88.72 +84.19 +84.40 +CCT [7] +81.14 +84.67 +74.20 +78.46 +CPS [5] +86.44 +88.71 +85.34 +86.69 +CMT (ours) +88.08 +88.61 +85.88 +86.83 +UCMT (ours) +88.68 +89.06 +87.30 +87.51 +Results on Colonoscopy. +We further conduct a comparative experiment on the polyp segmentation task from +colonoscopy images. Table 2 reports the quantitative results on Kvasir-SEG and CVC-ClinicDB datasets. Com- +pared with the adversarial learning-based [24, 32] and consistency learning-based [4, 7, 5] algorithms, the proposed +methods achieve the state-of-the-art performance. For example, both CMT and UCMT outperform AdvSemSeg [24] +and ColAdv [32] by large margins on Kvasir-SEG and CVC-ClinicDB, except that ColAdv shows the better perfor- +mance of 89.29% on CVC-ClinicDB under 30% labeled data. These results demonstrate that our uncertainty-guided +collaborative mean-teacher scheme is superior to the adversarial learning and consistency learning schemes used in +the compared approaches. In addition, CMT and UCMT show better performance on the low-data regime, i.e., 15% +DL, and the performance between 15% DL and 30% DL labeled data is close. This phenomenon reflects the capacity +of our method to produce high-quality pseudo labels from unlabeled data for semi-supervised learning, even with less +labeled data. +Results on Cardiac MRI. We further evaluate the proposed method in the 3D medical image segmentation task. Table +3 shows the comparison results on the 3D left atrium segmentation from cardiac MRI. The compared approaches are +based on consistency learning and pseudo labeling, including uncertainty-aware [18, 33], shape-aware [25], structure- +aware [34], dual-task [19], and mutual training [20] consistency. It can be observed that UCMT achieves the best +performance under 10% and 20% DL in terms of DSC and Jaccard over the state-of-the-art methods. For example, +compared with UA-MT [18] and MC-Net [20], UCMT shows 3.88% DSC and 0.43% DSC improvements on the 10% +labeled data. The results demonstrate the superiority of our UCMT for 3D medical image segmentation. +Table 3: Comparison with state-of-the-art methods on the 3D left atrial segmentation challenge dataset. 10% DL and +20% DL of the labeled data are used for training. +Method +10% DL +20% DL +DSC +Jaccard +95HD +ASD +DSC +Jaccard +95HD +ASD +UA-MT [18] +84.25 +73.48 +13.84 +3.36 +88.88 +80.21 +7.32 +2.26 +SASSNet [25] +87.32 +77.72 +9.62 +2.55 +89.54 +81.24 +8.24 +2.20 +LG-ER-MT [34] +85.54 +75.12 +13.29 +3.77 +89.62 +81.31 +7.16 +2.06 +DUWM [33] +85.91 +75.75 +12.67 +3.31 +89.65 +81.35 +7.04 +2.03 +DTC [19] +86.57 +76.55 +14.47 +3.74 +89.42 +80.98 +7.32 +2.10 +MC-Net [20] +87.71 +78.31 +9.36 +2.18 +90.34 +82.48 +6.00 +1.77 +MT [4] +86.15 +76.16 +11.37 +3.60 +89.81 +81.85 +6.08 +1.96 +CPS [5] +86.23 +76.22 +11.68 +3.65 +88.72 +80.01 +7.49 +1.91 +CMT (ours) +87.23 +77.83 +7.83 +2.23 +89.88 +81.74 +6.07 +1.94 +UCMT (ours) +88.13 +79.18 +9.14 +3.06 +90.41 +82.54 +6.31 +1.70 +4.3 +Ablation study +We conduct an ablation study in terms of network architectures, loss functions, and region mix to investigate the +effectiveness of each component and analyze the hyperparameters of the proposed method. There are three types of +network architectures: 1) teacher-student (TS) in MT [4], 2) student-student (SS) in CPS [5], and 3) student-teacher- +student in the proposed CMT. +8 + +UCMT +Effectiveness of each component. Table 4 reports the performance improvements over the baseline. It shows a trend +that the segmentation performance improves when the components, including the STS (student-teacher-student), Lcps, +Lmts, and UMIX are introduced into the baseline, and again confirms the necessity of encouraging model disagreement +and enhancing the quality of pseudo labels for semi-supervised segmentation. The semi-supervised segmentation +model is boosted for two reasons: 1) Lcps, Lmts and the STS architecture that force the model disagreement in CMT +for co-training, and 2) UMIX facilitating the model to produce high-confidence pseudo labels. All the components +contribute to UCMT to achieve 88.22% DSC. These results demonstrate their effectiveness and complementarity for +semi-supervised medical image segmentation. On the other hand, the two groups of comparisons between "TS (teacher- +student) + Lmts" (i.e., MT) and STS + Lmts (i.e., CMTv1), and between "SS (student-student) + Lcps" (i.e., CPS) and +"STS + Lcps" (i.e., CMTv2) show that the STS-based approaches yield the improvements of 0.17% and 0.64%, which +verifies the effectiveness of our STS for SSL. The performance gaps are not significant because the STS architecture +increases the co-training disagreement but decreases the confidences of pseudo labels. However, It can be easily found +that the results are improved to 87.86% by "STS + Lcps + Lmts" (i.e., CMTv3) and the relative improvements of 1.55% +and 1.41% DSC have been achieved by "STS + Lcps + Lmts + UMIX" (i.e., UCMT) compared with MT and CPS. +The results demonstrate our hypothesis that maintaining co-training with high-confidence pseudo labels can improve +the performance of semi-supervised learning. +Table 4: Ablation study of the different components combinations on ISIC dataset. All models are trained for 5% +labeled data. TS: teacher-student; SS: student-student; STS: student-teacher-student; Lcps: cross pseudo supervision; +Lmts: mean-teacher supervision; U: UMIX; +Method +TS +SS +STS +Lcps +Lmts +U +DSC +Baseline +83.31 +MT +√ +√ +86.67 +CPS +√ +√ +86.81 +CMTv1 +√ +√ +86.84 +CMTv2 +√ +√ +87.48 +CMTv3 +√ +√ +√ +87.86 +UCMT +√ +√ +√ +√ +88.22 +Comparison with CutMix. We further compare the proposed UMIX, component of our UCMT, with CutMix [14] +on ISIC and LA datasets with different labeled data to investigate their effects in semi-supervised medical image +segmentation. As illustrates in Figure 3, UMIX outperforms CutMix, especial in the low-data regime, i.e., 2% labeled +data. The reason for this phenomenon is that CutMix performs random region mix that inevitably introduces noise into +the new samples, which reduces the quality of the pseudo labels, while UMIX processes the image regions according +to the uncertainty of the model, which facilitates the model to generate more confident pseudo labels from the new +samples. +(a) ISIC +(b) LA +CMT+CutMix +CMT+UMIX +CMT +CMT+CutMix +CMT+UMIX +CMT +Figure 3: Comparison UCMT (UMIX) with CutMix on ISIC (a) and LA (b) dataset under 2%, 5%, 10%, and 20% +DL. +Parameter sensitivity analysis. UMIX has two hyperparamters, i.e., the top k regions for mix and the size of the +regions (patches) 1/r to the image size. We study the influence of these factors to UCMT on ISIC dataset with 5% DL. +It can be observed in Table 5 that reducing the patch size leads to a slight increase in performance. Moreover, varying +the number of k does not bring us any improvement. The reason for this phenomenon is that we can choose any value +9 + +UCMT +of K to eliminate outliers, thus bringing high-confidence pseudo labels for semi-supervised learning, indicating the +robustness of UMIX. +Table 5: Investigation on how the top k and region size affect the capacity of UMIX. All results are evaluated on ISIC +dataset with 5% labeled data. +1/r +k +1 +2 +3 +4 +5 +1/16 +87.95 +88.22 +88.12 +87.96 +88.08 +1/4 +87.65 +88.15 +87.80 +87.54 +87.90 +1/8 +87.86 +87.92 +88.03 +88.01 +87.87 +4.4 +Qualitative results +Figure 4 visualizes some example results of polyp segmentation, skin lesion segmentation, and left atrial segmentation. +As shown in Figure 4 (a), the supervised baseline insufficiently segments some lesion regions, mainly due to the limited +number of labeled data. Moreover, MT [Figure 4 (b)] and CPS [Figure 4 (c)] typically under-segment certain objects, +which can be attributed to the limited generalization capability. On the contrary, our CMT [Figure 4 (e)] corrects these +errors and produces smoother segment boundaries by gaining more effective supervision from unlabeled data. Besides, +our complete method UCMT [Figure 4 (f)] further generates more accurate results by recovering finer segmentation +details through more efficient training. These examples qualitatively verify the robustness of the proposed UCMT. In +addition, to clearly give an insight into the procedure of the pseudo label generation and utilization in the co-training +SSL method, we illustrate the uncertainty maps for two samples during the training in Figure 5. As shown, UCMT +generates the uncertainty maps with high uncertainty [Figure 5 (a)/(c)] in the early training stage whereas our model +produces relative higher confidence maps [Figure 5 (b)/(d)] from the UMIX images. During training, UCMT gradually +improves the confidence for the input images. These results prove that UMIX can facilitate SSL models to generate +high-confidence pseudo labels during training, guaranteeing that UCMT is able to maintain co-training in a more +proper way. +5 +Conclusion +We present an uncertainty-guided collaborative mean-teacher for semi-supervised medical image segmentation. Our +main ideas lies in maintaining co-training with high-confidence pseudo labels to improve the capability of the SSL +models to explore information from unlabeled data. Extensive experiments on four public datasets demonstrate the +effectiveness of this idea and show that the proposed UCMT can achieve state-of-the-art performance. In the future, +we will investigate more deeply the underlying mechanisms of co-training for more effective semi-supervised image +segmentation. +Acknowledgments +This work was supported by the National Natural Science Foundation of China under Grant 62076059 and the Natural +Science Foundation of Liaoning Province under Grant 2021-MS-105. +References +[1] Liang-Chieh Chen, Yukun Zhu, George Papandreou, Florian Schroff, and Hartwig Adam. Encoder-decoder with +atrous separable convolution for semantic image segmentation. 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Springer, 2020. +13 + diff --git a/GNE3T4oBgHgl3EQfWAqd/content/tmp_files/load_file.txt b/GNE3T4oBgHgl3EQfWAqd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..27fcacff40d5a1af34baf4da8250b0f0c5b260ff --- /dev/null +++ b/GNE3T4oBgHgl3EQfWAqd/content/tmp_files/load_file.txt @@ -0,0 +1,628 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf,len=627 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='04465v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='CV] 11 Jan 2023 CO-TRAINING WITH HIGH-CONFIDENCE PSEUDO LABELS FOR SEMI-SUPERVISED MEDICAL IMAGE SEGMENTATION Zhiqiang Shen1,2 Peng Cao1,2∗ Hua Yang3 Xiaoli Liu4 Jinzhu Yang1,2 Osmar R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Zaiane5 1College of Computer Science and Engineering, Northeastern University, Shenyang, China 2Key Laboratory of Intelligent Computing in Medical Image, Ministry of Education, Shenyang, China 3College of Photonic and Electronic Engineering, Fujian Normal University, Fuzhou, China 4DAMO Academy, Alibaba Group, China 5Alberta Machine Intelligence Institute, University of Alberta, Edmonton, Alberta, Canada xxszqyy@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='com, caopeng@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='neu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='cn ABSTRACT High-quality pseudo labels are essential for semi-supervised semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Consistency regularization and pseudo labeling-based semi-supervised methods perform co-training using the pseudo labels from multi-view inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' However, such co-training models tend to converge early to a consensus during training, so that the models degenerate to the self-training ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Besides, the multi-view inputs are generated by perturbing or augmenting the original images, which inevitably introduces noise into the input leading to low-confidence pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' To address these issues, we propose an Uncertainty-guided Collaborative Mean-Teacher (UCMT) for semi-supervised se- mantic segmentation with the high-confidence pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Concretely, UCMT consists of two main components: 1) collaborative mean-teacher (CMT) for encouraging model disagreement and performing co-training between the sub-networks, and 2) uncertainty-guided region mix (UMIX) for manipulating the input images according to the uncertainty maps of CMT and facilitating CMT to produce high-confidence pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Combining the strengths of UMIX with CMT, UCMT can retain model disagreement and enhance the quality of pseudo labels for the co-training seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Extensive experiments on four public medical image datasets including 2D and 3D modalities demonstrate the superiority of UCMT over the state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Code is available at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='com/Senyh/UCMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 1 Introduction Semantic segmentation is critical for medical image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Great progress has been made by deep learning-based segmentation models relying on a large amount of labeled data [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' However, labeling such pixel-level annotations is laborious and requires expert knowledge especially in medical images, resulting in that labeled data are expensive or simply unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Unlabeled data, on the contrary, are cheap and relatively easy to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Under this condition, semi-supervised learning (SSL) has been the dominant data-efficient strategy through exploiting information from a limited amount labeled data and an arbitrary amount of unlabeled data, so as to alleviate the label scarcity problem [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Consistency regularization [4] and pseudo labeling [6] are the two main methods for semi-supervised semantic seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Currently, combining consistency regularization and pseudo labeling via cross supervision between the sub-networks, has shown promising performance for semi-supervised segmentation [6, 7, 8, 5, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' One critical limita- tion of these approaches is that the sub-networks tend to converge early to a consensus situation causing the co-training model degenerating to the self-training [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Disagreement between the sub-networks is crucial for co-training, where the sub-networks initialized with different parameters or trained with different views have different biases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=', dis- agreement) ensuring that the information they provide is complementary to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Another key factor affecting ∗corresponding author UCMT (d) Co-training disagreement (e) Pseudo labels uncertainty � � �� � � �� � (a) MT � � �� � �� �� �� (b) CPS � � � � � �� � �� �� �� UMIX (c) UCMT (f) Semi-supervised Segmentation EMA EMA EMA Figure 1: Illustration of the architectures and curves for co-training based semi-supervised semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' (a) Mean-teacher [4], (b) Cross pseudo supervision [5], (c) Uncertainty-guided collaborative mean-teacher, (d) the disagreement between the pseudo labels in terms of dice loss of two branches (Y 1 and Y in MT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Y 1 and Y 2 in CPS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Y 1 and Y 2 in UCMT) from the co-training sub-networks (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' number of iterations), (e) the uncertainty variation of the pseudo labels in terms of entropy w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' number of iterations, and (f) the performance of MT, CPS, and UCMT on the semi-supervised skin lesion segmentation under different proportion of labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' the performance of these approaches is the quality of pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' More importantly, these two factors influence each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Intuitively, high quality pseudo labels should have low uncertainty [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' However, increasing the degree of the disagreement between the co-training sub-networks by different perturbations or augmentations could result in their opposite training directions, thus increasing the uncertainty of pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' To investigate the effect of the disagreement and the quality of pseudo labels for co-training based semi-supervised segmentation, which has not been studied in the literature, we conduct a pilot experiment to illustrate these correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' As shown in Figure 1, com- pared with mean-teacher (MT) [4] [Figure 1 (a)], cross pseudo supervision (CPS) [5] [Figure 1 (b)] with the higher model disagreement [(d)] and the lower uncertainty [Figure 1 (e)] produces higher performance [Figure 1 (f)] on semi- supervised segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Note that the dice loss of two branches are calculated to measure the disagreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' The question that comes to mind is: how to effectively improve the disagreement between the co-training sub-networks and the quality of pseudo labels jointly in a unified network for SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' In this paper, we focus on two major goals: maintaining model disagreement and the high-confidence pseudo labels at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' To this end, we propose the Uncertainty-guided Collaborative Mean Teacher (UCMT) framework that is capable of retaining higher disagreement between the co-training segmentation sub-networks [Figure 1 (d)] based on the higher confidence pseudo labels [Figure 1 (e)], thus achieving better semi-supervised segmentation performance under the same backbone network and task settings [Figure 1 (f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Specifically, UCMT involves two major compo- nents: 1) collaborative mean-teacher (CMT), and 2) uncertainty-guided region mix (UMIX), where UMIX operates the input images according to the uncertainty maps of CMT while CMT performs co-training under the supervision of the pseudo labels derived from the UMIX images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Inspired by the co-teaching [12, 10, 5] for struggling with early converging to a consensus situation and degrading into self-training, we introduce a third component, the teacher model, into the co-training framework as a regularizer to construct CMT for more effective SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' The teacher model acts as self-ensemble by averaging the student models, serving as a third part to guide the training of the two student models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Further, we develop UMIX to construct high-confident pseudo labels and perform regional dropout for learn- ing robust semi-supervised semantic segmentation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Instead of random region erasing or swapping [13, 14], UMIX manipulates the original image and its corresponding pseudo labels according to the epistemic uncertainty of the segmentation models, which not only reduces the uncertainty of the pseudo labels but also enlarges the training data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Finally, by combining the strengths of UMIX with CMT, the proposed approach UCMT significantly improves the state-of-the-art (sota) results in semi-supervised segmentation on multiple benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' For ex- ample, UCMT and UCMT(U-Net) achieve 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='22% and 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='14% Dice Similarity Coefficient (DSC) on ISIC dataset under 5% labeled data, outperforming our baseline model CPS [5] and the state-of-the-art UGCL [15] by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='41% and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='47%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' In a nutshell, our contributions mainly include: 2 UCMT We pinpoint the problem in existing co-training based semi-supervised segmentation methods: the insufficient disagreement among the sub-networks and the lower-confidence pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' To address the problem, we design an uncertainty-guidedcollaborative mean-teacher to maintain co-training with high-confidence pseudo labels, where we incorporate CMT and UMIX into a holistic framework for semi-supervised medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' To avoid introducing noise into the new samples, we propose an uncertainty-guided regional mix algorithm, UMIX, which encourages the segmentation model to yield high-confident pseudo labels and enlarge the training data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' We conduct extensive experiments on four public medical image segmentation datasets including 2D and 3D scenarios to investigate the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Comprehensive results demonstrate the effectiveness of each component of our method and the superiority of UCMT over the state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 2 Related work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='1 Semi-supervised learning Semi-supervised learning aims to improve performance in supervised learning by utilizing information generally asso- ciated with unsupervised learning, and vice versa [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' A common form of SSL is introducing a regularization term into the objective function of supervised learning to leverage unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' From this perspective, SSL-based methods can be divided into two main lines, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=', pseudo labeling and consistency regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Pseudo labeling attempts to generate pseudo labels similar to the ground truth, for which models are trained as in supervised learning [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Con- sistency regularization enforces the model’s outputs to be consistent for the inputs under different perturbations [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Current state-of-the-art approaches have incorporated these two strategies and shown superior performance for semi- supervised image classification [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Based on this line of research, we explore more effective consistency learning algorithms for semi-supervised semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='2 Semi-supervised semantic segmentation Compared with image classification, semantic segmentation requires much more intensively and costly labeling for pixel-level annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Semi-supervised semantic segmentation inherits the main ideas of semi-supervised image clas- sification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' The combination of consistency regularization and pseudo labeling, mainly conducting cross supervision between sub-networks using pseudo labels, has become the mainstream strategy for semi-supervised semantic seg- mentation in both natural images [7, 5] and medical images [18, 19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Specifically, these combined approaches enforce the consistency of the predictions under different perturbations, such as input perturbations [22, 23], feature perturbations [7], and network perturbations [4, 5, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' In addition, adversarial learning-based methods, rendering the distribution of model predictions from labeled data to be aligned with those from unlabeled data, can also be re- garded as a special form of consistency regularization [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' However, such cross supervision models may converge early to a consensus, thus degenerating to self-training ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' We hypothesize that enlarging the disagreement for the co-training models based on the high-confidence pseudo labels can improve the performance of SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Therefore, we propose a novel SSL framework, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=', UCMT, to generate more accurate pseudo labels and maintain co-training for semi-supervised medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='3 Uncertainty-guided semi-supervised semantic segmentation Model uncertainty (epistemic uncertainty) can guide the SSL models to capture information from the pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Two critical problems for leveraging model uncertainty are how to obtain and exploit model uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Recently, there are mainly two strategies to estimate model uncertainty: 1) using Monte Carlo dropout [26], and 2) calculating the variance among different predictions [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' For semi-supervised semantic segmentation, previous works exploit model uncertainty to re-weight the training loss [18] or selecting the contrastive samples [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' However, these methods require manually setting a threshold to neglect the low-confidence pseudo labels, where the fixed threshold is hard to determine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' In this paper, we obtain the epistemic uncertainty by the entropy of the predictions of CMT for the same input and exploit the uncertainty to guide the region mix for gradually exploring information from the unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 3 Methodology 3 UCMT EMA EMA � �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' �� � �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' �� � �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' � Collaborative mean-teacher UMIX � � � � MAA EMA EMA � �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' �� � �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' �� � �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' � Collaborative mean-teacher � � � � MAA First step: uncertainty estimation Second step: training with UMIX � �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' � Testing Phase Loss functions � �’ �� �� ��� ��� ���� ���� � � ��� ���� ����� ����� ����� ����� ����� ����� ����� ����� �� �� Training Phase �� = ���� + ����� +����� ���� = ����� + ����� ���� = ����� + ����� �� = ���� + ���� � = �� + ��� Figure 2: Overview of the proposed UCMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' CMT includes three sub-networks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=', the teacher sub-network (f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ)) and the two student sub-networks (f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ1) and f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' UMIX constructs each new samples X′ by replacing the top k most uncertain regions (red grids in V 1 and V 2) with the top k most certain regions (green grids in V 2 and V 1) in the original image X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Note that three sub-networks are collaboratively learning during the training stage, while only the teacher model is needed in the testing stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='1 Problem definition Before introducing our method, we first define the semi-supervised segmentation problem with some notations used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' The training set D = {DL, DU} contains a labeled set DL = {(Xi, Yi)N i=1} and a unlabeled set DU = {(Xj)M j=N+1}, where Xi/Xj denotes the ith/jth labeled/unlabeled image, Yi is the ground truth of the labeled image, and N and M − N are the number of labeled and unlabeled samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Given the training data D, the goal of semi-supervised semantic segmentation is to learn a model f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ) performing well on unseen test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='2 Overview To avoid the co-training degrading to the self-training, we propose to encourage model disagreement during train- ing and ensure pseudo labels with low uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' With this motivation, we propose uncertainty-guided collabo- rative mean-teacher for semi-supervised image segmentation, which includes 1) collaborative mean-teacher, and 2) uncertainty-guided region mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' As shown in Figure 1 (d), CMT and UCMT gradually enlarge the disagreement between the co-training sub-networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Meanwhile, CMT equipped with UMIX guarantees low-uncertainty for the pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' With the help of these conditions, we can safely maintain the co-training status to improve the effec- tiveness of SSL for exploring unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Details of CMT and UMIX are presented on Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='3 and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Figure 2 illustrates the schematic diagram of the proposed UCMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Generally, there are two steps in the training phase of UCMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' In the first step, we train CMT using the original labeled and unlabeled data to obtain the uncertainty maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Then, we perform UMIX to generate the new samples based on the uncertainty maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' In the second step, we re-train CMT using the UMIX samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Details of the training process of UCMT are shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Although UCMT includes three models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=', one teacher model and two student models, only the teacher model is required in the testing stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='3 Collaborative mean-teacher Current consistency learning-based SSL algorithms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=', Mean-teacher [4] and CPS [5], suggest to perform consis- tency regularization among the pseudo labels in a multi-model architecture rather than in a single model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' However, during the training process, the two-network SSL framework may converge early to a consensus and the co-training degenerate to the self-training [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' To tackle this issue, we design the collaborative mean teacher (CMT) framework by introducing a "arbitrator", i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=', the teacher model, into the co-training architecture [5] to guide the training of the 4 UCMT Algorithm 1 UCMT algorithm Input: DL = {{(Xi, Yi)}N i=1}, DU = {{Xj}M j=N+1} Parameter: θ, θ1, θ2 Output: f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ) 1: for T ∈ [1, numepochs] do 2: for each minibatch B do 3: // i/j is the index for labeled/unlabeled data 4: step 1: uncertainty estimation 5: ˆY 0 i ← f(Xi ∈ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ), ˆY 0 j ← f(Xj ∈ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ) 6: ˆY 1 i ← f(Xi ∈ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ1), ˆY 1 j ← f(Xj ∈ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ1) 7: ˆY 2 i ← f(Xi ∈ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ2), ˆY 2 j ← f(Xj ∈ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ2) 8: L ← Ls( ˆY 0 i , ˆY 1 i , ˆY 2 i , Yi) + λ(T )Lu( ˆY 0 j , ˆY 1 j , ˆY 2 j ) 9: Update f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ), f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ1), f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ2) using optimizer 10: U 1 i ← Uncertain(f(Xi ∈ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ1), f(Xi ∈ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ)) 11: U 2 i ← Uncertain(f(Xi ∈ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ2), f(Xi ∈ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ)) 12: U 1 j ← Uncertain(f(Xj ∈ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ1), f(Xj ∈ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ)) 13: U 2 j ← Uncertain(f(Xj ∈ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ2), f(Xj ∈ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ)) 14: step 2: training with UMIX 15: X′ i/Y ′ i ← UMIX(Xi/Yi, U 1 i , U 2 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' k, 1/r) 16: X′ j/ ˆY ′0 j ← UMIX(Xj/ ˆY 0 j , U 1 j , U 2 j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' k, 1/r) 17: Repeat 3-7 using X′ i, X′ j, Y ′ i , and ˆ Y ′0 j 18: end for 19: end for 20: return f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ) two student models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' As shown in Figure 2, CMT consists of one teacher model and two student models, where the teacher model is the self-ensemble of the average of the student models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' For labeled data, these models are all opti- mized by supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' For unlabeled data, there are two critical factors: 1) co-training between the two student models, and 2) direct supervision from the teacher to the student models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Formally, the data flow diagram of CMT can be illustrated as 2, ր f (·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ1) → ˆY 1 X → f (·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ) → ˆ Y 0 ց f (·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ2) → ˆY 2, (1) where X is an input image of the labeled or unlabeled data, ˆY 0/ ˆY 1/ ˆY 2 is the predicted segmentation map, and f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ)/f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ1)/f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ2) with parameters θ, θ1 and θ2 denote the teacher model and student models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' These models have the same architecture but initialized with different weights for network perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' To explore both the labeled and unlabeled data, the total loss L for training UCMT involves two parts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=', the super- vised loss Ls and the unsupervised loss Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' L = Ls + λLu, (2) where λ is a regularization parameter to balance the supervised and unsupervised learning losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' We adopt a Gaussian ramp-up function to gradually increase the coefficient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=', λ(t) = λm × exp [−5(1 − t tm )2], where λm scales the maximum value of the weighted function, t denotes the current iteration, and tm is the maximum iteration in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Supervised learning path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' For the labeled data, the supervised loss is formulated as, Ls = 1 N N � i=1 Lseg (f (Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ) , Yi) + Lseg (f (Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ1) , Yi) + Lseg (f (Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ2) , Yi) , (3) where Lseg can be any supervised semantic segmentation loss, such as cross entropy loss and dice loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Note that we choose dice loss in our experiments as its compelling performance in medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 2We omit the image index i to indicate that X can be the labeled data or unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 5 UCMT Unsupervised learning path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' The unsupervised loss Lu acts as a regularization term to explore potential knowledge for the labeled and unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Lu includes the cross pseudo supervision Lcps between the two student models and the mean-teacher supervision Lmts for guiding the student models from the teacher, as follow: Lu = Lcps + Lmts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' (4) 1) Cross pseudo supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' The purpose of Lcps is to promote two students to learn from each other and to enforce the consistency between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Lcps = Lcps1 + Lcps2 encourages bidirectional interaction for the two student sub- networks f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ1) and f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ2) as follows, Lcps1 = 1 M − N M−N � j=1 Lseg � f (Xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ1) , ˆY 2 j � Lcps2 = 1 M − N M−N � j=1 Lseg � f (Xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ2) , ˆY 1 j � , (5) where ˆY 1 j and ˆY 2 j are the pseudo labels (segmentation maps) for Xj predicted by f (·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ1) and f (·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ1) , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 2) Mean-teacher supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' To avoid the two students cross supervision in the wrong direction, we introduce a teacher model to guide the optimization of the student models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Specifically, the teacher model is updated by the exponential moving average (EMA) of the average of the student models: θt = αθt−1 + (1 − α)θt 1 + θt 2 2 , (6) where t represents the current training iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' α is the EMA decay that controls the parameters’ updating rate and we set α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='999 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' The loss of mean-teacher supervision Lmts = Lmts1 + Lmts2 is calculated from two branches: Lmts1 = 1 M − N M−N � j=1 Lseg � f (Xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ1) , ˆY 0 j � Lmts2 = 1 M − N M−N � j=1 Lseg � f (Xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ2) , ˆY 0 j � , (7) where ˆY 0 j refers to the predicted segmentation map derived from f (Xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='4 Uncertainty-guided Mix Although CMT can promote model disagreement for co-training, it also slightly increases the uncertainty of the pseudo labels as depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' On the other hand, random regional dropout can expand the training distribution and improve the generalization capability of models [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' However, such random perturbations to the input images in- evitably introduce noise into the new samples, thus deteriorating the quality of pseudo labels for SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' One sub-network may provide some incorrect pseudo labels to the other sub-networks, degrading their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' To overcome these limitations, we propose UMIX to manipulate image patches under the guidance of the uncertainty maps produced by CMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' The main idea of UMIX is constructing a new sample by replacing the top k most uncertain (low-confidence) re- gions with the top k most certain (high-confidence) regions in the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' As illustrated in Figure 2, for example, we obtain the most uncertain regions (the red grids) and the most certain regions (the green grids) from an uncertainty map U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Then, we replace the red regions with the green regions in the input image X to construct a new sample X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Formally, UMIX constructs a new sample X′ = UMIX(X, U 1, U 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' k, 1/r) by replacing the top k most uncertain regions (red grids in V 1 and V 2) with the top k most certain regions (green grids in V 2 and V 1) in X, where each region has size 1/r to the image size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' To ensure the reliability of the uncertainty evaluation, we obtain the uncertain maps by integrating the outputs of the teacher and the student model instead of performing T stochastic forward passes designed by Monte Carlo Dropout estimate model [26, 18], which is equivalent to sampling predictions from the previous and current iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' This process can be formulated as: U m = Uncertain(f(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θm), f(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ)) = − � c Pc log(Pc), Pc = 1 2(Softmax(f(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θm)) + Softmax(f(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' θ))), (8) where m = 1, 2 denotes the index of the student models and c refers to the class index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 6 UCMT Table 1: Comparison with state-of-the-art methods on ISIC dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 5% DL and 10% DL of the labeled data are used for training, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Results are measured by DSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Method 5% DL 10% DL MT [4] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='67 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='42 CCT [7] 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='97 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='43 CPS [5] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='81 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='70 UGCL(U-Net) [15] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='67 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='48 UCMT(U-Net) (ours) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='14 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='33 CMT (ours) 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='86 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='10 UCMT (ours) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='22 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='46 4 Experiments and results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='1 Experiments Settings Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' We conduct extensive experiments on different medical image segmentation tasks to evaluate the proposed method, including skin lesion segmentation from dermoscopy images, polyp segmentation from colonoscopy images, and the 3D left atrium segmentation from cardiac MRI images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Dermoscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' We validate our method on the ISIC dataset [28] including 2594 dermoscopy images and corresponding annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Following [15], we adopt 1815 images for training and 779 images for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Colonoscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' We evaluate the proposed method on the two public colonoscopy datasets, including Kvasir-SEG [29] and CVC-ClinicDB [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Kvasir-SEG and CVC-ClinicDB contain 1000 and 612 colonoscopy images with correspond- ing annotations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Cardiac MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' We evaluate our method on the 3D left atrial (LA) segmentation challenge dataset, which consists of 100 3D gadolinium-enhanced magnetic resonance images and LA segmentation masks for training and validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Following [18], we split the 100 scans into 80 samples for training and 20 samples for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='1 Implementation details We use DeepLabv3+ [1] equipped with ResNet50 as the baseline architecture for 2D image segmentation, whereas adopt VNet [31] as the baseline in the 3D scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' All images are resized to 256×256 for inference, while the outputs are recovered to the original size for evaluation, in the 2D scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' For 3D image segmentation, we randomly crop 80 × 112 × 112(Depth × Height × Width) patches for training and iteratively crop patches using a sliding window strategy to obtain the final segmentation mask for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' We implement our method using PyTorch framework on a NVIDIA Quadro RTX 6000 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' We adopt AdamW as an optimizer with the fixed learning rate of le-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' The batchsize is set to 16, including 8 labeled samples and 8 unlabeled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' All 2D models are trained for 50 epochs, while the 3D models are trained for 1000 epochs 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' We empirically set k = 2 and r = 16 for our method in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='2 Comparison with state of the arts We compare the proposed method with state-of-the art on the four public medical image segmentation datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' We re-implement MT [4], CCT [7], and CPS [5] by adopting implementations from [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' For other approaches, we directly use the results reported in their original papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Results on Dermoscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' In Table 1, we report the results of our methods on ISIC and compare them with other state- of-the-art approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' UCMT substantially outperforms all previous methods and sets new state-of-the-art of 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='22% DSC and 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='46 DSC under 5% and 10% labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' For fair comparison with UGCL [15], replace the backbone of UCMT with U-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' The results indicate that our UCMT(U-Net) exceeds UGCL by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Moreover, our CMT version also outperforms other approaches under the two labeled data rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' For example, CMT surpasses MT and CPS by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='19% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='08% on 5% DL labeled data, showing the superiority of collaborative mean-teacher against the current consistency learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' By introducing UMIX, UCMT consistently increases the performance under different labeled data rates, which implies that promoting model disagreement and guaranteeing high-confident pseudo labels are beneficial for semi-supervised segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 3Since UCMT performs the two-step training within one iteration, it is trained for half of the epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 7 UCMT Table 2: Comparison with state-of-the-art methods on Kvasir-SEG and CVC-ClinicDB datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 15% DL and 30% DL of the labeled data are individually used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Results are measured by DSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Method Kvasir-SEG CVC-ClinicDB 15% DL 30% DL 15% DL 30% DL AdvSemSeg [24] 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='88 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='09 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='39 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='93 ColAdv [32] 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='76 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='95 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='18 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='29 MT [4] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='44 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='72 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='19 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='40 CCT [7] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='14 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='67 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='20 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='46 CPS [5] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='44 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='71 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='34 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='69 CMT (ours) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='08 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='61 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='88 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='83 UCMT (ours) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='68 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='06 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='30 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='51 Results on Colonoscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' We further conduct a comparative experiment on the polyp segmentation task from colonoscopy images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Table 2 reports the quantitative results on Kvasir-SEG and CVC-ClinicDB datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Com- pared with the adversarial learning-based [24, 32] and consistency learning-based [4, 7, 5] algorithms, the proposed methods achieve the state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' For example, both CMT and UCMT outperform AdvSemSeg [24] and ColAdv [32] by large margins on Kvasir-SEG and CVC-ClinicDB, except that ColAdv shows the better perfor- mance of 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='29% on CVC-ClinicDB under 30% labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' These results demonstrate that our uncertainty-guided collaborative mean-teacher scheme is superior to the adversarial learning and consistency learning schemes used in the compared approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' In addition, CMT and UCMT show better performance on the low-data regime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=', 15% DL, and the performance between 15% DL and 30% DL labeled data is close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' This phenomenon reflects the capacity of our method to produce high-quality pseudo labels from unlabeled data for semi-supervised learning, even with less labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Results on Cardiac MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' We further evaluate the proposed method in the 3D medical image segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Table 3 shows the comparison results on the 3D left atrium segmentation from cardiac MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' The compared approaches are based on consistency learning and pseudo labeling, including uncertainty-aware [18, 33], shape-aware [25], structure- aware [34], dual-task [19], and mutual training [20] consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' It can be observed that UCMT achieves the best performance under 10% and 20% DL in terms of DSC and Jaccard over the state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' For example, compared with UA-MT [18] and MC-Net [20], UCMT shows 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='88% DSC and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='43% DSC improvements on the 10% labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' The results demonstrate the superiority of our UCMT for 3D medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Table 3: Comparison with state-of-the-art methods on the 3D left atrial segmentation challenge dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 10% DL and 20% DL of the labeled data are used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Method 10% DL 20% DL DSC Jaccard 95HD ASD DSC Jaccard 95HD ASD UA-MT [18] 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='25 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='48 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='84 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='36 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='88 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='21 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='26 SASSNet [25] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='32 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='72 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='62 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='55 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='54 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='24 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='20 LG-ER-MT [34] 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='54 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='12 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='29 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='77 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='62 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='31 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='06 DUWM [33] 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='91 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='75 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='67 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='31 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='65 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='35 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='03 DTC [19] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='57 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='55 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='74 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='42 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='98 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='10 MC-Net [20] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='71 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='31 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='36 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='18 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='34 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='48 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='77 MT [4] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='15 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='16 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='37 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='60 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='81 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='85 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='96 CPS [5] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='23 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='22 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='68 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='65 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='72 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='01 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='91 CMT (ours) 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='23 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='83 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='83 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='23 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='88 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='74 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='94 UCMT (ours) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='13 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='18 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='06 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='41 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='54 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='70 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='3 Ablation study We conduct an ablation study in terms of network architectures, loss functions, and region mix to investigate the effectiveness of each component and analyze the hyperparameters of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' There are three types of network architectures: 1) teacher-student (TS) in MT [4], 2) student-student (SS) in CPS [5], and 3) student-teacher- student in the proposed CMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 8 UCMT Effectiveness of each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Table 4 reports the performance improvements over the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' It shows a trend that the segmentation performance improves when the components, including the STS (student-teacher-student), Lcps, Lmts, and UMIX are introduced into the baseline, and again confirms the necessity of encouraging model disagreement and enhancing the quality of pseudo labels for semi-supervised segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' The semi-supervised segmentation model is boosted for two reasons: 1) Lcps, Lmts and the STS architecture that force the model disagreement in CMT for co-training, and 2) UMIX facilitating the model to produce high-confidence pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' All the components contribute to UCMT to achieve 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='22% DSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' These results demonstrate their effectiveness and complementarity for semi-supervised medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' On the other hand, the two groups of comparisons between "TS (teacher- student) + Lmts" (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=', MT) and STS + Lmts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=', CMTv1), and between "SS (student-student) + Lcps" (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=', CPS) and "STS + Lcps" (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=', CMTv2) show that the STS-based approaches yield the improvements of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='17% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='64%, which verifies the effectiveness of our STS for SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' The performance gaps are not significant because the STS architecture increases the co-training disagreement but decreases the confidences of pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' However, It can be easily found that the results are improved to 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='86% by "STS + Lcps + Lmts" (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=', CMTv3) and the relative improvements of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='55% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='41% DSC have been achieved by "STS + Lcps + Lmts + UMIX" (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=', UCMT) compared with MT and CPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' The results demonstrate our hypothesis that maintaining co-training with high-confidence pseudo labels can improve the performance of semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Table 4: Ablation study of the different components combinations on ISIC dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' All models are trained for 5% labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' TS: teacher-student;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' SS: student-student;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' STS: student-teacher-student;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Lcps: cross pseudo supervision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Lmts: mean-teacher supervision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' U: UMIX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Method TS SS STS Lcps Lmts U DSC Baseline 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='31 MT √ √ 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='67 CPS √ √ 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='81 CMTv1 √ √ 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='84 CMTv2 √ √ 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='48 CMTv3 √ √ √ 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='86 UCMT √ √ √ √ 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='22 Comparison with CutMix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' We further compare the proposed UMIX, component of our UCMT, with CutMix [14] on ISIC and LA datasets with different labeled data to investigate their effects in semi-supervised medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' As illustrates in Figure 3, UMIX outperforms CutMix, especial in the low-data regime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=', 2% labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' The reason for this phenomenon is that CutMix performs random region mix that inevitably introduces noise into the new samples, which reduces the quality of the pseudo labels, while UMIX processes the image regions according to the uncertainty of the model, which facilitates the model to generate more confident pseudo labels from the new samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' (a) ISIC (b) LA CMT+CutMix CMT+UMIX CMT CMT+CutMix CMT+UMIX CMT Figure 3: Comparison UCMT (UMIX) with CutMix on ISIC (a) and LA (b) dataset under 2%, 5%, 10%, and 20% DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Parameter sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' UMIX has two hyperparamters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=', the top k regions for mix and the size of the regions (patches) 1/r to the image size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' We study the influence of these factors to UCMT on ISIC dataset with 5% DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' It can be observed in Table 5 that reducing the patch size leads to a slight increase in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Moreover, varying the number of k does not bring us any improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' The reason for this phenomenon is that we can choose any value 9 UCMT of K to eliminate outliers, thus bringing high-confidence pseudo labels for semi-supervised learning, indicating the robustness of UMIX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Table 5: Investigation on how the top k and region size affect the capacity of UMIX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' All results are evaluated on ISIC dataset with 5% labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 1/r k 1 2 3 4 5 1/16 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='95 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='22 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='12 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='96 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='08 1/4 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='65 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='15 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='80 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='54 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='90 1/8 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='86 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='92 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='03 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='01 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='87 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content='4 Qualitative results Figure 4 visualizes some example results of polyp segmentation, skin lesion segmentation, and left atrial segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' As shown in Figure 4 (a), the supervised baseline insufficiently segments some lesion regions, mainly due to the limited number of labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Moreover, MT [Figure 4 (b)] and CPS [Figure 4 (c)] typically under-segment certain objects, which can be attributed to the limited generalization capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' On the contrary, our CMT [Figure 4 (e)] corrects these errors and produces smoother segment boundaries by gaining more effective supervision from unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Besides, our complete method UCMT [Figure 4 (f)] further generates more accurate results by recovering finer segmentation details through more efficient training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' These examples qualitatively verify the robustness of the proposed UCMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' In addition, to clearly give an insight into the procedure of the pseudo label generation and utilization in the co-training SSL method, we illustrate the uncertainty maps for two samples during the training in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' As shown, UCMT generates the uncertainty maps with high uncertainty [Figure 5 (a)/(c)] in the early training stage whereas our model produces relative higher confidence maps [Figure 5 (b)/(d)] from the UMIX images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' During training, UCMT gradually improves the confidence for the input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' These results prove that UMIX can facilitate SSL models to generate high-confidence pseudo labels during training, guaranteeing that UCMT is able to maintain co-training in a more proper way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 5 Conclusion We present an uncertainty-guided collaborative mean-teacher for semi-supervised medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Our main ideas lies in maintaining co-training with high-confidence pseudo labels to improve the capability of the SSL models to explore information from unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Extensive experiments on four public datasets demonstrate the effectiveness of this idea and show that the proposed UCMT can achieve state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' In the future, we will investigate more deeply the underlying mechanisms of co-training for more effective semi-supervised image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant 62076059 and the Natural Science Foundation of Liaoning Province under Grant 2021-MS-105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' References [1] Liang-Chieh Chen, Yukun Zhu, George Papandreou, Florian Schroff, and Hartwig Adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Encoder-decoder with atrous separable convolution for semantic image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' In Proceedings of the European conference on computer vision (ECCV), pages 801–818, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' 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images for the first example, while (c) and (d) are for the second example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' [14] Sangdoo Yun, Dongyoon Han, Seong Joon Oh, Sanghyuk Chun, Junsuk Choe, and Youngjoon Yoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' Cutmix: Regularization strategy to train strong classifiers with localizable features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF international conference on computer vision, pages 6023–6032, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQfWAqd/content/2301.04465v1.pdf'} +page_content=' [15] Tao Wang, Jianglin Lu, Zhihui Lai, Jiajun Wen, and Heng Kong.' metadata={'source': 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--git a/HNE5T4oBgHgl3EQfWQ9u/content/tmp_files/2301.05557v1.pdf.txt b/HNE5T4oBgHgl3EQfWQ9u/content/tmp_files/2301.05557v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..09f5798907d73e6d40b70fc94454e7dc264cf612 --- /dev/null +++ b/HNE5T4oBgHgl3EQfWQ9u/content/tmp_files/2301.05557v1.pdf.txt @@ -0,0 +1,2132 @@ +Vibronic Effects on the Quantum Tunnelling of Magnetisation in Single-Molecule Magnets +Andrea Mattioni,1, ∗ Jakob K. Staab,1 William J. A. Blackmore,1 Daniel +Reta,1, 2 Jake Iles-Smith,3, 4 Ahsan Nazir,3 and Nicholas F. Chilton1, † +1Department of Chemistry, School of Natural Sciences, +The University of Manchester, Oxford Road, Manchester, M13 9PL, UK +2Faculty of Chemistry, UPV/EHU & Donostia International Physics Center DIPC, +Ikerbasque, Basque Foundation for Science, Bilbao, Spain +3Department of Physics and Astronomy, School of Natural Sciences, +The University of Manchester, Oxford Road, Manchester M13 9PL, UK +4Department of Electrical and Electronic Engineering, School of Engineering, +The University of Manchester, Sackville Street Building, Manchester M1 3BB, UK +Single-molecule magnets are among the most promising platforms for achieving molecular-scale data stor- +age and processing. Their magnetisation dynamics are determined by the interplay between electronic and +vibrational degrees of freedom, which can couple coherently, leading to complex vibronic dynamics. Building +on an ab initio description of the electronic and vibrational Hamiltonians, we formulate a non-perturbative vi- +bronic model of the low-energy magnetic degrees of freedom in a single-molecule magnet, which we benchmark +against field-dependent magnetisation measurements. Describing the low-temperature magnetism of the com- +plex in terms of magnetic polarons, we are able to quantify the vibronic contribution to the quantum tunnelling +of the magnetisation. Despite collectively enhancing magnetic relaxation, we observe that specific vibrations +suppress quantum tunnelling by enhancing the magnetic axiality of the complex. Finally, we discuss how this +observation might impact the current paradigm to chemical design of new high-performance single-molecule +magnets, promoting vibrations to an active role rather than just regarding them as sources of noise and decoher- +ence. +I. +INTRODUCTION +Single-molecule magnets (SMMs) hold the potential for +realising high-density data storage and quantum informa- +tion processing [1–4]. +These molecules exhibit a doubly- +degenerate ground state, comprising two states supporting a +large magnetic moment with opposite orientation, which rep- +resents an ideal platform for storing digital data. Slow reori- +entation of this magnetic moment results in magnetic hystere- +sis at the single-molecule level at sufficiently low tempera- +tures [5]. The main obstacle to extending this behaviour to +room temperature is the coupling of the magnetic degrees of +freedom to molecular and lattice vibrations, often referred to +as spin-phonon coupling. Thermal excitation of the molec- +ular vibrations cause transitions between different magnetic +states, ultimately leading to a complete loss of magnetisation. +Advances in design, synthesis and characterisation of SMMs +have shed light on the microscopic mechanisms underlying +their desirable magnetic properties, extending this behaviour +to increasingly higher temperatures [6–8]. +The mechanism responsible for magnetic relaxation in +SMMs strongly depends on temperature. At higher temper- +atures, relaxation is driven by one (Orbach) and two (Raman) +phonon transitions between magnetic sublevels [9]. +When +temperatures approach absolute zero, all vibrations are pre- +dominantly found in their ground state. +Thus, both Or- +bach and Raman transitions become negligible and the dom- +inant mechanism is quantum tunnelling of the magnetisation +∗ andrea.mattioni@manchester.ac.uk +† nicholas.chilton@manchester.ac.uk +(QTM) between the two degenerate ground states [10, 11]. +This process relies on the presence of a coherent coupling +mixing the two otherwise degenerate ground states, opening +a tunnelling gap, and allowing population to redistribute be- +tween them, thus leading to facile magnetic reorientation. +While the role of vibrations in high-temperature magnetic +relaxation is well understood in terms of weak-coupling rate +equations for the electronic populations [12–15], the connec- +tion between QTM and spin-phonon coupling is still unclear. +Some analyses have looked at the influence of vibrations on +QTM in integer-spin SMMs, where a model spin system was +used to show that spin-phonon coupling could open a tunnel- +ing gap [16, 17]. However, QTM remains more elusive to +grasp in half-integer spin complexes, such as monometallic +Dy(III) SMMs, since it is observed experimentally despite +being forbidden by Kramers theorem [18]. +In this case, a +magnetic field is needed to break the time-reversal symmetry +of the molecular Hamiltonian and lift the degeneracy of the +ground doublet. This magnetic field can be provided by hy- +perfine interaction with nuclear spins or by dipolar coupling +to other SMMs; both these effects have been shown to af- +fect tunnelling behaviour [19–25]. Once the tunnelling gap +is opened by a magnetic field, molecular vibrations can in +principle affect its magnitude in a nontrivial way. In a re- +cent work, Ortu et al. analysed the magnetic hysteresis of a +series of Dy(III) SMMs, suggesting that QTM efficiency cor- +relates with molecular flexibility [22]. In another work, hyper- +fine coupling was proposed to assists QTM by facilitating the +interaction between molecular vibrations and spin sublevels +[26]. However, a clear and unambiguous demonstration of the +influence of the spin-phonon coupling on QTM beyond toy- +model approaches is still lacking to this date. +In this work we present a theoretical analysis of the effect of +arXiv:2301.05557v1 [quant-ph] 13 Jan 2023 + +2 +molecular vibrations on the tunnelling dynamics in a Dy(III) +SMM. In contrast to previous treatments, our approach is +based on a fully ab initio description of the SMM vibrational +environment and accounts for the spin-phonon coupling in a +non perturbative way, overcoming the standard weak-coupling +master equation approach commonly used to determine the +high-temperature magnetisation dynamics. After deriving an +effective low-energy model for the relevant vibronic degrees +of freedom based on a polaron approach [27], we demon- +strate that vibrations can either enhance or reduce the quantum +tunnelling gap, depending on the orientation of the magnetic +field relative to the main anisotropy axis of the SMM. More- +over, we validate our vibronic model against frozen solution, +field-dependent magnetisation measurements and show that +vibronic effects on QTM survive the orientational averaging +imposed by amorphous samples, leading, on average, to a sig- +nificant enhancement of the tunnelling probability. Lastly, we +argue that not all vibrations lead to faster QTM; depending on +how strongly vibrations impact the axiality of the lowest en- +ergy magnetic doublet, we show that they can play a benign +role by suppressing tunnelling, and discuss first steps in that +direction. +II. +MODEL +The compound investigated in this work is [Dy(Cpttt)2]+, +shown in Fig. 1a [6]. The complex consists of a dyspro- +sium ion Dy(III) enclosed between two negatively charged +cyclopentadienyl rings with tert-butyl groups at positions 1, 2 +and 4 (Cpttt). The crystal field generated by the axial ligands +makes the states with larger angular momentum energetically +favourable, resulting in the energy level diagram sketched in +Fig. 1b. The energy barrier separating the two degenerate +ground states results in magnetic hysteresis, which was ob- +served up to T = 60 K [6]. Magnetic hysteresis is hindered +by QTM, which leads to a characteristic sudden drop of the +magnetisation at zero magnetic field. +To single out the contribution of molecular vibrations, we +focus on a magnetically diluted sample in a frozen solution +of dichloromethane (DCM). Thus, our computational model +consists of a solvated [Dy(Cpttt)2]+ cation (see Section S1 for +details; Fig. 1a), which provides a realistic description of the +low-frequency vibrational environment, comprised of pseudo- +acoustic vibrational modes (Fig. 1c). These constitute the ba- +sis to consider further contributions of dipolar and hyperfine +interactions to QTM (Fig. 1b). +Once the equilibrium geometry and vibrational modes of +the solvated SMM (which are in general combinations of +molecular and solvent vibrations) are obtained at the density- +functional level of theory (see Section S1), we proceed to de- +termine the equilibrium electronic structure via complete ac- +tive space self-consistent field spin-orbit (CASSCF-SO) cal- +culations. The electronic structure is projected onto an effec- +tive crystal-field Hamiltonian, parametrised in terms of crys- +tal field parameters. The spin-phonon couplings are obtained +from a single CASSCF calculation, by computing the analytic +derivatives of the molecular Hamiltonian with respect to the +nuclear coordinates [14] (see Section S1 for more details). +The +lowest-energy +angular +momentum +multiplet +of +[Dy(Cpttt)2]+ (J = 15/2) can thus be described by the ab ini- +tio vibronic Hamiltonian +ˆH = ∑ +m +Em|m⟩⟨m|+∑ +j +ˆVj ⊗(ˆbj + ˆb† +j)+∑ +j +ωj ˆb† +j ˆbj, +(1) +where Em denotes the energy associated with the electronic +state |m⟩ and ˆVj represent the spin-phonon coupling opera- +tors. The harmonic vibrational modes of the DCM-solvated +[Dy(Cpttt)2]+ are described in terms of their bosonic annihi- +lation (creation) operators ˆbj (ˆb† +j) and frequencies ωj. +In the absence of magnetic fields, the Hamiltonian (1) is +symmetric under time reversal. This symmetry results in a +two-fold degeneracy of the energy levels Em, whose corre- +sponding eigenstates |m⟩ and | ¯m⟩ form a time-reversal conju- +gate Kramers doublet. The degeneracy is lifted by introducing +a magnetic field B, which couples to the electronic degrees of +freedom via the Zeeman interaction ˆHZee = µBgJB · ˆJ, where +gJ is the Landé g-factor and ˆJ is the total angular momentum +operator. To linear order in the magnetic field, each Kramers +doublet splits into two energy levels Em±∆m/2 corresponding +to the states +|m+⟩ = cos θm +2 |m⟩+eiφm sin θm +2 | ¯m⟩ +(2) +|m−⟩ = −sin θm +2 |m⟩+eiφm cos θm +2 | ¯m⟩ +(3) +where the energy splitting ∆m and the mixing angles θm and +φm are determined by the matrix elements of the Zeeman +Hamiltonian on the subspace {|m⟩,| ¯m⟩}. In addition to the +intra-doublet mixing described by Eqs. (2) and (3), the Zee- +man interaction also mixes Kramers doublets at different ener- +gies. The ground doublet acquires contributions from higher- +lying states +|1′ +±⟩ = |1±⟩+ ∑ +m̸=1,¯1 +|m⟩⟨m| ˆHZee|1±⟩ +E1 −Em ++O(B2). +(4) +These states no longer form a time-reversal conjugate doublet, +meaning that the spin-phonon coupling can now contribute to +transitions between them. +Since QTM is typically observed at much lower tempera- +tures than the energy gap between the lowest and first excited +doublets (which here is ∼ 660 K [6]), we focus on the per- +turbed ground doublet |1′ +±⟩. Within this subspace, the Hamil- +tonian ˆH + ˆHZee takes the form +ˆHeff = E1 + ∆1 +2 σ′ +z +∑ +j +ωj ˆb† +j ˆbj +(5) ++ ∑ +j +� +⟨1| ˆVj|1⟩−wz +jσ′ +z +�� +ˆbj + ˆb† +j +� +− ∑ +j +� +wx +jσ′ +x +wy +jσ′ +y +�� +ˆbj + ˆb† +j +� +. +This Hamiltonian describes the interaction between vi- +brational +modes +and +an +effective +spin +one-half +rep- +resented +by +the +Pauli +matrices +σ′ = (σ′ +x,σ′ +y,σ′ +z), + +3 +QTM +electronic +vibronic +b) +a) +Energy +[Dy(Cpttt)2]+ +c) +vibrational DOS +DCM +z +d) +polarons +FIG. 1. +Quantum tunnelling in single-molecule magnets. (a) Molecular structure of a Dy(III) single-molecule magnet surrounded by a +dichloromethane bath. (b) Equilibrium energy level diagram of the lowest-energy angular momentum multiplet with J = 15/2. The second- +lowest doublet at E2 is 524 cm−1 higher than the ground doublet at E1, while the highest doublet is 1523 cm−1 above E1. Dipolar and hyperfine +magnetic fields (Bint) can lift the degeneracy of the doublets and cause quantum tunnelling, which results in avoided crossings when sweeping +an external magnetic field Bext. Molecular vibrations can influence the magnitude of the avoided crossing. (c) Spin-phonon coupling for the +solvated complex shown above, as a function of the vibrational frequency (vibrations with ωj > 1500 cm−1 not shown), calculated as the +Frobenius norm of the operator ˆVj. The grey dashed line represents the vibrational density of states, obtained by assigning to each molecular +vibration a (anti-symmetrised) Lorentzian lineshape with full width at half-maximum 10 cm−1 (corresponding to a typical timescale of ∼ 1 ps). +(d) Idea behind the polaron transformation of Eq. (6). Each spin state |1′±⟩ is accompanied by a vibrational distortion (greatly exaggerated +for visualisation), thus forming a magnetic polaron. Vibrational states |ν⟩ are now described in terms of harmonic displacements around the +deformed structure, which depends on the state of the spin. Polarons provide an accurate physical picture when the spin-phonon coupling is +strong and mostly modulates the energy of different spin states but not the coupling between them. +where σ′ +z = |1′ ++⟩⟨1′ ++| − |1′ +−⟩⟨1′ +−|. +The vector w j = +(ℜ⟨1−| ˆWj|1+⟩,ℑ⟨1−| ˆWj|1+⟩,⟨1+| ˆWj|1+⟩) is defined in terms +of the operator ˆWj = ∑m̸=1,¯1 ˆVj|m⟩⟨m| ˆHZee/(Em −E1)+ h.c., +describing the effect of the Zeeman interaction on the +spin-phonon coupling. Due to the strong magnetic axiality of +the complex considered here, the longitudinal component of +the spin-phonon coupling wz +j dominates over the transverse +part wx +j, wy +j (see Section S3). +In this case, we can get a +better physical picture of the system by transforming the +Hamiltonian (5) to the polaron frame defined by the unitary +operator +ˆS = exp +� +∑ +s=± +|1′ +s⟩⟨1′ +s| ∑ +j +ξ s +j +� +ˆb† +j − ˆbj +�� +, +(6) +which mixes electronic and vibrational degrees of freedom by +displacing the mode operators by ξ ± +j = (⟨1| ˆVj|1⟩ ∓ wz +j)/ωj +depending on the state of the effective spin one-half [27]. +The idea behind this transformation is to allow nuclei to re- +lax around a new equilibrium geometry, which may be differ- +ent for every spin state. This lowers the energy of the system +and provides a good description of the vibronic eigenstates +when the spin-phonon coupling is approximately diagonal in +the spin basis (Fig. 1d). In the polaron frame, the longitu- +dinal spin-phonon coupling is fully absorbed into the purely +electronic part of the Hamiltonian, while the transverse com- +ponents can be approximated by their thermal average over +vibrations, neglecting their vanishingly small quantum fluc- +tuations (see Section S2). +After transforming back to the +original frame, we are left with an effective spin one-half +Hamiltonian with no residual spin-phonon coupling Heff ≈ +ˆH(pol) +eff ++∑j ωj ˆb† +j ˆbj, where +ˆH(pol) +eff += E1 + ∆1 +2 σ′′ +z +2∑ +j +⟨1| ˆVj|1⟩ +ωj +w j ·σ′′. +(7) +The set of Pauli matrices σ′′ = ˆS†(σ′ ⊗ 1lvib) ˆS describe the +two-level system formed by the magnetic polarons of the +form ˆS†|1′ +±⟩|{νj}⟩vib, where {νj} is a set of occupation num- +bers for the vibrational modes of the solvent-SMM system. +These magnetic polarons can be thought as magnetic elec- +tronic states strongly coupled to a distortion of the molecular +geometry. They inherit the magnetic properties of the cor- +responding electronic states, and can be seen as the molecu- + +4 +lar equivalent of the magnetic polarons observed in a range +of magnetic materials [28–30]. +Polaron representations of +vibronic systems have been employed in a wide variety of +settings, ranging from spin-boson models [27, 31] to photo- +synthetic complexes [32–34], to quantum dots [35–37], pro- +viding a convenient basis to describe the dynamics of quan- +tum systems strongly coupled to a vibrational environment. +These methods are particularly well suited for condensed +matter systems where the electron-phonon coupling is strong +but causes very slow transitions between different electronic +states, allowing exact treatment of the pure-dephasing part +of the electron-phonon coupling and renormalising the elec- +tronic parameters. For this reason, the polaron transformation +is especially effective for describing our system (as detailed in +Section S3). The most striking advantage of this approach is +that the average effect of the spin-phonon coupling is included +non-perturbatively into the electronic part of the Hamiltonian, +leaving behind a vanishingly small residual spin-phonon cou- +pling. +As a last step, we bring the Hamiltonian in Eq. (7) into a +more familiar form by expressing it in terms of an effective g- +matrix. We recall that the quantities ∆1 and w j depend linearly +on the magnetic field B via the Zeeman Hamiltonian ˆHZee. An +additional dependence on the orientation of the magnetic field +comes from the mixing angles θ1 and φ1 introduced in Eqs. +(2) and (3), appearing in the states |1±⟩ used in the definition +of w j. This further dependence is removed by transforming +the Pauli operators back to the basis {|1⟩,|¯1⟩} via a three- +dimensional rotation σ = Rθ1,φ1 ·σ′′. Finally, we obtain +ˆH(pol) +eff += E1 + µBB· +� +gel +∑ +j +gvib +j +� +· σ +2 , +(8) +for appropriately defined electronic and single-mode vibronic +g-matrices gel and gvib +j . These are directly related to the elec- +tronic splitting term ∆1 and to the vibronic corrections de- +scribed by w j in Eq. (7), respectively (see Section S2 for +a thorough derivation). The main advantage of representing +the ground Kramers doublet with an effective spin one-half +Hamiltonian is that it provides a conceptually simple founda- +tion for studying low-temperature magnetic behaviour of the +complex, confining all microscopic details, including vibronic +effects, to an effective g-matrix. +III. +RESULTS +We begin by considering the influence of vibrations on the +Zeeman splitting of the lowest doublet. The Zeeman splitting +in absence of vibrations is simply given by ∆1 = µB|B · gel|. +In the presence of vibrations, the electronic g-matrix gel is +modified by adding the vibronic correction ∑j gvib +j , resulting +in the Zeeman splitting ∆vib +1 . In Fig. 2 we show the Zee- +man splittings as a function of the orientation of the mag- +netic field B, parametrised in terms of the polar angles (θ,φ). +Depending on the field orientation, vibrations can lead to ei- +ther an increase or decrease of the Zeeman splitting. These +changes seem rather small when compared to the largest elec- +tronic splitting, obtained when B is oriented along the z-axis +(Fig. 1a), as expected for a complex with easy-axis anisotropy. +However, they become quite significant for field orientations +close to the xy-plane, where the purely electronic splitting ∆1 +becomes vanishingly small and ∆vib +1 +can be dominated by the +vibronic contribution. This is clearly shown in Fig. 2b and +2c, where we decompose the total field B = Bint + Bext in a +fixed internal component Bint originating from dipolar and hy- +perfine interactions, responsible for opening a tunnelling gap, +and an external part Bext which we sweep along a fixed direc- +tion across zero. We note that this effect is specific to states +with easy-axis magnetic anisotropy, however this is the defin- +ing feature of SMMs, such that our results should be generally +applicable to all Kramers SMMs. A more in-depth discussion +on the origin and magnitude of the internal field can be found +in Section S5. When these fields lie in the plane perpendicu- +lar to the purely electronic easy axis, i.e. the hard plane, the +vibronic splitting can be four orders of magnitude larger than +the electronic one (Fig. 2b). The situation is reversed when +the fields lie in the hard plane of the vibronic g-matrix (Fig. +2c). +So far we have seen that spin-phonon coupling can either +enhance or reduce the tunnelling gap in the presence of a mag- +netic field depending on its orientation. For this reason, it is +not immediately clear whether its effects survive ensemble av- +eraging in a collection of randomly oriented SMMs, such as +the frozen solutions considered in magnetometry experiments. +In order to check this, let us consider an ideal field-dependent +magnetisation measurement. When sweeping a magnetic field +Bext at a constant rate from positive to negative values along +a given direction, QTM is typically observed as a sharp step +in the magnetisation of the sample when crossing the region +around Bext = 0 [10]. This sudden change of the magnetisa- +tion is due to a non-adiabatic spin-flip transition between the +two lowest energy spin states, that occurs when traversing an +avoided crossing (see diagram in Fig. 1b, right). The spin-flip +probability is given by the celebrated Landau-Zener expres- +sion [38–43], which in our case takes the form +PLZ = 1−exp +� +−π|∆⊥|2 +2|v| +� +, +(9) +where we have defined v = µBdBext/dt ·g, and ∆⊥ is the com- +ponent of ∆ = µBBint · g perpendicular to v, while g denotes +the total electronic-vibrational g-matrix appearing in Eq. (8) +(see Section S2 for a derivation of Eq. (9)). We account for +orientational disorder by averaging Eq. (9) over all possible +orientations of internal and external magnetic fields, yielding +the ensemble average ⟨PLZ⟩. +The effect of spin-phonon coupling on the spin-flip dynam- +ics of an ensemble of SMMs can be clearly seen in Fig. 3. In- +cluding the vibronic correction to the ground doublet g-matrix +leads to enhanced spin-flip probabilities across a wide range +of internal field strengths and field sweep rates. This is in line +with previous results suggesting that molecular flexibility cor- +relates with QTM [22]. To further corroborate our model, we +test its predictions against experimental data. We extracted the +average spin-flip probability from published hysteresis data + +5 +a) +b) +c) +FIG. 2. +Zeeman splitting of the ground Kramers doublet. +(a) +Electronic ground doublet splitting (∆1, top) and vibronic correction +(∆vib +1 +− ∆1, bottom) as a function of the orientation of the magnetic +field B = (sinθ cosφ,sinθ sinφ,cosθ), with magnitude fixed to 1 T. +The dashed (solid) line corresponds to the electronic (vibronic) hard +plane. (b–c) Electronic (dashed) and vibronic (solid) Zeeman split- +ting of the ground doublet as a function of the external field magni- +tude Bext in the presence of a transverse internal field Bint = 1 mT. +External and internal fields are perpendicular to each other and were +both chosen to lie in the hard plane of either the electronic (b) or +vibronic (c) g-matrix. The orientation of the external (internal) field +is shown for both cases as circles (crosses) in the inset in (a), with +colors matching the ones in (b) and (c). +of [Dy(Cpttt)2][B(C6F5)4] in DCM with sweep rates ranging +between 10–20 Oe/s [6], yielding a value of ⟨PLZ⟩ = 0.27, +indicated by the pink line in Fig. 3. We then checked what +strength of the internal field Bint is required to reproduce such +spin-flip probability based on Eq. (9). In Fig. 3, we observe +that the values of Bint required by the vibronic model to re- +produce the observed spin-flip probability are perfectly con- +sistent with the dipolar fields naturally occurring in the sam- +ple, whereas the purely electronic model necessitates internal +fields that are one order of magnitude larger. These results +clearly demonstrate the significance of spin-phonon coupling +for QTM in a disordered ensemble of SMMs. A detailed dis- +cussion on the estimation of spin-flip probabilities and internal +fields from magnetisation measurements is presented in Sec- +FIG. 3. +Landau-Zener spin-flip probability. Ensemble-averaged +spin-flip probability as a function of the internal field strength Bint +causing tunnelling within the ground Kramers doublet, shown for +different sweep rates dBext/dt. Results for the vibronic model of Eq. +(8) are shown as orange solid lines, together with the spin-flip prob- +abilities predicted by a purely electronic model obtained by setting +the spin-phonon coupling to zero, shown as blue dashed lines. The +horizontal pink line indicates ⟨PLZ⟩ = 0.27, extracted from hysteresis +data from Ref. [6] (Section S4). The green shaded area indicates the +range of values for typical dipolar fields in the corresponding sample +(Section S5). +tions S4 and S5. +IV. +DISCUSSION +As shown above, the combined effect of all vibrations in +a randomly oriented ensemble of solvated SMMs is to en- +hance QTM. However, not all vibrations contribute to the +same extent. Based on the polaron model introduced above, +vibrations with large spin-phonon coupling and low frequency +have a larger impact on the magnetic properties of the ground +Kramers doublet. This can be seen from Eq. (7), where the +vibronic correction to the effective ground Kramers Hamil- +tonian is weighted by the factor ⟨1| ˆVj|1⟩/ω j. Another prop- +erty of vibrations that can influence QTM is their symmetry. +In monometallic SMMs, QTM has generally been correlated +with a reduction of the axial symmetry of the complex, either +by the presence of flexible ligands or by transverse magnetic +fields. Since we are interested in symmetry only as long as it +influences magnetism, it is useful to introduce a measure of +axiality on the g-matrix, such as +A(g) = +��g− 1 +3Tr g +�� +� +2 +3Tr g +, +(10) +where ∥·∥ denotes the Frobenius norm. This measure yields 1 +for a perfect easy-axis complex, 1/2 for an easy plane system, +and 0 for the perfectly isotropic case. The axiality of an indi- +vidual vibrational mode can be quantified as Aj = A(gel+gvib +j ) +by building a single-mode vibronic g-matrix, analogous to + +△1 (cm-1 +元 +10 +8 +6 +4 +2 +2 +0 +0 +0 +一 +元 +2 +2 +ΦAvib - △1 (cm-1) +0.2 +0.1 +0 +2 +-0.1 +-0.2 +0 +元 +0 +一元 +2 +2(cm +0.2 +0.1 +0 +2 +-0.1 +-0.2 +0 +一元 +2 +2(cm +0.2 +0.1 +0 +2 +-0.1 +-0.2 +0 +一元 +2 +26 +a) +b) +FIG. 4. +Single-mode contributions to tunnelling of the magnetisation. (a) Single-mode vibronic Landau-Zener probabilities plotted for +each vibrational mode, shown as a function of the mode axiality relative to the axiality of the purely electronic g-matrix (∆Aj = Aj −Ael). The +magnitude of the internal field is fixed to Bint = 1 mT and the external field sweep rate is 10 Oe/s. The color coding represents the spin-phonon +coupling strength ∥ ˆVj∥. Grey dashed lines corresponds to the purely electronic model. (b) Visual representation of the displacements induced +by the vibrational modes indicated by arrows in (a). Solvent motion is only shown for modes 2 and 6, which have negligible amplitude on the +SMM. +the multi-mode one introduced in Eq. +(8). +We might be +tempted to intuitively conclude that vibrational motion al- +ways decreases the axiality with respect to its electronic value +Ael = A(gel), given that the collective effect of vibrations is to +enhance QTM. However, when considered individually, some +vibrations can have the opposite effect, of effectively increas- +ing the magnetic axiality. +In order to see how axiality correlates to QTM, we calcu- +late the single-mode Landau-Zener probabilities ⟨Pj⟩. These +are obtained by replacing the multi-mode vibronic g-matrix in +Eq. (8) with the single-mode one gel +gvib +j , and following the +same procedure detailed in Section S2. The single-mode con- +tribution to the spin-flip probability unambiguously correlates +with mode axiality, as shown in Fig. 4a. Vibrational modes +that lead to a larger QTM probability are likely to reduce the +magnetic axiality of the complex (top-left sector). Vice versa, +those vibrational modes that enhance axiality also suppress +QTM (bottom-right sector). +As a first step towards uncovering the microscopic basis +of this unexpected behaviour, we single out the three vibra- +tional modes that have the largest impact on axiality and spin- +flip probability in both directions. These vibrational modes, +labelled 1–6, represent a range of qualitatively distinct vibra- +tions, as can be observed in Fig. 4b. Modes 4 and 5 are among +the ones exhibiting the strongest spin-phonon coupling. Both +of them are mainly localised on one of the Cpttt ligands and +involve atomic displacements along the easy axis and, to a +lesser extent, rotations of the methyl groups. Modes 1 and +3 are among the ones with largest amplitude on the Dy ion, +which in both cases mainly moves in the hard plane, disrupt- +ing axial symmetry and enhancing tunnelling. Lastly, modes +2 and 6 predominantly correspond to solvent vibrations, and +are thus very low energy and so give a large contribution via +the small denominator in Eq. (7). +This analysis shows that the effect of vibrational modes on +QTM is more nuanced than what both intuition and previous +work would suggest. Despite leading to an overall increase +of the spin-flip probability on average, coupling the spin to +specific vibrations can increase the magnetic axiality of the +complex and suppress QTM. This opens a new avenue for the +improvement of magnetic relaxation times in SMMs, shifting +the role of vibrations from purely antagonistic to potentially +beneficial. +According to the results shown above, the ideal candidates +to observe vibronic suppression of QTM are systems exhibit- +ing strongly axial, low frequency vibrations, strongly coupled +to the electronic effective spin. Strong spin-phonon coupling +and low frequency ensure a significant change in magnetic +properties according to Eq. (7), but may not be enough to hin- +der tunnelling. In order to be beneficial, vibrations also need +to enhance the axiality of the ground doublet g-matrix. The +relation between magnetic axiality and vibrational symmetry +remains yet to be explored, and might lead to new insights +regarding rational design of ideal ligands. + +17 +V. +CONCLUSIONS +In conclusion, we have presented a detailed description of +the effect of molecular and solvent vibrations on the quan- +tum tunnelling between low-energy spin states in a single-ion +Dy(III) SMM. Our theoretical results, based on an ab initio +approach, are complemented by a polaron treatment of the rel- +evant vibronic degrees of freedom, which does not suffer from +any weak spin-phonon coupling assumption and is therefore +well-suited to other strong coupling scenarios. We have been +able to derive a non-perturbative vibronic correction to the ef- +fective g-matrix of the lowest-energy Kramers doublet, which +we have used as a basis to determine the tunnelling dynamics +in a magnetic field sweep experiment. This has allowed us to +formulate the key observation that, vibrations collectively en- +hance QTM, but some particular vibrational modes unexpect- +edly suppress QTM. This behaviour correlates to the axiality +of each mode, which can be used as a proxy for determining +whether a specific vibration enhances or hinders tunnelling. +The observation that individual vibrational modes can sup- +press QTM challenges the paradigm that dismisses vibrations +as detrimental, a mere obstacle to achieving long-lasting in- +formation storage on SMMs, and forces us instead to recon- +sider them under a new light, as tools that can be actively en- +gineered to our advantage to keep tunnelling at bay and ex- +tend relaxation timescales in molecular magnets. This idea +suggests parallelisms with other seemingly unrelated chem- +ical systems where electron-phonon coupling plays an im- +portant role. +For example, the study of electronic energy +transfer across photosynthetic complexes was radically trans- +formed by the simple observation that vibrations could play +an active role, maintaining quantum coherence in noisy room- +temperature environments, rather than just passively causing +decoherence between electronic states [44]. Identifying these +beneficial vibrations and amplifying their effect via chemical +design of new SMMs remains an open question, whose solu- +tion we believe could greatly benefit from the results and the +methods introduced in this work. +ACKNOWLEDGEMENTS +This work was made possible thanks to the ERC +grant 2019-STG-851504 and Royal Society fellowship +URF191320. 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Zhu, Using coherence to enhance +function in chemical and biophysical systems, Nature 543, 647 +(2017). + +1 +Supplementary Information: +Vibronic Effects on the Quantum Tunnelling of Magnetisation +in Single-Molecule Magnets +Andrea Mattioni,1,∗ Jakob K. Staab,1 William J. A. Blackmore,1 Daniel Reta,1,2 +Jake Iles-Smith,3,4 Ahsan Nazir,3 and Nicholas F. Chilton1,† +1Department of Chemistry, School of Natural Sciences, +The University of Manchester, Oxford Road, Manchester, M13 9PL, UK +2 Faculty of Chemistry, UPV/EHU & Donostia International Physics Center DIPC, +Ikerbasque, Basque Foundation for Science, Bilbao, Spain +3Department of Physics and Astronomy, School of Natural Sciences, +The University of Manchester, Oxford Road, Manchester M13 9PL, UK +4Department of Electrical and Electronic Engineering, School of Engineering, +The University of Manchester, Sackville Street Building, Manchester M1 3BB, UK +∗ andrea.mattioni@manchester.ac.uk +† nicholas.chilton@manchester.ac.uk + +2 +S1. +AB INITIO CALCULATIONS +The ab initio model of the DCM-solvated [Dy(Cpttt)2]+ molecule is constructed using a multi-layer approach. During ge- +ometry optimisation and frequency calculation the system is partitioned into two layers following the ONIOM scheme [1]. +The high-level layer, consisting of the SMM itself and the first solvation shell of 26 DCM molecules, is described by Density +Functional Theory (DFT) while the outer bulk of the DCM ball constitutes the low-level layer modelled by the semi-empirical +PM6 method. All DFT calculations are carried out using the pure PBE exchange-correlation functional [2] with Grimme’s D3 +dispersion correction. Dysprosium is replaced by its diamagnetic analogue yttrium for which the Stuttgart RSC 1997 ECP basis +is employed [3]. Cp ring carbons directly coordinated to the central ion are equipped with Dunning’s correlation consistent +triple-zeta polarised cc-pVTZ basis set and all remaining atoms with its double-zeta analogue cc-pVDZ [4]. Subsequently, the +electronic spin states and spin-phonon coupling parameters are calculated at the CASSCF-SO level explicitly accounting for the +strong static correlation present in the f-shell of Dy(III) ions. At this level, environmental effects are treated using an electrostatic +point charge representation of all DCM atoms. All DFT/PM6 calculations are carried out with GAUSSIAN version 9 revision +D.01 [5] and the CASSCF calculations are carried out with OPENMOLCAS version 21.06 [6]. +The starting [Dy(Cpttt)2]+ solvated system was obtained using the solvate program belonging to the AmberTool suite of +packages, with box as method and CHCL3BOX as solvent model. Chloroform molecules were subsequently converted to +DCM. From this large system, only molecules falling within 9 Å from the central metal atom are considered from now on. +The initial disordered system of 160 DCM molecules packed around the [Dy(Cpttt)2]+ crystal structure [7] is pre-optimised +in steps, starting by only optimising the high-level layer atoms and freezing the rest of the system. The low-layer atoms are +pre-optimised along the same lines starting with DCM molecules closest to the SMM and working in shells towards the outside. +Subsequently, the whole system is geometry optimised until RMS (maximum) values in force and displacement corresponding +to 0.00045 au (0.0003 au) and 0.0018 au (0.0012 au) are reached, respectively. After adjusting the isotopic mass of yttrium to +that of dysprosium mDy = 162.5u, vibrational normal modes and frequencies of the entire molecular aggregate are computed +within the harmonic approximation. +Electrostatic atomic point charge representations of the environment DCM molecules are evaluated for each isolated solvent +molecule independently at the DFT level of theory employing the CHarges from ELectrostatic Potentials using a Grid-based +(ChelpG) method [8], which serve as a classical model of environmental effects in the subsequent CASSCF calculations. +The evaluation of equilibrium electronic states and spin-phonon coupling parameters is carried out at the CASSCF level +including scalar relativistic effects using the second-order Douglas-Kroll Hamiltonian and spin-orbit coupling through the atomic +mean field approximation implemented in the restricted active space state interaction approach [9, 10]. The dysprosium atom is +equipped with the ANO-RCC-VTZP, the Cp ring carbons with the ANO-RCC-VDZP and the remaining atoms with the ANO- +RCC-VDZ basis set [11]. The resolution of the identity approximation with an on-the-fly acCD auxiliary basis is employed to +handle the two-electron integrals [12]. The active space of 9 electrons in 7 orbitals, spanned by 4f atomic orbitals, is employed +in a state-average CASSCF calculation including the 18 lowest lying sextet roots which span the 6H and 6F atomic terms. +We use our own implementation of spin Hamiltonian parameter projection to obtain the crystal field parameters Bq +k entering +the Hamiltonian +ˆHCF = ∑ +k=2,4,6 +k +∑ +q=−k +θkBq +kOq +k(ˆJ), +(S1) +describing the 6H15/2 ground state multiplet. Operator equivalent factors and Stevens operators are denoted by θk and Oq +k(ˆJ), +where ˆJ = ( ˆJx, ˆJy, ˆJz) are the angular momentum components. Spin-phonon coupling arises from changes to the Hamiltonian +(S1) due to slight distortions of the molecular geometry, parametrised as +Bq +k({Xj}) = Bq +k + +M +∑ +j=1 +∂Bq +k +∂Xj +Xj +..., +(S2) +where Xj denotes the dimensionless j-th normal coordinate of the complex under consideration. The derivatives ∂Bq +k/∂Xj are +calculated using the Linear Vibronic Coupling (LVC) approach described in Ref. [13] based on the state-average CASSCF +density-fitting gradients and non-adiabatic coupling involving all 18 sextet roots. +The final step leading to Eq. (1) in the main text is to quantise the normal modes and express them in terms of bosonic +annihilation and creation operators satisfying [ˆbi, ˆb† +j] = δij as +ˆXj = +ˆbj + ˆb† +j +√ +2 +. +(S3) + +3 +Defining the spin-phonon coupling operators +ˆVj = 1 +√ +2 ∑ +k,q +θk +∂Bq +k +∂Xj +Oq +k(ˆJ), +(S4) +we can finally write down the crystal field Hamiltonian including linear spin-phonon coupling as +ˆH = ˆHCF +∑ +j +ˆVj ⊗(ˆbj + ˆb† +j)+∑ +j +ωj ˆb† +j ˆb j. +(S5) + +4 +S2. +DERIVATION OF THE EFFECTIVE VIBRONIC DOUBLET HAMILTONIAN +A. +Electronic perturbation Theory +The starting point for our analysis of vibronic effects on QTM is the vibronic Hamiltonian +ˆH = ∑ +m>0 +Em(|m⟩⟨m|+| ¯m⟩⟨ ¯m|)+ ˆHZee +∑ +j +ˆVj ⊗(ˆbj + ˆb† +j)+∑ +j +ωj ˆb† +j ˆb j, +(S6) +where ˆHZee = µBgJB · ˆJ. is the Zeeman interaction with a magnetic field B. The doubly degenerate eigenstates of the crystal +field Hamiltonian HCF = ∑m>0 Em(|m⟩⟨m|+| ¯m⟩⟨ ¯m|) are related by time-reversal symmetry, i.e. ˆΘ|m⟩ ∝ | ¯m⟩ with ˆΘ2|m⟩ = −|m⟩, +where ˆΘ is the time-reversal operator. In the case of [Dy(Cpttt)2]+, the total electronic angular momentum is J = 15/2, leading +to 2J + 1 = 16 electronic states. We label these states in ascending energy with integers m = ±1,...,±8, using the compact +notation |−m⟩ = | ¯m⟩. +We momentarily neglect the spin-phonon coupling and focus on the purely electronic Hamiltonian Hel = HCF +HZee. Within +each degenerate subspace, the Zeeman term selects a specific electronic basis and lifts its degeneracy. This can be seen by +projecting the electronic Hamitonian onto the m-th subspace and diagonalising the 2×2 matrix +H(m) +el += Em + µBgJ +� +⟨m|B· ˆJ|m⟩ ⟨m|B· ˆJ| ¯m⟩ +⟨ ¯m|B· ˆJ|m⟩ ⟨ ¯m|B· ˆJ| ¯m⟩ +� +. +(S7) +For each individual cartesian component of the angular momentum, we decompose the corresponding 2 × 2 matrix in terms of +Pauli spin operators, which allows to rewrite the Hamiltonian of the m-th doublet as H(m) +el += Em + µBB·g(m) +el ·σ(m)/2, where +g(m) +el += 2gJ +� +� +ℜ⟨ ¯m| ˆJx|m⟩ ℑ⟨ ¯m| ˆJx|m⟩ ⟨m| ˆJx|m⟩ +ℜ⟨ ¯m| ˆJy|m⟩ ℑ⟨ ¯m| ˆJy|m⟩ ⟨m| ˆJy|m⟩ +ℜ⟨ ¯m| ˆJz|m⟩ ℑ⟨ ¯m| ˆJz|m⟩ ⟨m| ˆJz|m⟩ +� +� +(S8) +is the g-matrix for an effective spin 1/2 and σ(m) = (σ(m) +x +,σ(m) +y +,σ(m) +z +), with σ(m) +z += |m⟩⟨m|−| ¯m⟩⟨ ¯m|. We note that in general the +g-matrix in Eq. (S8) is not hermitean, but can be brought to such form by transforming the spin operators σ(m) to an appropriate +basis [14]. An easier prescription to find the hermitean form af any g-matrix g is to redefine it as +� +gg†. +To lowest order in the magnetic field, the Zeeman interaction lifts the two-fold degeneracy by selecting the basis +|m+⟩ = cos θm +2 |m⟩+eiφm sin θm +2 | ¯m⟩ +(S9) +|m−⟩ = −sin θm +2 |m⟩+eiφm cos θm +2 | ¯m⟩ +(S10) +and shifting the energies according to Em,± = Em ±∆m/2, where the gap +∆m = ⟨m+| ˆHZee|m+⟩−⟨m−| ˆHZee|m−⟩ +(S11) += 2µBgJ +� +⟨m|B· ˆJ|m⟩2 +|⟨m|B· ˆJ| ¯m⟩|2 +can be obtained as the norm of the vector jm = µBB·g(m) +el +and the phase and mixing angles are defined as +eiφm = ⟨ ¯m|B· ˆJ|m⟩ +|⟨ ¯m|B· ˆJ|m⟩|, +tanθm = |⟨ ¯m|B· ˆJ|m⟩| +⟨m|B· ˆJ|m⟩ , +(S12) +or equivalently as the azimuthal and polar angles determining the direction of jm. +Besides selecting a preferred basis and lifting the degeneracy of each doublet, the Zeeman interaction also causes mixing +between different doublets. In particular, the lowest doublet will change according to +|1′ +±⟩ = |1±⟩+ ∑ +m̸=1,¯1 +|m⟩⟨m| ˆHZee|1±⟩ +E1 −Em ++O(B2) ≈ +� +1− ˆQ1 ˆHZee +� +|1±⟩, +(S13) +with +ˆQ1 = ∑ +m̸=1,¯1 +|m⟩ +1 +Em −E1 +⟨m|. +(S14) + +5 +B. +Spin-boson Hamiltonian for the ground doublet +Now that we have an approximate expression for the relevant electronic states, we reintroduce the spin-phonon coupling into +the picture. First, we project the vibronic Hamiltonian (S6) onto the subspace spanned by |1′ +±⟩, yielding +ˆHeff = E1 + +� ∆1 +2 +0 +0 +− ∆1 +2 +� ++∑ +j +� +⟨1′ ++| ˆVj|1′ ++⟩ ⟨1′ ++| ˆVj|1′ +−⟩ +⟨1′ +−| ˆVj|1′ ++⟩ ⟨1′ +−| ˆVj|1′ +−⟩ +� +⊗(ˆbj + ˆb† +j)+∑ +j +ωj ˆb† +j ˆb j. +(S15) +On this basis, the purely electronic part ˆHCF + ˆHZee is diagonal with eigenvalues E1 ± ∆1/2, and the purely vibrational part is +trivially unaffected. On the other hand, the spin-phonon couplings can be calculated to lowest order in the magnetic field strength +B as +⟨1′ +±| ˆVj|1′ +±⟩ = ⟨1±| +� +1− ˆHZee ˆQ1 +� ˆVj +� +1− ˆQ1 ˆHZee +� +|1±⟩+O(B2) +(S16) += ⟨1±| ˆVj|1±⟩−⟨1±| +� ˆVj ˆQ1 ˆHZee + ˆHZee ˆQ1 ˆVj +� +|1±⟩+O(B2) += ⟨1| ˆVj|1⟩−⟨1±| ˆWj|1±⟩+O(B2), +⟨1′ +∓| ˆVj|1′ +±⟩ = ⟨1∓| +� +1− ˆHZee ˆQ1 +� ˆVj +� +1− ˆQ1 ˆHZee +� +|1±⟩+O(B2) +(S17) += ⟨1∓| ˆVj|1±⟩−⟨1∓| +� ˆVj ˆQ1 ˆHZee + ˆHZee ˆQ1 ˆVj +� +|1±⟩+O(B2) += −⟨1∓| ˆWj|1±⟩+O(B2), +where we have defined +ˆWj = ˆVj ˆQ1 ˆHZee + ˆHZee ˆQ1 ˆVj +(S18) +and used the time-reversal invariance of the spin-phonon coupling operators to obtain ⟨1±| ˆVj|1±⟩ = ⟨1| ˆVj|1⟩ and ⟨1∓| ˆVj|1±⟩ = 0. +The two states |1±⟩ form a conjugate pair under time reversal, meaning that ˆΘ|1±⟩ = ∓eiα|1∓⟩ for some α ∈ R. Using the +fact that for any two states ψ, ϕ, and for any operator ˆO we have ⟨ψ| ˆO|ϕ⟩ = ⟨ ˆΘϕ| ˆΘ ˆO† ˆΘ−1| ˆΘψ⟩, and recalling that the angular +momentum operator is odd under time reversal, i.e. ˆΘˆJ ˆΘ−1 = −ˆJ, we can show that +⟨1−| ˆWj|1−⟩ = ⟨ ˆΘ1−| ˆΘ ˆWj ˆΘ−1| ˆΘ1−⟩ = −⟨1+| ˆWj|1+⟩. +Keeping in mind these observations, and defining the vector +w j = +� +� +wx +j +wy +j +wz +j +� +� = +� +� +ℜ ⟨1−| ˆWj|1+⟩ +ℑ ⟨1−| ˆWj|1+⟩ +⟨1+| ˆWj|1+⟩ +� +�, +(S19) +we can rewrite the spin-phonon coupling operators in Eq. (S15) as +� +⟨1′ ++| ˆVj|1′ ++⟩ ⟨1′ ++| ˆVj|1′ +−⟩ +⟨1′ +−| ˆVj|1′ ++⟩ ⟨1′ +−| ˆVj|1′ +−⟩ +� += ⟨1| ˆVj|1⟩− +� +⟨1+| ˆWj|1+⟩ ⟨1−| ˆWj|1+⟩∗ +⟨1−| ˆWj|1+⟩ −⟨1+| ˆWj|1+⟩ +� += ⟨1| ˆVj|1⟩−w j ·σ′ +(S20) +where σ′ is a vector whose entries are the Pauli matrices in the basis |1′ +±⟩, i.e. σ′ +z = |1′ ++⟩⟨1′ ++| − |1′ +−⟩⟨1′ +−|. Plugging this back +into Eq. (S15) and explicitly singling out the diagonal components of ˆHeff in the basis |1′ +±⟩, we obtain +ˆHeff = |1′ ++⟩⟨1′ ++| +� +E1 + ∆1 +2 +∑ +j +� +⟨1| ˆVj|1⟩−wz +j +�� +ˆbj + ˆb† +j +� ++∑ +j +ω j ˆb† +j ˆbj +� +(S21) ++ |1′ +−⟩⟨1′ +−| +� +E1 − ∆1 +2 +∑ +j +� +⟨1| ˆVj|1⟩+wz +j +�� +ˆbj + ˆb† +j +� ++∑ +j +ω j ˆb† +j ˆbj +� +− ∑ +j +� +wx +jσ′ +x +wy +jσ′ +y +�� +ˆbj + ˆb† +j +� +. +At this point, we apply a unitary polaron transformation to the Hamiltonian (S21) +ˆS = exp +� +∑ +s=± +|1′ +s⟩⟨1′ +s| ∑ +j +1 +ωj +� +⟨1| ˆVj|1⟩−swz +j +�� +ˆb† +j − ˆbj +�� +(S22) += ∑ +s=± +|1′ +s⟩⟨1′ +s| ∏ +j +ˆD j(ξ s +j) + +6 +where ξ s +j = +� +⟨1| ˆVj|1⟩−swz +j +� +/ω j and +ˆD j(ξ s +j) = eξ s +j +� +ˆb† +j−ˆb j +� +(S23) +is the bosonic displacement operator acting on mode j, i.e. ˆD j(ξ)ˆbj ˆD† +j(ξ) = ˆbj −ξ. The Hamiltonian thus becomes +ˆS ˆHeff ˆS† = ∑ +s=± +|1′ +s⟩⟨1′ +s| +� +E1 +s∆1 +2 −∑ +j +ωj|ξ s +j|2 +� ++∑ +j +ωj ˆb† +j ˆbj −∑ +j +ˆS +� +wx +jσ′ +x +wy +jσ′ +y +�� +ˆb j + ˆb† +j +� +ˆS†. +(S24) +The polaron transformation reabsorbes the diagonal component of the spin-phonon coupling (S20) proportional to wz +j into the +energy shifts ωj|ξ ± +j |2, leaving a residual off-diagonal spin-phonon coupling proportional to wx +j and wy +j. Note that the polaron +transformation exactly diagonalises the Hamiltonian (S15) if wx +j = wy +j = 0. In Section S3, we argue in detail that in our case +|wx +j|,|wy +j| ≪ |wz +j| to a very good approximation. Based on this argument, we could decide to neglect the residual spin-phonon +coupling in the polaron frame. The energies of the states belonging to the lowest doublet are shifted by a vibronic correction +E1′± = E1 ± ∆1 +2 −∑ +j +1 +ω j +� +⟨1| ˆVj|1⟩∓wz +j +�2 +(S25) += E1 ± ∆1 +2 −∑ +j +1 +ω j +� +⟨1| ˆVj|1⟩2 ∓2⟨1| ˆVj|1⟩wz +j +O(B2) +� +, +(S26) +leading to a redefinition of the energy gap +E1′+ −E1′− = ∆1 +4∑ +j +⟨1| ˆVj|1⟩ +ωj +wz +j. +(S27) +Although the off-diagonal components of the spin-phonon coupling wx +j and wy +j are several orders of magnitude smaller than +the diagonal one wz +j (see Section S3), the sheer number of vibrational modes could still lead to an observable effect on the +electronic degrees of freedom. We can estimate this effect by averaging the residual spin-phonon coupling over a thermal +phonon distribution in the polaron frame. Making use of Eq. (S22), the off-diagonal coupling in Eq. (S24) can be written as +ˆH(pol) +sp-ph = −∑ +j +ˆS +� +wx +jσ′ +x +wy +jσ′ +y +�� +ˆbj + ˆb† +j +� +ˆS† +(S28) += −∑ +j +|1′ +−⟩⟨1−| ˆWj|1+⟩⟨1′ ++| ˆD j(ξ − +j ) +� +ˆbj + ˆb† +j +� +ˆD† +j(ξ + +j )+h.c. +Assuming the vibrations to be in a thermal state at temperature T in the polaron frame +ρ(th) +ph = ∏ +j +ρ(th) +j += ∏ +j +e−ωj ˆb† +j ˆb j/kBT +Tr +� +e−ωj ˆb† +j ˆb j/kBT�, +(S29) +obtaining the average of Eq. (S28) reduces to calculating the dimensionless quantity +κj = −Tr +� +ˆD j(ξ − +j ) +� +ˆbj + ˆb† +j +� +ˆD† +j(ξ + +j )ρ(th) +j +� +(S30) += +� +ξ + +j +ξ − +j +� +e− 1 +2 +� +ξ + +j −ξ − +j +�2 +coth +� ωj +2kBT +� += 2⟨1| ˆVj|1⟩ +ωj +e +−2 +(wz +j)2 +ω2 +j +coth +� ωj +2kBT +� += 2⟨1| ˆVj|1⟩ +ωj +� +1+O(B2), +� +which appears as a multiplicative rescaling factor for the off-diagonal couplings ⟨1∓| ˆWj|1±⟩. Note that, when neglecting second +and higher order terms in the magnetic field, κj does not show any dependence on temperature or on the magnetic field orientation +via θ1 and φ1. + +7 +After thermal averaging, the effective electronic Hamiltonian for the lowest energy doublet becomes +ˆHel = Trph +� +ˆS ˆHeff ˆS†ρ(th) +ph +� += E1 +δE1 + +� +2∑ +j +⟨1| ˆVj|1⟩ +ωj +wx +j,2∑ +j +⟨1| ˆVj|1⟩ +ωj +wy +j, ∆1 +2 +2∑ +j +⟨1| ˆVj|1⟩ +ωj +wz +j +� +· +� +� +σ′ +x +σ′ +y +σ′ +z +� +� +(S31) +where the energy of the lowest doublet is shifted by +δE1 = −∑ +j +⟨1| ˆVj|1⟩2 +ωj ++∑ +j +ω j +eωj/kBT −1 +(S32) +due to the spin-phonon coupling and to the thermal phonon energy. Eq. (S31) thus represents a refined description of the lowest +effective spin-1/2 doublet in the presence of spin-phonon coupling. +We can finally recast the Hamiltonian (S31) in terms of a g-matrix for an effective spin 1/2, similarly to what we did earlier +in the case of no spin-phonon coupling. In order to do so, we first recall from Eq. (S11) and (S19) that the quantities ∆1 and +(wx +j,wy +j,wz +j) appearing in Eq. (S31) depend on the magnetic field orientation via the states |1±⟩, and on both orientation and +intensity via ˆHZee. We can get rid of the first dependence by expressing the Zeeman eigenstates |1±⟩ in terms of the original +crystal field eigenstates |1⟩, |¯1⟩. For the spin-phonon coupling vector wj, we obtain +w j = +� +� +ℜ⟨1−| ˆWj|1+⟩ +ℑ⟨1−| ˆWj|1+⟩ +⟨1+| ˆWj|1+⟩ +� +� = +� +� +cosθ1 cosφ1 cosθ1 sinφ1 −sinθ1 +−sinφ1 +cosφ1 +0 +sinθ1 cosφ1 +sinθ1 sinφ1 +cosθ1 +� +� +� +� +ℜ⟨¯1| ˆWj|1⟩ +ℑ⟨¯1| ˆWj|1⟩ +⟨1| ˆWj|1⟩ +� +� = R(θ1,φ1)· ˜w j. +(S33) +where R(θ1,φ1) is a rotation matrix. Similarly, the elctronic contribution ∆1 transforms as +(0,0,∆1) = j1 ·R(θ1,φ1)T,= µBB·g(1) +el ·R(θ1,φ1)T. +(S34) +The Pauli spin operators need to be changed accordingly to ˜σ = R(θ1,φ1)T · σ′. Lastly, we single out explicitly the magnetic +field dependence of ˆWj, defined in Eq. (S18), by introducing a three-component operator ˆKj = ( ˆKx +j, ˆKy +j, ˆKz +j), such that +ˆWj = µBgJB· +� ˆVj ˆQ1ˆJ+ ˆJ ˆQ1 ˆVj +� +(S35) += µBgJB· ˆK j. +Thus, the effective electronic Hamiltonian in Eq. (S31) can be finally rewritten as +ˆHel = E1 +δE1 + µBB· +� +g(1) +el +gvib +� +· ˜σ/2 +(S36) +where g(1) +el is the electronic g-matrix defined in Eq. (S8), and +gvib = 4gJ∑ +j +⟨1| ˆVj|1⟩ +ωj +� +� +ℜ⟨¯1| ˆKx +j|1⟩ ℑ⟨¯1| ˆKx +j|1⟩ ⟨1| ˆKx +j|1⟩ +ℜ⟨¯1| ˆKy +j|1⟩ ℑ⟨¯1| ˆKy +j|1⟩ ⟨1| ˆKy +j|1⟩ +ℜ⟨¯1| ˆKz +j|1⟩ ℑ⟨¯1| ˆKz +j|1⟩ ⟨1| ˆKz +j|1⟩ +� +� +(S37) +is a vibronic correction. +Note that this correction is non-perturbative in the spin-phonon coupling, despite only containing quadratic terms in ˆVj (recall +that ˆK j depends linearly on ˆVj). The only approximations leading to Eq. (S36) are a linear perturbative expansion in the magnetic +field B and neglecting quantum fluctuations of the off-diagonal spin-phonon coupling in the polaron frame, which is accounted +for only via its thermal expectation value. This approximation relies on the fact that the off-diagonal couplings are much smaller +than the diagonal spin-phonon coupling that is treated exactly by the polaron transformation (see Section S3). +C. +Landau-Zener probability +Let us consider a situation in which the magnetic field comprises a time-independent contribution arising from internal dipolar +or hyperfine fields Bint and a time dependent external field Bext(t). Let us fix the orientation of the external field and vary its +magnitude at a constant rate, such that the field switches direction at t = 0. Under these circumstances, the Hamiltonian of Eq. +(S36) becomes +ˆHel(t) = E1 +δE1 + µB +� +Bint + dBext +dt +t +� +·g· ˜σ +2 , +(S38) + +8 +where g = g(1) +el +gvib. Neglecting the constant energy shift and introducing the vectors +∆ += µBBext ·g, +(S39) +v = µBdBext/dt ·g, +(S40) +the Hamiltonian then becomes +ˆHel(t) = ∆ +2 · ˜σ + vt +2 · ˜σ = ∆⊥ +2 +· ˜σ + vt +∆∥ +2 +· ˜σ. +(S41) +In the second equality, we have split the vector ∆ = ∆⊥ +∆∥ into a perpendicular and a parallel component to v. Choosing an +appropriate reference frame, we can write +ˆHel(t′) = ∆⊥ +2 ˜σx + vt′ +2 ˜σz, +(S42) +in terms of the new time variable t′ = t +∆∥/v. Assuming that the spin is initialised in its ground state at t′ → −∞, the probability +of observing a spin flip at t′ → +∞ is given by the Landau-Zener formula [15–20] +PLZ = 1−exp +� +−π∆2 +⊥ +2v +� +. +(S43) +We remark that tunnelling is only made possible by the presence of ∆⊥, which stems from internal fields that have a perpen- +dicular component to the externally applied field. We also observe that a perfectly axial system would not exhibit tunnelling +behaviour, since in that case the direction of B · g would always point along the easy axis (i.e. along the only eigenvector of +g with a non-vanishing eigenvalue), and therefore v and ∆ would always be parallel. Thus, deviations from axiality and the +presence of transverse fields are both required for QTM to occur. + +9 +S3. +DISTRIBUTION OF SPIN-PHONON COUPLING VECTORS +The effective polaron Hamiltonian presented in Eq. (7) and derived in the previous section provides a good description of +the ground doublet only if the spin-phonon coupling operators are approximately diagonal in the electronic eigenbasis. This is +equivalent to requiring that the components of the vectors w j defined in Eq. (S19) satisfy +|wx +j|,|wy +j| ≪ |wz +j|. +(S44) +Fig. S1a shows the distribution of points {w j, j = 1,...,M} (where M is the number of vibrational modes) in 3D space for +different orientations of the magnetic field. As a consequence of the strong magnetic axiality of the complex under consideration, +we see that these points are mainly distributed along the z-axis, therefore satisfying the criterion expressed in Eq. (S44) (note +the different scale on the xy-plane). +b) +a) +FIG. S1. Distribution of spin-phonon coupling vectors wj. (a) The points w j distribute along a straight line in 3D space (units: cm−1) when +the magnetic field is oriented along x, y, z. The magnitude is fixed to 1 T. Note that, owing to the definition of w j, a different magnitude would +yield a uniformly rescaled distribution of points, leaving the shape unchanged. (b) Variance of the points w j in the xy-plane in units of the total +variance, as a function of magnetic field orientation. +In order to confirm that the points w j maintain a similar distribution regardless of the magnetic field orientation, we calculate +their variances along different directions of the 3D space they inhabit. We define +σ2 +α = var(wα +j ) = +1 +M −1 +M +∑ +j=1 +� +wα +j − µα +�2 , +(S45) +where α = x,y,z and µα = 1 +M ∑M +j=1 wα +j . The dependence of these variances on the field orientation is made evident by recalling +that the points w j are related via a rotation R(θ1,φ1) to the set of points ˜w j, which only depend linearly on the field B, as shown +in Eqs. (S33) and (S35). If the points are mainly distributed along z for any field orientation, we expect the combined variance +in the xy-plane to be much smaller than the total variance of the dataset, i.e. +σ2 +x +σ2 +y ≪ σ2 +x +σ2 +y +σ2 +z . +(S46) +Fig. S1b provides a direct confirmation of this hypothesis, showing that the variance in the xy-plane is at most 6 × 10−4 times +smaller than the total variance. Therefore, we conclude that the approach followed in Section S2 is fully justified. + +0.05 +0.00 +-0.0002 +-0.05 +0.0000 +0.0002 +0.0000 +0.0002 +-0.00020.05 +0.00 +-0.0002 +-0.05 +0.0000 +0.0002 +0.0000 +0.0002 +-0.00020.005 +0.000 +-0.0002 +-0.005 +0.0000 +0.0002 +0.0000 +0.0002 +-0.0002元 +0.0006 +0.0004 +元 +2 +0.0002 +0 +0 +爪 +0 +一元 +元 +2 +210 +S4. +EXPERIMENTAL ESTIMATE OF THE SPIN-FLIP PROBABILITY +In order to provide experimental support for our vibronic model of QTM, we compare the calculated spin-flip probabilities +with values extracted from previously reported measurements of magnetic hysteresis. We use data from field-dependent mag- +netisation measurements reported in Ref. [7](Fig. S35, sample 4), reproduced here in Fig. S2. The sample consisted of a 83 µL +volume of a 170 mM solution of [Dy(Cpttt)2][B(C6F5)4] in dichloromethane (DCM). The field-dependent magnetisation was +measured at T = 2 K while sweeping an external magnetic field Bext from +7 T to −7 T and back again to +7 T. The result- +ing hysteresis loop is shown in Fig. S2a. The sweep rate dBext/dt is not constant throughout the hysteresis loop, as shown in +Fig. S2b. In particular, it takes values between 10 Oe/s and 20 Oe/s across the zero field region where QTM takes place. +QTM results in a characteristic step around the zero field region in magnetic hysteresis curves (Fig. S2a). The spin-flip +probability across the tunnelling transition can be easily related to the height of this step via the expression [21] +P↑→↓ = 1 +2 +� M +Msat +− M′ +Msat +� +. +(S47) +The value of the magnetisation before (M) and after (M′) the QTM drop is estimated by performing a linear fit of the field- +dependent magnetisation close to the zero field region, for both Bext > 0 and Bext < 0, and extrapolating the magnetisation at +Bext = 0 (Fig. S2a, inset). The saturation value of the magnetisation Msat is obtained by measuring the magnetisation at low +temperature in a strong external magnetic field (T = 2 K, Bext = 7 T). Following this method, we obtain a spin-flip probability +P↑→↓ = 0.27, which is shown as a purple horizontal line in Fig. 4 in the main text. +b) +a) +FIG. S2. Magnetic hysteresis of [Dy(Cpttt)2]+ from Ref. [7]. (a) Field-dependent magnetisation was measured on a 170 mM frozen solution +of [Dy(Cpttt)2]+ (counter ion [B(C6F5)4]−) in DCM at T = 2 K. Data presented in [7] (Fig. S35, sample 4). The loop is traversed in the +direction indicated by the blue arrows. The sudden drop of the magnetisation from M to M′ around Bext = 0 is a characteristic signature +of QTM. The slow magnetisation decay around the QTM step can be ascribed to other magnetic relaxation mechanisms (Raman). (b) Time +dependence of the magnetic field Bext (top) and instantaneous sweep rate (bottom). Note that the sweep rate is not constant around the avoided +crossing at Bext = 0, but assumes values in the range 10–20 Oe/s. + +11 +S5. +ESTIMATE OF THE INTERNAL FIELDS IN A FROZEN SOLUTION +A. +Dipolar fields +In this section we provide an estimate of the internal fields Bint in a disordered ensemble of SMMs, based on field-dependent +magnetisation data introduced in Section S4. +When a SMM with strongly axial magnetic anisotropy is placed in a strong external magnetic field Bext, it gains a non-zero +magnetic dipole moment along its easy axis. Once the external field is removed, the SMM partially retains its magnetisation +µ = µ ˆµ, which produces a microscopic dipolar field +Bdip(r) = µ0µ +4πr3 [3ˆr( ˆµ· ˆr)− ˆµ] +(S48) +at a point r = rˆr in space. This field can then cause a tunnelling gap to open in neighboring SMMs, depending on their relative +distance and orientation. +In order to estimate the strength of typical dipolar fields, we need to determine the average distance between SMMs in the +sample, and the magnetic dipole moment associated with a single SMM. Since we know both volume V and concentration of +Dy centres in the sample (see previous section), we can easily obtain the number of SMMs in solution N. The average distance +between SMMs can then be obtained simply by taking the cubic root of the volume per particle, as +r = +�V +N +�1/3 +≈ 21.4 Å. +(S49) +The magnetic moment can be obtained from the hysteresis curve shown in Fig. S2a, by reading the value of the magnetisation +M right before the QTM step. This amounts to an average magnetic moment per molecule +⟨µ∥⟩ = M +N ≈ 4.07µB +(S50) +along the direction of the external field Bext, where ⟨·⟩ denotes the average over the ensemble of SMMs. Since the orientation +of SMMs in a frozen solution is random, the component of the magnetisation µ perpendicular to the applied field averages to +zero, i.e. ⟨µ⊥⟩ = 0. However, it still contributes to the formation of the microscopic dipolar field (S48), which depends on +µ = µ∥ +µ⊥. Since the sample consists of many randomly oriented SMMs, the average magnetisation in Eq. (S50) can also be +expressed in terms of µ = |µ| via the orientational average +⟨µ∥⟩ = +� π/2 +0 +dθ sinθ µ∥(θ) = µ +2 , +(S51) +where µ∥(θ) = µ cosθ is the component of the magnetisation of a SMM along the direction of the external field Bext. Thus, the +magnetic moment responsible for the microscopic dipolar field is twice as big as the measured value (S50). +Based on these estimates, the magnitude of dipolar fields experienced by a Dy atom in the sample is +Bdip = 0.77 mT× +� +|3( ˆµ· ˆr)2 −1|. +(S52) +The square root averages to 1.38 for randomly oriented µ and r and can take values between 1 and 2, represented by the green +shaded area in Fig. 4 in the main text. +B. +Hyperfine coupling +Another possible source of microscopic magnetic fields are nuclear spins. Among the different isotopes of dysprosium, only +161Dy and 163Dy have non-zero nuclear spin (I = 5/2), making up for approximately 44 % of naturally occurring dysprosium. +The nucear spin degrees of freedom are described by the Hamiltonian +ˆHnuc = ˆHQ + ˆHHF = ˆI·P· ˆI+ ˆI·A· ˆJ, +(S53) +where the first term is the quadrupole Hamiltonian ˆHQ = ˆI·P· ˆI, accounting for the zero-field splitting of the nuclear spin states, +and the second term ˆHHF = ˆI·A· ˆJ accounts for the hyperfine coupling between nuclear spin ˆI and electronic angular momentum + +12 +ˆJ operators. In analogy with the electronic Zeeman Hamiltonian ˆHZee = µBgJB· ˆJ, we define the effective nuclear magnetic field +operator +µBgJ ˆBnuc = AT · ˆI, +(S54) +so that the hyperfine coupling Hamiltonian takes the form of a Zeeman interaction ˆHHF = µBgJ ˆB† +nuc · ˆJ. If we consider the nuclear +spin to be in a thermal state at temperature T with respect to the quadrupole Hamiltonian ˆHQ, the resulting expectation value of +the nuclear magnetic field vanishes, since the nuclear spin is completely unpolarised. However, the external field Bext will tend +to polarise the nuclear spin via the nuclear Zeeman Hamiltonian +ˆHnuc, Zee = µNg Bext · ˆI, +(S55) +where µN is the nuclear magneton and g is the nuclear g-factor of a Dy nucleus. In this case, the nuclear spin is described by the +thermal state +ρ(th) +nuc = +e−( ˆHQ+ ˆHnuc, Zee)/kBT +Tr +� +e−( ˆHQ+ ˆHnuc, Zee)/kBT� +(S56) +and the effective nuclear magnetic field can be calculated as +Bnuc = Tr +� ˆBnucρ(th) +nuc +� +. +(S57) +To the best of our knowledge, quadrupole and hyperfine coupling tensors for Dy in [Dy(Cpttt)2]+ have not been reported in +the literature. However, ab initio calculations of hyperfine coupling tensors have been performed on DyPc2 [22]. Although the +dysprosium atom in DyPc2 and [Dy(Cpttt)2]+ interacts with different ligands, the crystal field is qualitatively similar for these +two complexes, therefore we expect the nuclear spin Hamiltonian to be sufficiently close to the one for [Dy(Cpttt)2]+, at least for +the purpose of obtaining an approximate estimate. Using the quadrupolar and hyperfine tensors determined for DyPc2 [22] and +the nuclear g-factors measured for 161Dy and 163Dy [23], we can compute Bnuc = |Bnuc| from Eq. (S57) for different orientations +of the external magnetic field. As shown in Table S1, the effective nuclear magnetic fields at T = 2 K are at least one order of +magnitude smaller than the dipolar fields calculated in the previous section, regardless of the orientation of the external field. +161Dy +163Dy +Bext//ˆx 2.82×10−8 5.34×10−8 +Bext//ˆy 1.77×10−8 3.38×10−8 +Bext//ˆz 5.51×10−5 1.08×10−4 +TABLE S1. Effective Dy nuclear magnetic field Bnuc (T) at T = 2 K. + +13 +S6. +RESULTS FOR A DIFFERENT SOLVENT CONFIGURATION +In this section we show that the results presented in the main text are robust against variations of the solvent environment on +a qualitative level. In order to show this, we consider a smaller and rounder solvent ball consisting of 111 DCM molecules, and +reproduce the results shown in the main text, as shown in Fig. S3. It is worth noting that the vibronic spin-flip probabilities are +significantly smaller for the smaller solvent ball, confirming the importance of the low-frequency vibrational modes associated +to the solvent for determining QTM behaviour. The general tendency of vibrations to enhance QTM, however, is correctly +reproduced. +vibrational DOS +a) +b) +c) +d) +e) +j +el +j +FIG. S3. +Results for a different solvent configuration. (a) Alternative arrangement of 111 DCM molecules around [Dy(Cpttt)2]+. (b) +Spin-phonon coupling strength and vibrational density of states (see Fig. 1c). (c) Vibronic correction to the energy splitting of the ground +Kramers doublet (∆vib +1 +− ∆1) for different orientations of the magnetic field (see Fig. 2a). (d) Ensemble-averaged spin-flip probability for +different field sweep rates as a function of the internal field strength (see Fig. 3). (e) Orientationally averaged single-mode spin-flip probability +⟨Pj⟩ vs change in magnetic axiality ∆Aj/Ael (see Fig. 4). +The most evident difference between these results and the ones presented in the main text is the shape of the single-mode +axiality distribution (Fig. S3e). In this case, single-mode spin-flip probability ⟨Pj⟩ still correlates to relative single-mode axiality +∆A j/Ael. However, instead of taking values on a continuous range, the relative axiality seems to cluster around discrete values. +In an attempt to clarify the origin of this strange behaviour, we looked at the composition of the vibrational modes belonging +to the different clusters. Vibrational modes belonging to the same cluster were not found to share any evident common feature. +Rather than in the structure of the vibrational modes, this behaviour seems to originate from the equilibrium electronic g-matrix +gel. This can be seen by computing the single-mode axiality Aj = A(gel +gvib +j ) for slightly different choices of gel. In particular, +we checked how axiality of the electronic g-matrix affects the mode axiality. In order to do that, we considered the singular +value decomposition of the electronic g-matrix +gel = U·diag(g1,g2,g3)·V†, +(S58) + +△1 +-A1 +(cm +0.2 +0.1 +2 +0 +0.1 +- +0.2 +0 +-元 +0 +元 +2 +2(cm +0.2 +0.1 +0 +2 +-0.1 +-0.2 +0 +一元 +2 +2A1 +VID +A1 +(cm +0.2 +0.1 +2 +0 +0.1 +0.2 +-元 +0 +元 +2 +214 +the matrices U and V contain its left and right eigenvectors. The singular values are g1 = 19.99, g2 = 3.40 × 10−6, g3 = +2.98 × 10−6, and the axiality is very close to one, i.e. 1 − Ael = 4.79 × 10−7. We artificially change the axiality of gel by +rescaling the hard-plane g-values by a factor α and redefining the electronic g-matrix as +gel +α = U·diag(g1,αg2,αg3)·V†. +(S59) +The results are shown in Fig. S4. The three different colours distinguish the vibrational modes belonging to the three clusters +visible in Fig. S3e (corresponding to α = 1). When α = 0, the g-matrix has perfect easy-axis anisotropy. In this case, the +vibronic correction to the g-matrix is too small to cause significant changes in the magnetic axiality, and all the vibrational +modes align around A j ≈ Ael. Increasing α to 0.9, clusters begin to appear. For α = 1.3, the single-mode axiality distribution +begins to look like the one shown in Fig. 4a in the main text. The electronic g-matrix obtained for the solvent ball considered in +the main text has a lower axiality than the one used throughout this section, i.e. 1−Ael = 1.12×10−6. Therefore, it makes sense +that for α sufficiently larger than 1 we recover the same type of distribution as in the main text, since increasing α corresponds +to lowering the electronic axiality A(gel +α). +FIG. S4. Impact of electronic axiality on single-mode axiality. Distribution of single-mode spin-flip probability ⟨Pj⟩ and g-matrix axiality +A j = A(gelα + gvib +j ) relative to the axiality of the modified electronic g-matrix A(gelα) defined in Eq. (S59). Vibrational modes belonging to +different clusters in Fig. S3e (α = 1) are labelled with different colors. + +15 +[1] M. Svensson, S. Humbel, R. D. J. Froese, T. Matsubara, S. Sieber, and K. Morokuma, ONIOM: a multilayered integrated MO + MM +method for geometry optimizations and single point energy predictions. a test for diels-alder reactions and Pt(P(t-Bu)3)2 + H2 oxidative +addition, The Journal of Physical Chemistry 100, 19357 (1996). +[2] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Physical Review Letters 77, 3865 (1996). +[3] D. Andrae, U. Häußermann, M. Dolg, H. Stoll, and H. Preuß, Energy-adjusted ab initio pseudopotentials for the second and third row +transition elements, Theor. Chim. Acta 77, 123 (1990). +[4] T. H. Dunning, Gaussian basis sets for use in correlated molecular calculations. i. the atoms boron through neon and hydrogen, The +Journal of Chemical Physics 90, 1007 (1989). +[5] M. J. Frisch, G. W. Trucks, H. B. 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Gebauer, Hyperfine structure investigations in DyI with the atomic beam magnetic resonance method, +Physics Letters A 49, 287 (1974). + diff --git a/HNE5T4oBgHgl3EQfWQ9u/content/tmp_files/load_file.txt b/HNE5T4oBgHgl3EQfWQ9u/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c78649c61e86c255b01621290ab57017bbba57b2 --- /dev/null +++ b/HNE5T4oBgHgl3EQfWQ9u/content/tmp_files/load_file.txt @@ -0,0 +1,1346 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf,len=1345 +page_content='Vibronic Effects on the Quantum Tunnelling of Magnetisation in Single-Molecule Magnets Andrea Mattioni,1, ∗ Jakob K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Staab,1 William J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Blackmore,1 Daniel Reta,1, 2 Jake Iles-Smith,3, 4 Ahsan Nazir,3 and Nicholas F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Chilton1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' † 1Department of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' School of Natural Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The University of Manchester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Oxford Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Manchester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' M13 9PL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' UK 2Faculty of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' UPV/EHU & Donostia International Physics Center DIPC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Ikerbasque,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Basque Foundation for Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Bilbao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Spain 3Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' School of Natural Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The University of Manchester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Oxford Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Manchester M13 9PL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' UK 4Department of Electrical and Electronic Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' School of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The University of Manchester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Sackville Street Building,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Manchester M1 3BB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' UK Single-molecule magnets are among the most promising platforms for achieving molecular-scale data stor- age and processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Their magnetisation dynamics are determined by the interplay between electronic and vibrational degrees of freedom, which can couple coherently, leading to complex vibronic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Building on an ab initio description of the electronic and vibrational Hamiltonians, we formulate a non-perturbative vi- bronic model of the low-energy magnetic degrees of freedom in a single-molecule magnet, which we benchmark against field-dependent magnetisation measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Describing the low-temperature magnetism of the com- plex in terms of magnetic polarons, we are able to quantify the vibronic contribution to the quantum tunnelling of the magnetisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Despite collectively enhancing magnetic relaxation, we observe that specific vibrations suppress quantum tunnelling by enhancing the magnetic axiality of the complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Finally, we discuss how this observation might impact the current paradigm to chemical design of new high-performance single-molecule magnets, promoting vibrations to an active role rather than just regarding them as sources of noise and decoher- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' INTRODUCTION Single-molecule magnets (SMMs) hold the potential for realising high-density data storage and quantum informa- tion processing [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' These molecules exhibit a doubly- degenerate ground state, comprising two states supporting a large magnetic moment with opposite orientation, which rep- resents an ideal platform for storing digital data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Slow reori- entation of this magnetic moment results in magnetic hystere- sis at the single-molecule level at sufficiently low tempera- tures [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The main obstacle to extending this behaviour to room temperature is the coupling of the magnetic degrees of freedom to molecular and lattice vibrations, often referred to as spin-phonon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Thermal excitation of the molec- ular vibrations cause transitions between different magnetic states, ultimately leading to a complete loss of magnetisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Advances in design, synthesis and characterisation of SMMs have shed light on the microscopic mechanisms underlying their desirable magnetic properties, extending this behaviour to increasingly higher temperatures [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The mechanism responsible for magnetic relaxation in SMMs strongly depends on temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' At higher temper- atures, relaxation is driven by one (Orbach) and two (Raman) phonon transitions between magnetic sublevels [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' When temperatures approach absolute zero, all vibrations are pre- dominantly found in their ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Thus, both Or- bach and Raman transitions become negligible and the dom- inant mechanism is quantum tunnelling of the magnetisation ∗ andrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='mattioni@manchester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='uk † nicholas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='chilton@manchester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='uk (QTM) between the two degenerate ground states [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This process relies on the presence of a coherent coupling mixing the two otherwise degenerate ground states, opening a tunnelling gap, and allowing population to redistribute be- tween them, thus leading to facile magnetic reorientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' While the role of vibrations in high-temperature magnetic relaxation is well understood in terms of weak-coupling rate equations for the electronic populations [12–15], the connec- tion between QTM and spin-phonon coupling is still unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Some analyses have looked at the influence of vibrations on QTM in integer-spin SMMs, where a model spin system was used to show that spin-phonon coupling could open a tunnel- ing gap [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' However, QTM remains more elusive to grasp in half-integer spin complexes, such as monometallic Dy(III) SMMs, since it is observed experimentally despite being forbidden by Kramers theorem [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In this case, a magnetic field is needed to break the time-reversal symmetry of the molecular Hamiltonian and lift the degeneracy of the ground doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This magnetic field can be provided by hy- perfine interaction with nuclear spins or by dipolar coupling to other SMMs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' both these effects have been shown to af- fect tunnelling behaviour [19–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Once the tunnelling gap is opened by a magnetic field, molecular vibrations can in principle affect its magnitude in a nontrivial way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In a re- cent work, Ortu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' analysed the magnetic hysteresis of a series of Dy(III) SMMs, suggesting that QTM efficiency cor- relates with molecular flexibility [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In another work, hyper- fine coupling was proposed to assists QTM by facilitating the interaction between molecular vibrations and spin sublevels [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' However, a clear and unambiguous demonstration of the influence of the spin-phonon coupling on QTM beyond toy- model approaches is still lacking to this date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In this work we present a theoretical analysis of the effect of arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='05557v1 [quant-ph] 13 Jan 2023 2 molecular vibrations on the tunnelling dynamics in a Dy(III) SMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In contrast to previous treatments, our approach is based on a fully ab initio description of the SMM vibrational environment and accounts for the spin-phonon coupling in a non perturbative way, overcoming the standard weak-coupling master equation approach commonly used to determine the high-temperature magnetisation dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' After deriving an effective low-energy model for the relevant vibronic degrees of freedom based on a polaron approach [27], we demon- strate that vibrations can either enhance or reduce the quantum tunnelling gap, depending on the orientation of the magnetic field relative to the main anisotropy axis of the SMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' More- over, we validate our vibronic model against frozen solution, field-dependent magnetisation measurements and show that vibronic effects on QTM survive the orientational averaging imposed by amorphous samples, leading, on average, to a sig- nificant enhancement of the tunnelling probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Lastly, we argue that not all vibrations lead to faster QTM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' depending on how strongly vibrations impact the axiality of the lowest en- ergy magnetic doublet, we show that they can play a benign role by suppressing tunnelling, and discuss first steps in that direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' MODEL The compound investigated in this work is [Dy(Cpttt)2]+, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 1a [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The complex consists of a dyspro- sium ion Dy(III) enclosed between two negatively charged cyclopentadienyl rings with tert-butyl groups at positions 1, 2 and 4 (Cpttt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The crystal field generated by the axial ligands makes the states with larger angular momentum energetically favourable, resulting in the energy level diagram sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The energy barrier separating the two degenerate ground states results in magnetic hysteresis, which was ob- served up to T = 60 K [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Magnetic hysteresis is hindered by QTM, which leads to a characteristic sudden drop of the magnetisation at zero magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' To single out the contribution of molecular vibrations, we focus on a magnetically diluted sample in a frozen solution of dichloromethane (DCM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Thus, our computational model consists of a solvated [Dy(Cpttt)2]+ cation (see Section S1 for details;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 1a), which provides a realistic description of the low-frequency vibrational environment, comprised of pseudo- acoustic vibrational modes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' These constitute the ba- sis to consider further contributions of dipolar and hyperfine interactions to QTM (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Once the equilibrium geometry and vibrational modes of the solvated SMM (which are in general combinations of molecular and solvent vibrations) are obtained at the density- functional level of theory (see Section S1), we proceed to de- termine the equilibrium electronic structure via complete ac- tive space self-consistent field spin-orbit (CASSCF-SO) cal- culations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The electronic structure is projected onto an effec- tive crystal-field Hamiltonian, parametrised in terms of crys- tal field parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The spin-phonon couplings are obtained from a single CASSCF calculation, by computing the analytic derivatives of the molecular Hamiltonian with respect to the nuclear coordinates [14] (see Section S1 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The lowest-energy angular momentum multiplet of [Dy(Cpttt)2]+ (J = 15/2) can thus be described by the ab ini- tio vibronic Hamiltonian ˆH = ∑ m Em|m⟩⟨m|+∑ j ˆVj ⊗(ˆbj + ˆb† j)+∑ j ωj ˆb† j ˆbj, (1) where Em denotes the energy associated with the electronic state |m⟩ and ˆVj represent the spin-phonon coupling opera- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The harmonic vibrational modes of the DCM-solvated [Dy(Cpttt)2]+ are described in terms of their bosonic annihi- lation (creation) operators ˆbj (ˆb† j) and frequencies ωj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In the absence of magnetic fields, the Hamiltonian (1) is symmetric under time reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This symmetry results in a two-fold degeneracy of the energy levels Em, whose corre- sponding eigenstates |m⟩ and | ¯m⟩ form a time-reversal conju- gate Kramers doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The degeneracy is lifted by introducing a magnetic field B, which couples to the electronic degrees of freedom via the Zeeman interaction ˆHZee = µBgJB · ˆJ, where gJ is the Landé g-factor and ˆJ is the total angular momentum operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' To linear order in the magnetic field, each Kramers doublet splits into two energy levels Em±∆m/2 corresponding to the states |m+⟩ = cos θm 2 |m⟩+eiφm sin θm 2 | ¯m⟩ (2) |m−⟩ = −sin θm 2 |m⟩+eiφm cos θm 2 | ¯m⟩ (3) where the energy splitting ∆m and the mixing angles θm and φm are determined by the matrix elements of the Zeeman Hamiltonian on the subspace {|m⟩,| ¯m⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In addition to the intra-doublet mixing described by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (2) and (3), the Zee- man interaction also mixes Kramers doublets at different ener- gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The ground doublet acquires contributions from higher- lying states |1′ ±⟩ = |1±⟩+ ∑ m̸=1,¯1 |m⟩⟨m| ˆHZee|1±⟩ E1 −Em +O(B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (4) These states no longer form a time-reversal conjugate doublet, meaning that the spin-phonon coupling can now contribute to transitions between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Since QTM is typically observed at much lower tempera- tures than the energy gap between the lowest and first excited doublets (which here is ∼ 660 K [6]), we focus on the per- turbed ground doublet |1′ ±⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Within this subspace, the Hamil- tonian ˆH + ˆHZee takes the form ˆHeff = E1 + ∆1 2 σ′ z +∑ j ωj ˆb† j ˆbj (5) + ∑ j � ⟨1| ˆVj|1⟩−wz jσ′ z �� ˆbj + ˆb† j � − ∑ j � wx jσ′ x +wy jσ′ y �� ˆbj + ˆb† j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This Hamiltonian describes the interaction between vi- brational modes and an effective spin one-half rep- resented by the Pauli matrices σ′ = (σ′ x,σ′ y,σ′ z), 3 QTM electronic vibronic b) a) Energy [Dy(Cpttt)2]+ c) vibrational DOS DCM z d) polarons FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Quantum tunnelling in single-molecule magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (a) Molecular structure of a Dy(III) single-molecule magnet surrounded by a dichloromethane bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (b) Equilibrium energy level diagram of the lowest-energy angular momentum multiplet with J = 15/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The second- lowest doublet at E2 is 524 cm−1 higher than the ground doublet at E1, while the highest doublet is 1523 cm−1 above E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Dipolar and hyperfine magnetic fields (Bint) can lift the degeneracy of the doublets and cause quantum tunnelling, which results in avoided crossings when sweeping an external magnetic field Bext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Molecular vibrations can influence the magnitude of the avoided crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (c) Spin-phonon coupling for the solvated complex shown above, as a function of the vibrational frequency (vibrations with ωj > 1500 cm−1 not shown), calculated as the Frobenius norm of the operator ˆVj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The grey dashed line represents the vibrational density of states, obtained by assigning to each molecular vibration a (anti-symmetrised) Lorentzian lineshape with full width at half-maximum 10 cm−1 (corresponding to a typical timescale of ∼ 1 ps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (d) Idea behind the polaron transformation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Each spin state |1′±⟩ is accompanied by a vibrational distortion (greatly exaggerated for visualisation), thus forming a magnetic polaron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Vibrational states |ν⟩ are now described in terms of harmonic displacements around the deformed structure, which depends on the state of the spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Polarons provide an accurate physical picture when the spin-phonon coupling is strong and mostly modulates the energy of different spin states but not the coupling between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' where σ′ z = |1′ +⟩⟨1′ +| − |1′ −⟩⟨1′ −|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The vector w j = (ℜ⟨1−| ˆWj|1+⟩,ℑ⟨1−| ˆWj|1+⟩,⟨1+| ˆWj|1+⟩) is defined in terms of the operator ˆWj = ∑m̸=1,¯1 ˆVj|m⟩⟨m| ˆHZee/(Em −E1)+ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=', describing the effect of the Zeeman interaction on the spin-phonon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Due to the strong magnetic axiality of the complex considered here, the longitudinal component of the spin-phonon coupling wz j dominates over the transverse part wx j, wy j (see Section S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In this case, we can get a better physical picture of the system by transforming the Hamiltonian (5) to the polaron frame defined by the unitary operator ˆS = exp � ∑ s=± |1′ s⟩⟨1′ s| ∑ j ξ s j � ˆb† j − ˆbj �� , (6) which mixes electronic and vibrational degrees of freedom by displacing the mode operators by ξ ± j = (⟨1| ˆVj|1⟩ ∓ wz j)/ωj depending on the state of the effective spin one-half [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The idea behind this transformation is to allow nuclei to re- lax around a new equilibrium geometry, which may be differ- ent for every spin state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This lowers the energy of the system and provides a good description of the vibronic eigenstates when the spin-phonon coupling is approximately diagonal in the spin basis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In the polaron frame, the longitu- dinal spin-phonon coupling is fully absorbed into the purely electronic part of the Hamiltonian, while the transverse com- ponents can be approximated by their thermal average over vibrations, neglecting their vanishingly small quantum fluc- tuations (see Section S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' After transforming back to the original frame, we are left with an effective spin one-half Hamiltonian with no residual spin-phonon coupling Heff ≈ ˆH(pol) eff +∑j ωj ˆb† j ˆbj, where ˆH(pol) eff = E1 + ∆1 2 σ′′ z +2∑ j ⟨1| ˆVj|1⟩ ωj w j ·σ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (7) The set of Pauli matrices σ′′ = ˆS†(σ′ ⊗ 1lvib) ˆS describe the two-level system formed by the magnetic polarons of the form ˆS†|1′ ±⟩|{νj}⟩vib, where {νj} is a set of occupation num- bers for the vibrational modes of the solvent-SMM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' These magnetic polarons can be thought as magnetic elec- tronic states strongly coupled to a distortion of the molecular geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' They inherit the magnetic properties of the cor- responding electronic states, and can be seen as the molecu- 4 lar equivalent of the magnetic polarons observed in a range of magnetic materials [28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Polaron representations of vibronic systems have been employed in a wide variety of settings, ranging from spin-boson models [27, 31] to photo- synthetic complexes [32–34], to quantum dots [35–37], pro- viding a convenient basis to describe the dynamics of quan- tum systems strongly coupled to a vibrational environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' These methods are particularly well suited for condensed matter systems where the electron-phonon coupling is strong but causes very slow transitions between different electronic states, allowing exact treatment of the pure-dephasing part of the electron-phonon coupling and renormalising the elec- tronic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' For this reason, the polaron transformation is especially effective for describing our system (as detailed in Section S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The most striking advantage of this approach is that the average effect of the spin-phonon coupling is included non-perturbatively into the electronic part of the Hamiltonian, leaving behind a vanishingly small residual spin-phonon cou- pling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' As a last step, we bring the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (7) into a more familiar form by expressing it in terms of an effective g- matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' We recall that the quantities ∆1 and w j depend linearly on the magnetic field B via the Zeeman Hamiltonian ˆHZee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' An additional dependence on the orientation of the magnetic field comes from the mixing angles θ1 and φ1 introduced in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (2) and (3), appearing in the states |1±⟩ used in the definition of w j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This further dependence is removed by transforming the Pauli operators back to the basis {|1⟩,|¯1⟩} via a three- dimensional rotation σ = Rθ1,φ1 ·σ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Finally, we obtain ˆH(pol) eff = E1 + µBB· � gel +∑ j gvib j � σ 2 , (8) for appropriately defined electronic and single-mode vibronic g-matrices gel and gvib j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' These are directly related to the elec- tronic splitting term ∆1 and to the vibronic corrections de- scribed by w j in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (7), respectively (see Section S2 for a thorough derivation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The main advantage of representing the ground Kramers doublet with an effective spin one-half Hamiltonian is that it provides a conceptually simple founda- tion for studying low-temperature magnetic behaviour of the complex, confining all microscopic details, including vibronic effects, to an effective g-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' RESULTS We begin by considering the influence of vibrations on the Zeeman splitting of the lowest doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The Zeeman splitting in absence of vibrations is simply given by ∆1 = µB|B · gel|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In the presence of vibrations, the electronic g-matrix gel is modified by adding the vibronic correction ∑j gvib j , resulting in the Zeeman splitting ∆vib 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 2 we show the Zee- man splittings as a function of the orientation of the mag- netic field B, parametrised in terms of the polar angles (θ,φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Depending on the field orientation, vibrations can lead to ei- ther an increase or decrease of the Zeeman splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' These changes seem rather small when compared to the largest elec- tronic splitting, obtained when B is oriented along the z-axis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 1a), as expected for a complex with easy-axis anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' However, they become quite significant for field orientations close to the xy-plane, where the purely electronic splitting ∆1 becomes vanishingly small and ∆vib 1 can be dominated by the vibronic contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This is clearly shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 2b and 2c, where we decompose the total field B = Bint + Bext in a fixed internal component Bint originating from dipolar and hy- perfine interactions, responsible for opening a tunnelling gap, and an external part Bext which we sweep along a fixed direc- tion across zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' We note that this effect is specific to states with easy-axis magnetic anisotropy, however this is the defin- ing feature of SMMs, such that our results should be generally applicable to all Kramers SMMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' A more in-depth discussion on the origin and magnitude of the internal field can be found in Section S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' When these fields lie in the plane perpendicu- lar to the purely electronic easy axis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' the hard plane, the vibronic splitting can be four orders of magnitude larger than the electronic one (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The situation is reversed when the fields lie in the hard plane of the vibronic g-matrix (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' So far we have seen that spin-phonon coupling can either enhance or reduce the tunnelling gap in the presence of a mag- netic field depending on its orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' For this reason, it is not immediately clear whether its effects survive ensemble av- eraging in a collection of randomly oriented SMMs, such as the frozen solutions considered in magnetometry experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In order to check this, let us consider an ideal field-dependent magnetisation measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' When sweeping a magnetic field Bext at a constant rate from positive to negative values along a given direction, QTM is typically observed as a sharp step in the magnetisation of the sample when crossing the region around Bext = 0 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This sudden change of the magnetisa- tion is due to a non-adiabatic spin-flip transition between the two lowest energy spin states, that occurs when traversing an avoided crossing (see diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 1b, right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The spin-flip probability is given by the celebrated Landau-Zener expres- sion [38–43], which in our case takes the form PLZ = 1−exp � −π|∆⊥|2 2|v| � , (9) where we have defined v = µBdBext/dt ·g, and ∆⊥ is the com- ponent of ∆ = µBBint · g perpendicular to v, while g denotes the total electronic-vibrational g-matrix appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (8) (see Section S2 for a derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' We account for orientational disorder by averaging Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (9) over all possible orientations of internal and external magnetic fields, yielding the ensemble average ⟨PLZ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The effect of spin-phonon coupling on the spin-flip dynam- ics of an ensemble of SMMs can be clearly seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In- cluding the vibronic correction to the ground doublet g-matrix leads to enhanced spin-flip probabilities across a wide range of internal field strengths and field sweep rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This is in line with previous results suggesting that molecular flexibility cor- relates with QTM [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' To further corroborate our model, we test its predictions against experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' We extracted the average spin-flip probability from published hysteresis data 5 a) b) c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Zeeman splitting of the ground Kramers doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (a) Electronic ground doublet splitting (∆1, top) and vibronic correction (∆vib 1 − ∆1, bottom) as a function of the orientation of the magnetic field B = (sinθ cosφ,sinθ sinφ,cosθ), with magnitude fixed to 1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The dashed (solid) line corresponds to the electronic (vibronic) hard plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (b–c) Electronic (dashed) and vibronic (solid) Zeeman split- ting of the ground doublet as a function of the external field magni- tude Bext in the presence of a transverse internal field Bint = 1 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' External and internal fields are perpendicular to each other and were both chosen to lie in the hard plane of either the electronic (b) or vibronic (c) g-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The orientation of the external (internal) field is shown for both cases as circles (crosses) in the inset in (a), with colors matching the ones in (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' of [Dy(Cpttt)2][B(C6F5)4] in DCM with sweep rates ranging between 10–20 Oe/s [6], yielding a value of ⟨PLZ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='27, indicated by the pink line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' We then checked what strength of the internal field Bint is required to reproduce such spin-flip probability based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 3, we observe that the values of Bint required by the vibronic model to re- produce the observed spin-flip probability are perfectly con- sistent with the dipolar fields naturally occurring in the sam- ple, whereas the purely electronic model necessitates internal fields that are one order of magnitude larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' These results clearly demonstrate the significance of spin-phonon coupling for QTM in a disordered ensemble of SMMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' A detailed dis- cussion on the estimation of spin-flip probabilities and internal fields from magnetisation measurements is presented in Sec- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Landau-Zener spin-flip probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Ensemble-averaged spin-flip probability as a function of the internal field strength Bint causing tunnelling within the ground Kramers doublet, shown for different sweep rates dBext/dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Results for the vibronic model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (8) are shown as orange solid lines, together with the spin-flip prob- abilities predicted by a purely electronic model obtained by setting the spin-phonon coupling to zero, shown as blue dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The horizontal pink line indicates ⟨PLZ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='27, extracted from hysteresis data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' [6] (Section S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The green shaded area indicates the range of values for typical dipolar fields in the corresponding sample (Section S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' tions S4 and S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' DISCUSSION As shown above, the combined effect of all vibrations in a randomly oriented ensemble of solvated SMMs is to en- hance QTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' However, not all vibrations contribute to the same extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Based on the polaron model introduced above, vibrations with large spin-phonon coupling and low frequency have a larger impact on the magnetic properties of the ground Kramers doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This can be seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (7), where the vibronic correction to the effective ground Kramers Hamil- tonian is weighted by the factor ⟨1| ˆVj|1⟩/ω j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Another prop- erty of vibrations that can influence QTM is their symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In monometallic SMMs, QTM has generally been correlated with a reduction of the axial symmetry of the complex, either by the presence of flexible ligands or by transverse magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Since we are interested in symmetry only as long as it influences magnetism, it is useful to introduce a measure of axiality on the g-matrix, such as A(g) = ��g− 1 3Tr g �� � 2 3Tr g , (10) where ∥·∥ denotes the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This measure yields 1 for a perfect easy-axis complex, 1/2 for an easy plane system, and 0 for the perfectly isotropic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The axiality of an indi- vidual vibrational mode can be quantified as Aj = A(gel+gvib j ) by building a single-mode vibronic g-matrix, analogous to △1 (cm-1 元 10 8 6 4 2 2 0 0 0 一 元 2 2 ΦAvib - △1 (cm-1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='1 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='2 0 元 0 一元 2 2(cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='1 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='2 0 一元 2 2(cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='1 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='2 0 一元 2 26 a) b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Single-mode contributions to tunnelling of the magnetisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (a) Single-mode vibronic Landau-Zener probabilities plotted for each vibrational mode, shown as a function of the mode axiality relative to the axiality of the purely electronic g-matrix (∆Aj = Aj −Ael).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The magnitude of the internal field is fixed to Bint = 1 mT and the external field sweep rate is 10 Oe/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The color coding represents the spin-phonon coupling strength ∥ ˆVj∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Grey dashed lines corresponds to the purely electronic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (b) Visual representation of the displacements induced by the vibrational modes indicated by arrows in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Solvent motion is only shown for modes 2 and 6, which have negligible amplitude on the SMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' the multi-mode one introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' We might be tempted to intuitively conclude that vibrational motion al- ways decreases the axiality with respect to its electronic value Ael = A(gel), given that the collective effect of vibrations is to enhance QTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' However, when considered individually, some vibrations can have the opposite effect, of effectively increas- ing the magnetic axiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In order to see how axiality correlates to QTM, we calcu- late the single-mode Landau-Zener probabilities ⟨Pj⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' These are obtained by replacing the multi-mode vibronic g-matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (8) with the single-mode one gel +gvib j , and following the same procedure detailed in Section S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The single-mode con- tribution to the spin-flip probability unambiguously correlates with mode axiality, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Vibrational modes that lead to a larger QTM probability are likely to reduce the magnetic axiality of the complex (top-left sector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Vice versa, those vibrational modes that enhance axiality also suppress QTM (bottom-right sector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' As a first step towards uncovering the microscopic basis of this unexpected behaviour, we single out the three vibra- tional modes that have the largest impact on axiality and spin- flip probability in both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' These vibrational modes, labelled 1–6, represent a range of qualitatively distinct vibra- tions, as can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Modes 4 and 5 are among the ones exhibiting the strongest spin-phonon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Both of them are mainly localised on one of the Cpttt ligands and involve atomic displacements along the easy axis and, to a lesser extent, rotations of the methyl groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Modes 1 and 3 are among the ones with largest amplitude on the Dy ion, which in both cases mainly moves in the hard plane, disrupt- ing axial symmetry and enhancing tunnelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Lastly, modes 2 and 6 predominantly correspond to solvent vibrations, and are thus very low energy and so give a large contribution via the small denominator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This analysis shows that the effect of vibrational modes on QTM is more nuanced than what both intuition and previous work would suggest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Despite leading to an overall increase of the spin-flip probability on average, coupling the spin to specific vibrations can increase the magnetic axiality of the complex and suppress QTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This opens a new avenue for the improvement of magnetic relaxation times in SMMs, shifting the role of vibrations from purely antagonistic to potentially beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' According to the results shown above, the ideal candidates to observe vibronic suppression of QTM are systems exhibit- ing strongly axial, low frequency vibrations, strongly coupled to the electronic effective spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Strong spin-phonon coupling and low frequency ensure a significant change in magnetic properties according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (7), but may not be enough to hin- der tunnelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In order to be beneficial, vibrations also need to enhance the axiality of the ground doublet g-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The relation between magnetic axiality and vibrational symmetry remains yet to be explored, and might lead to new insights regarding rational design of ideal ligands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 17 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' CONCLUSIONS In conclusion, we have presented a detailed description of the effect of molecular and solvent vibrations on the quan- tum tunnelling between low-energy spin states in a single-ion Dy(III) SMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Our theoretical results, based on an ab initio approach, are complemented by a polaron treatment of the rel- evant vibronic degrees of freedom, which does not suffer from any weak spin-phonon coupling assumption and is therefore well-suited to other strong coupling scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' We have been able to derive a non-perturbative vibronic correction to the ef- fective g-matrix of the lowest-energy Kramers doublet, which we have used as a basis to determine the tunnelling dynamics in a magnetic field sweep experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This has allowed us to formulate the key observation that, vibrations collectively en- hance QTM, but some particular vibrational modes unexpect- edly suppress QTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This behaviour correlates to the axiality of each mode, which can be used as a proxy for determining whether a specific vibration enhances or hinders tunnelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The observation that individual vibrational modes can sup- press QTM challenges the paradigm that dismisses vibrations as detrimental, a mere obstacle to achieving long-lasting in- formation storage on SMMs, and forces us instead to recon- sider them under a new light, as tools that can be actively en- gineered to our advantage to keep tunnelling at bay and ex- tend relaxation timescales in molecular magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This idea suggests parallelisms with other seemingly unrelated chem- ical systems where electron-phonon coupling plays an im- portant role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' For example, the study of electronic energy transfer across photosynthetic complexes was radically trans- formed by the simple observation that vibrations could play an active role, maintaining quantum coherence in noisy room- temperature environments, rather than just passively causing decoherence between electronic states [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Identifying these beneficial vibrations and amplifying their effect via chemical design of new SMMs remains an open question, whose solu- tion we believe could greatly benefit from the results and the methods introduced in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This work was made possible thanks to the ERC grant 2019-STG-851504 and Royal Society fellowship URF191320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The authors also acknowledge support from the Computational Shared Facility at the University of Manch- ester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' DATA AVAILABILITY The data that support the findings of this study are available at http://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='48420/21892887.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Leuenberger and D.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Ogilvie, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Olaya-Castro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Ratner, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Spano, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Whaley, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Zhu, Using coherence to enhance function in chemical and biophysical systems, Nature 543, 647 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 1 Supplementary Information: Vibronic Effects on the Quantum Tunnelling of Magnetisation in Single-Molecule Magnets Andrea Mattioni,1,∗ Jakob K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Staab,1 William J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Blackmore,1 Daniel Reta,1,2 Jake Iles-Smith,3,4 Ahsan Nazir,3 and Nicholas F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Chilton1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='† 1Department of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' School of Natural Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The University of Manchester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Oxford Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Manchester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' M13 9PL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' UK 2 Faculty of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' UPV/EHU & Donostia International Physics Center DIPC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Ikerbasque,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Basque Foundation for Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Bilbao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Spain 3Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' School of Natural Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The University of Manchester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Oxford Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Manchester M13 9PL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' UK 4Department of Electrical and Electronic Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' School of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The University of Manchester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Sackville Street Building,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Manchester M1 3BB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' UK ∗ andrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='mattioni@manchester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='uk † nicholas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='chilton@manchester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='uk 2 S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' AB INITIO CALCULATIONS The ab initio model of the DCM-solvated [Dy(Cpttt)2]+ molecule is constructed using a multi-layer approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' During ge- ometry optimisation and frequency calculation the system is partitioned into two layers following the ONIOM scheme [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The high-level layer, consisting of the SMM itself and the first solvation shell of 26 DCM molecules, is described by Density Functional Theory (DFT) while the outer bulk of the DCM ball constitutes the low-level layer modelled by the semi-empirical PM6 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' All DFT calculations are carried out using the pure PBE exchange-correlation functional [2] with Grimme’s D3 dispersion correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Dysprosium is replaced by its diamagnetic analogue yttrium for which the Stuttgart RSC 1997 ECP basis is employed [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Cp ring carbons directly coordinated to the central ion are equipped with Dunning’s correlation consistent triple-zeta polarised cc-pVTZ basis set and all remaining atoms with its double-zeta analogue cc-pVDZ [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Subsequently, the electronic spin states and spin-phonon coupling parameters are calculated at the CASSCF-SO level explicitly accounting for the strong static correlation present in the f-shell of Dy(III) ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' At this level, environmental effects are treated using an electrostatic point charge representation of all DCM atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' All DFT/PM6 calculations are carried out with GAUSSIAN version 9 revision D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='01 [5] and the CASSCF calculations are carried out with OPENMOLCAS version 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='06 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The starting [Dy(Cpttt)2]+ solvated system was obtained using the solvate program belonging to the AmberTool suite of packages, with box as method and CHCL3BOX as solvent model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Chloroform molecules were subsequently converted to DCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' From this large system, only molecules falling within 9 Å from the central metal atom are considered from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The initial disordered system of 160 DCM molecules packed around the [Dy(Cpttt)2]+ crystal structure [7] is pre-optimised in steps, starting by only optimising the high-level layer atoms and freezing the rest of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The low-layer atoms are pre-optimised along the same lines starting with DCM molecules closest to the SMM and working in shells towards the outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Subsequently, the whole system is geometry optimised until RMS (maximum) values in force and displacement corresponding to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='00045 au (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='0003 au) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='0018 au (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='0012 au) are reached, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' After adjusting the isotopic mass of yttrium to that of dysprosium mDy = 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='5u, vibrational normal modes and frequencies of the entire molecular aggregate are computed within the harmonic approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Electrostatic atomic point charge representations of the environment DCM molecules are evaluated for each isolated solvent molecule independently at the DFT level of theory employing the CHarges from ELectrostatic Potentials using a Grid-based (ChelpG) method [8], which serve as a classical model of environmental effects in the subsequent CASSCF calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The evaluation of equilibrium electronic states and spin-phonon coupling parameters is carried out at the CASSCF level including scalar relativistic effects using the second-order Douglas-Kroll Hamiltonian and spin-orbit coupling through the atomic mean field approximation implemented in the restricted active space state interaction approach [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The dysprosium atom is equipped with the ANO-RCC-VTZP, the Cp ring carbons with the ANO-RCC-VDZP and the remaining atoms with the ANO- RCC-VDZ basis set [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The resolution of the identity approximation with an on-the-fly acCD auxiliary basis is employed to handle the two-electron integrals [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The active space of 9 electrons in 7 orbitals, spanned by 4f atomic orbitals, is employed in a state-average CASSCF calculation including the 18 lowest lying sextet roots which span the 6H and 6F atomic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' We use our own implementation of spin Hamiltonian parameter projection to obtain the crystal field parameters Bq k entering the Hamiltonian ˆHCF = ∑ k=2,4,6 k ∑ q=−k θkBq kOq k(ˆJ), (S1) describing the 6H15/2 ground state multiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Operator equivalent factors and Stevens operators are denoted by θk and Oq k(ˆJ), where ˆJ = ( ˆJx, ˆJy, ˆJz) are the angular momentum components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Spin-phonon coupling arises from changes to the Hamiltonian (S1) due to slight distortions of the molecular geometry, parametrised as Bq k({Xj}) = Bq k + M ∑ j=1 ∂Bq k ∂Xj Xj +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=', (S2) where Xj denotes the dimensionless j-th normal coordinate of the complex under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The derivatives ∂Bq k/∂Xj are calculated using the Linear Vibronic Coupling (LVC) approach described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' [13] based on the state-average CASSCF density-fitting gradients and non-adiabatic coupling involving all 18 sextet roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The final step leading to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (1) in the main text is to quantise the normal modes and express them in terms of bosonic annihilation and creation operators satisfying [ˆbi, ˆb† j] = δij as ˆXj = ˆbj + ˆb† j √ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S3) 3 Defining the spin-phonon coupling operators ˆVj = 1 √ 2 ∑ k,q θk ∂Bq k ∂Xj Oq k(ˆJ), (S4) we can finally write down the crystal field Hamiltonian including linear spin-phonon coupling as ˆH = ˆHCF +∑ j ˆVj ⊗(ˆbj + ˆb† j)+∑ j ωj ˆb† j ˆb j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S5) 4 S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' DERIVATION OF THE EFFECTIVE VIBRONIC DOUBLET HAMILTONIAN A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Electronic perturbation Theory The starting point for our analysis of vibronic effects on QTM is the vibronic Hamiltonian ˆH = ∑ m>0 Em(|m⟩⟨m|+| ¯m⟩⟨ ¯m|)+ ˆHZee +∑ j ˆVj ⊗(ˆbj + ˆb† j)+∑ j ωj ˆb† j ˆb j, (S6) where ˆHZee = µBgJB · ˆJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' is the Zeeman interaction with a magnetic field B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The doubly degenerate eigenstates of the crystal field Hamiltonian HCF = ∑m>0 Em(|m⟩⟨m|+| ¯m⟩⟨ ¯m|) are related by time-reversal symmetry, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' ˆΘ|m⟩ ∝ | ¯m⟩ with ˆΘ2|m⟩ = −|m⟩, where ˆΘ is the time-reversal operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In the case of [Dy(Cpttt)2]+, the total electronic angular momentum is J = 15/2, leading to 2J + 1 = 16 electronic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' We label these states in ascending energy with integers m = ±1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=',±8, using the compact notation |−m⟩ = | ¯m⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' We momentarily neglect the spin-phonon coupling and focus on the purely electronic Hamiltonian Hel = HCF +HZee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Within each degenerate subspace, the Zeeman term selects a specific electronic basis and lifts its degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This can be seen by projecting the electronic Hamitonian onto the m-th subspace and diagonalising the 2×2 matrix H(m) el = Em + µBgJ � ⟨m|B· ˆJ|m⟩ ⟨m|B· ˆJ| ¯m⟩ ⟨ ¯m|B· ˆJ|m⟩ ⟨ ¯m|B· ˆJ| ¯m⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S7) For each individual cartesian component of the angular momentum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' we decompose the corresponding 2 × 2 matrix in terms of Pauli spin operators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' which allows to rewrite the Hamiltonian of the m-th doublet as H(m) el = Em + µBB·g(m) el ·σ(m)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' where g(m) el = 2gJ � � ℜ⟨ ¯m| ˆJx|m⟩ ℑ⟨ ¯m| ˆJx|m⟩ ⟨m| ˆJx|m⟩ ℜ⟨ ¯m| ˆJy|m⟩ ℑ⟨ ¯m| ˆJy|m⟩ ⟨m| ˆJy|m⟩ ℜ⟨ ¯m| ˆJz|m⟩ ℑ⟨ ¯m| ˆJz|m⟩ ⟨m| ˆJz|m⟩ � � (S8) is the g-matrix for an effective spin 1/2 and σ(m) = (σ(m) x ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='σ(m) y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='σ(m) z ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' with σ(m) z = |m⟩⟨m|−| ¯m⟩⟨ ¯m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' We note that in general the g-matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S8) is not hermitean, but can be brought to such form by transforming the spin operators σ(m) to an appropriate basis [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' An easier prescription to find the hermitean form af any g-matrix g is to redefine it as � gg†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' To lowest order in the magnetic field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' the Zeeman interaction lifts the two-fold degeneracy by selecting the basis |m+⟩ = cos θm 2 |m⟩+eiφm sin θm 2 | ¯m⟩ (S9) |m−⟩ = −sin θm 2 |m⟩+eiφm cos θm 2 | ¯m⟩ (S10) and shifting the energies according to Em,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='± = Em ±∆m/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' where the gap ∆m = ⟨m+| ˆHZee|m+⟩−⟨m−| ˆHZee|m−⟩ (S11) = 2µBgJ � ⟨m|B· ˆJ|m⟩2 +|⟨m|B· ˆJ| ¯m⟩|2 can be obtained as the norm of the vector jm = µBB·g(m) el and the phase and mixing angles are defined as eiφm = ⟨ ¯m|B· ˆJ|m⟩ |⟨ ¯m|B· ˆJ|m⟩|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' tanθm = |⟨ ¯m|B· ˆJ|m⟩| ⟨m|B· ˆJ|m⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S12) or equivalently as the azimuthal and polar angles determining the direction of jm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Besides selecting a preferred basis and lifting the degeneracy of each doublet, the Zeeman interaction also causes mixing between different doublets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In particular, the lowest doublet will change according to |1′ ±⟩ = |1±⟩+ ∑ m̸=1,¯1 |m⟩⟨m| ˆHZee|1±⟩ E1 −Em +O(B2) ≈ � 1− ˆQ1 ˆHZee � |1±⟩, (S13) with ˆQ1 = ∑ m̸=1,¯1 |m⟩ 1 Em −E1 ⟨m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S14) 5 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Spin-boson Hamiltonian for the ground doublet Now that we have an approximate expression for the relevant electronic states, we reintroduce the spin-phonon coupling into the picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' First, we project the vibronic Hamiltonian (S6) onto the subspace spanned by |1′ ±⟩, yielding ˆHeff = E1 + � ∆1 2 0 0 − ∆1 2 � +∑ j � ⟨1′ +| ˆVj|1′ +⟩ ⟨1′ +| ˆVj|1′ −⟩ ⟨1′ −| ˆVj|1′ +⟩ ⟨1′ −| ˆVj|1′ −⟩ � ⊗(ˆbj + ˆb† j)+∑ j ωj ˆb† j ˆb j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S15) On this basis, the purely electronic part ˆHCF + ˆHZee is diagonal with eigenvalues E1 ± ∆1/2, and the purely vibrational part is trivially unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' the spin-phonon couplings can be calculated to lowest order in the magnetic field strength B as ⟨1′ ±| ˆVj|1′ ±⟩ = ⟨1±| � 1− ˆHZee ˆQ1 � ˆVj � 1− ˆQ1 ˆHZee � |1±⟩+O(B2) (S16) = ⟨1±| ˆVj|1±⟩−⟨1±| � ˆVj ˆQ1 ˆHZee + ˆHZee ˆQ1 ˆVj � |1±⟩+O(B2) = ⟨1| ˆVj|1⟩−⟨1±| ˆWj|1±⟩+O(B2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' ⟨1′ ∓| ˆVj|1′ ±⟩ = ⟨1∓| � 1− ˆHZee ˆQ1 � ˆVj � 1− ˆQ1 ˆHZee � |1±⟩+O(B2) (S17) = ⟨1∓| ˆVj|1±⟩−⟨1∓| � ˆVj ˆQ1 ˆHZee + ˆHZee ˆQ1 ˆVj � |1±⟩+O(B2) = −⟨1∓| ˆWj|1±⟩+O(B2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' where we have defined ˆWj = ˆVj ˆQ1 ˆHZee + ˆHZee ˆQ1 ˆVj (S18) and used the time-reversal invariance of the spin-phonon coupling operators to obtain ⟨1±| ˆVj|1±⟩ = ⟨1| ˆVj|1⟩ and ⟨1∓| ˆVj|1±⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The two states |1±⟩ form a conjugate pair under time reversal, meaning that ˆΘ|1±⟩ = ∓eiα|1∓⟩ for some α ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Using the fact that for any two states ψ, ϕ, and for any operator ˆO we have ⟨ψ| ˆO|ϕ⟩ = ⟨ ˆΘϕ| ˆΘ ˆO† ˆΘ−1| ˆΘψ⟩, and recalling that the angular momentum operator is odd under time reversal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' ˆΘˆJ ˆΘ−1 = −ˆJ, we can show that ⟨1−| ˆWj|1−⟩ = ⟨ ˆΘ1−| ˆΘ ˆWj ˆΘ−1| ˆΘ1−⟩ = −⟨1+| ˆWj|1+⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Keeping in mind these observations, and defining the vector w j = � � wx j wy j wz j � � = � � ℜ ⟨1−| ˆWj|1+⟩ ℑ ⟨1−| ˆWj|1+⟩ ⟨1+| ˆWj|1+⟩ � �, (S19) we can rewrite the spin-phonon coupling operators in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S15) as � ⟨1′ +| ˆVj|1′ +⟩ ⟨1′ +| ˆVj|1′ −⟩ ⟨1′ −| ˆVj|1′ +⟩ ⟨1′ −| ˆVj|1′ −⟩ � = ⟨1| ˆVj|1⟩− � ⟨1+| ˆWj|1+⟩ ⟨1−| ˆWj|1+⟩∗ ⟨1−| ˆWj|1+⟩ −⟨1+| ˆWj|1+⟩ � = ⟨1| ˆVj|1⟩−w j ·σ′ (S20) where σ′ is a vector whose entries are the Pauli matrices in the basis |1′ ±⟩, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' σ′ z = |1′ +⟩⟨1′ +| − |1′ −⟩⟨1′ −|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Plugging this back into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S15) and explicitly singling out the diagonal components of ˆHeff in the basis |1′ ±⟩, we obtain ˆHeff = |1′ +⟩⟨1′ +| � E1 + ∆1 2 +∑ j � ⟨1| ˆVj|1⟩−wz j �� ˆbj + ˆb† j � +∑ j ω j ˆb† j ˆbj � (S21) + |1′ −⟩⟨1′ −| � E1 − ∆1 2 +∑ j � ⟨1| ˆVj|1⟩+wz j �� ˆbj + ˆb† j � +∑ j ω j ˆb† j ˆbj � − ∑ j � wx jσ′ x +wy jσ′ y �� ˆbj + ˆb† j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' At this point, we apply a unitary polaron transformation to the Hamiltonian (S21) ˆS = exp � ∑ s=± |1′ s⟩⟨1′ s| ∑ j 1 ωj � ⟨1| ˆVj|1⟩−swz j �� ˆb† j − ˆbj �� (S22) = ∑ s=± |1′ s⟩⟨1′ s| ∏ j ˆD j(ξ s j) 6 where ξ s j = � ⟨1| ˆVj|1⟩−swz j � /ω j and ˆD j(ξ s j) = eξ s j � ˆb† j−ˆb j � (S23) is the bosonic displacement operator acting on mode j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' ˆD j(ξ)ˆbj ˆD† j(ξ) = ˆbj −ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The Hamiltonian thus becomes ˆS ˆHeff ˆS† = ∑ s=± |1′ s⟩⟨1′ s| � E1 +s∆1 2 −∑ j ωj|ξ s j|2 � +∑ j ωj ˆb† j ˆbj −∑ j ˆS � wx jσ′ x +wy jσ′ y �� ˆb j + ˆb† j � ˆS†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S24) The polaron transformation reabsorbes the diagonal component of the spin-phonon coupling (S20) proportional to wz j into the energy shifts ωj|ξ ± j |2, leaving a residual off-diagonal spin-phonon coupling proportional to wx j and wy j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Note that the polaron transformation exactly diagonalises the Hamiltonian (S15) if wx j = wy j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In Section S3, we argue in detail that in our case |wx j|,|wy j| ≪ |wz j| to a very good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Based on this argument, we could decide to neglect the residual spin-phonon coupling in the polaron frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The energies of the states belonging to the lowest doublet are shifted by a vibronic correction E1′± = E1 ± ∆1 2 −∑ j 1 ω j � ⟨1| ˆVj|1⟩∓wz j �2 (S25) = E1 ± ∆1 2 −∑ j 1 ω j � ⟨1| ˆVj|1⟩2 ∓2⟨1| ˆVj|1⟩wz j +O(B2) � , (S26) leading to a redefinition of the energy gap E1′+ −E1′− = ∆1 +4∑ j ⟨1| ˆVj|1⟩ ωj wz j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S27) Although the off-diagonal components of the spin-phonon coupling wx j and wy j are several orders of magnitude smaller than the diagonal one wz j (see Section S3), the sheer number of vibrational modes could still lead to an observable effect on the electronic degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' We can estimate this effect by averaging the residual spin-phonon coupling over a thermal phonon distribution in the polaron frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Making use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S22), the off-diagonal coupling in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S24) can be written as ˆH(pol) sp-ph = −∑ j ˆS � wx jσ′ x +wy jσ′ y �� ˆbj + ˆb† j � ˆS† (S28) = −∑ j |1′ −⟩⟨1−| ˆWj|1+⟩⟨1′ +| ˆD j(ξ − j ) � ˆbj + ˆb† j � ˆD† j(ξ + j )+h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Assuming the vibrations to be in a thermal state at temperature T in the polaron frame ρ(th) ph = ∏ j ρ(th) j = ∏ j e−ωj ˆb† j ˆb j/kBT Tr � e−ωj ˆb† j ˆb j/kBT�, (S29) obtaining the average of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S28) reduces to calculating the dimensionless quantity κj = −Tr � ˆD j(ξ − j ) � ˆbj + ˆb† j � ˆD† j(ξ + j )ρ(th) j � (S30) = � ξ + j +ξ − j � e− 1 2 � ξ + j −ξ − j �2 coth � ωj 2kBT � = 2⟨1| ˆVj|1⟩ ωj e −2 (wz j)2 ω2 j coth � ωj 2kBT � = 2⟨1| ˆVj|1⟩ ωj � 1+O(B2), � which appears as a multiplicative rescaling factor for the off-diagonal couplings ⟨1∓| ˆWj|1±⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Note that, when neglecting second and higher order terms in the magnetic field, κj does not show any dependence on temperature or on the magnetic field orientation via θ1 and φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 7 After thermal averaging, the effective electronic Hamiltonian for the lowest energy doublet becomes ˆHel = Trph � ˆS ˆHeff ˆS†ρ(th) ph � = E1 +δE1 + � 2∑ j ⟨1| ˆVj|1⟩ ωj wx j,2∑ j ⟨1| ˆVj|1⟩ ωj wy j, ∆1 2 +2∑ j ⟨1| ˆVj|1⟩ ωj wz j � � � σ′ x σ′ y σ′ z � � (S31) where the energy of the lowest doublet is shifted by δE1 = −∑ j ⟨1| ˆVj|1⟩2 ωj +∑ j ω j eωj/kBT −1 (S32) due to the spin-phonon coupling and to the thermal phonon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S31) thus represents a refined description of the lowest effective spin-1/2 doublet in the presence of spin-phonon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' We can finally recast the Hamiltonian (S31) in terms of a g-matrix for an effective spin 1/2, similarly to what we did earlier in the case of no spin-phonon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In order to do so, we first recall from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S11) and (S19) that the quantities ∆1 and (wx j,wy j,wz j) appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S31) depend on the magnetic field orientation via the states |1±⟩, and on both orientation and intensity via ˆHZee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' We can get rid of the first dependence by expressing the Zeeman eigenstates |1±⟩ in terms of the original crystal field eigenstates |1⟩, |¯1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' For the spin-phonon coupling vector wj, we obtain w j = � � ℜ⟨1−| ˆWj|1+⟩ ℑ⟨1−| ˆWj|1+⟩ ⟨1+| ˆWj|1+⟩ � � = � � cosθ1 cosφ1 cosθ1 sinφ1 −sinθ1 −sinφ1 cosφ1 0 sinθ1 cosφ1 sinθ1 sinφ1 cosθ1 � � � � ℜ⟨¯1| ˆWj|1⟩ ℑ⟨¯1| ˆWj|1⟩ ⟨1| ˆWj|1⟩ � � = R(θ1,φ1)· ˜w j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S33) where R(θ1,φ1) is a rotation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Similarly, the elctronic contribution ∆1 transforms as (0,0,∆1) = j1 ·R(θ1,φ1)T,= µBB·g(1) el ·R(θ1,φ1)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S34) The Pauli spin operators need to be changed accordingly to ˜σ = R(θ1,φ1)T · σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Lastly, we single out explicitly the magnetic field dependence of ˆWj, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S18), by introducing a three-component operator ˆKj = ( ˆKx j, ˆKy j, ˆKz j), such that ˆWj = µBgJB· � ˆVj ˆQ1ˆJ+ ˆJ ˆQ1 ˆVj � (S35) = µBgJB· ˆK j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Thus, the effective electronic Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S31) can be finally rewritten as ˆHel = E1 +δE1 + µBB· � g(1) el +gvib � ˜σ/2 (S36) where g(1) el is the electronic g-matrix defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S8), and gvib = 4gJ∑ j ⟨1| ˆVj|1⟩ ωj � � ℜ⟨¯1| ˆKx j|1⟩ ℑ⟨¯1| ˆKx j|1⟩ ⟨1| ˆKx j|1⟩ ℜ⟨¯1| ˆKy j|1⟩ ℑ⟨¯1| ˆKy j|1⟩ ⟨1| ˆKy j|1⟩ ℜ⟨¯1| ˆKz j|1⟩ ℑ⟨¯1| ˆKz j|1⟩ ⟨1| ˆKz j|1⟩ � � (S37) is a vibronic correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Note that this correction is non-perturbative in the spin-phonon coupling, despite only containing quadratic terms in ˆVj (recall that ˆK j depends linearly on ˆVj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The only approximations leading to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S36) are a linear perturbative expansion in the magnetic field B and neglecting quantum fluctuations of the off-diagonal spin-phonon coupling in the polaron frame, which is accounted for only via its thermal expectation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This approximation relies on the fact that the off-diagonal couplings are much smaller than the diagonal spin-phonon coupling that is treated exactly by the polaron transformation (see Section S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Landau-Zener probability Let us consider a situation in which the magnetic field comprises a time-independent contribution arising from internal dipolar or hyperfine fields Bint and a time dependent external field Bext(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Let us fix the orientation of the external field and vary its magnitude at a constant rate, such that the field switches direction at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Under these circumstances, the Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S36) becomes ˆHel(t) = E1 +δE1 + µB � Bint + dBext dt t � g· ˜σ 2 , (S38) 8 where g = g(1) el +gvib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Neglecting the constant energy shift and introducing the vectors ∆ = µBBext ·g, (S39) v = µBdBext/dt ·g, (S40) the Hamiltonian then becomes ˆHel(t) = ∆ 2 · ˜σ + vt 2 · ˜σ = ∆⊥ 2 ˜σ + vt +∆∥ 2 ˜σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S41) In the second equality, we have split the vector ∆ = ∆⊥ +∆∥ into a perpendicular and a parallel component to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Choosing an appropriate reference frame, we can write ˆHel(t′) = ∆⊥ 2 ˜σx + vt′ 2 ˜σz, (S42) in terms of the new time variable t′ = t +∆∥/v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Assuming that the spin is initialised in its ground state at t′ → −∞, the probability of observing a spin flip at t′ → +∞ is given by the Landau-Zener formula [15–20] PLZ = 1−exp � −π∆2 ⊥ 2v � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S43) We remark that tunnelling is only made possible by the presence of ∆⊥, which stems from internal fields that have a perpen- dicular component to the externally applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' We also observe that a perfectly axial system would not exhibit tunnelling behaviour, since in that case the direction of B · g would always point along the easy axis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' along the only eigenvector of g with a non-vanishing eigenvalue), and therefore v and ∆ would always be parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Thus, deviations from axiality and the presence of transverse fields are both required for QTM to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 9 S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' DISTRIBUTION OF SPIN-PHONON COUPLING VECTORS The effective polaron Hamiltonian presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (7) and derived in the previous section provides a good description of the ground doublet only if the spin-phonon coupling operators are approximately diagonal in the electronic eigenbasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This is equivalent to requiring that the components of the vectors w j defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S19) satisfy |wx j|,|wy j| ≪ |wz j|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S44) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' S1a shows the distribution of points {w j, j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=',M} (where M is the number of vibrational modes) in 3D space for different orientations of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' As a consequence of the strong magnetic axiality of the complex under consideration, we see that these points are mainly distributed along the z-axis, therefore satisfying the criterion expressed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S44) (note the different scale on the xy-plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' b) a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Distribution of spin-phonon coupling vectors wj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (a) The points w j distribute along a straight line in 3D space (units: cm−1) when the magnetic field is oriented along x, y, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The magnitude is fixed to 1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Note that, owing to the definition of w j, a different magnitude would yield a uniformly rescaled distribution of points, leaving the shape unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (b) Variance of the points w j in the xy-plane in units of the total variance, as a function of magnetic field orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In order to confirm that the points w j maintain a similar distribution regardless of the magnetic field orientation, we calculate their variances along different directions of the 3D space they inhabit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' We define σ2 α = var(wα j ) = 1 M −1 M ∑ j=1 � wα j − µα �2 , (S45) where α = x,y,z and µα = 1 M ∑M j=1 wα j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The dependence of these variances on the field orientation is made evident by recalling that the points w j are related via a rotation R(θ1,φ1) to the set of points ˜w j, which only depend linearly on the field B, as shown in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S33) and (S35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' If the points are mainly distributed along z for any field orientation, we expect the combined variance in the xy-plane to be much smaller than the total variance of the dataset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' σ2 x +σ2 y ≪ σ2 x +σ2 y +σ2 z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S46) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' S1b provides a direct confirmation of this hypothesis, showing that the variance in the xy-plane is at most 6 × 10−4 times smaller than the total variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Therefore, we conclude that the approach followed in Section S2 is fully justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='0002元 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='0004 元 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='0002 0 0 爪 0 一元 元 2 210 S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' EXPERIMENTAL ESTIMATE OF THE SPIN-FLIP PROBABILITY In order to provide experimental support for our vibronic model of QTM, we compare the calculated spin-flip probabilities with values extracted from previously reported measurements of magnetic hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' We use data from field-dependent mag- netisation measurements reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' [7](Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' S35, sample 4), reproduced here in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The sample consisted of a 83 µL volume of a 170 mM solution of [Dy(Cpttt)2][B(C6F5)4] in dichloromethane (DCM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The field-dependent magnetisation was measured at T = 2 K while sweeping an external magnetic field Bext from +7 T to −7 T and back again to +7 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The result- ing hysteresis loop is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' S2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The sweep rate dBext/dt is not constant throughout the hysteresis loop, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' S2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In particular, it takes values between 10 Oe/s and 20 Oe/s across the zero field region where QTM takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' QTM results in a characteristic step around the zero field region in magnetic hysteresis curves (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' S2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The spin-flip probability across the tunnelling transition can be easily related to the height of this step via the expression [21] P↑→↓ = 1 2 � M Msat − M′ Msat � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S47) The value of the magnetisation before (M) and after (M′) the QTM drop is estimated by performing a linear fit of the field- dependent magnetisation close to the zero field region, for both Bext > 0 and Bext < 0, and extrapolating the magnetisation at Bext = 0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' S2a, inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The saturation value of the magnetisation Msat is obtained by measuring the magnetisation at low temperature in a strong external magnetic field (T = 2 K, Bext = 7 T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Following this method, we obtain a spin-flip probability P↑→↓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='27, which is shown as a purple horizontal line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 4 in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' b) a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Magnetic hysteresis of [Dy(Cpttt)2]+ from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (a) Field-dependent magnetisation was measured on a 170 mM frozen solution of [Dy(Cpttt)2]+ (counter ion [B(C6F5)4]−) in DCM at T = 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Data presented in [7] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' S35, sample 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The loop is traversed in the direction indicated by the blue arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The sudden drop of the magnetisation from M to M′ around Bext = 0 is a characteristic signature of QTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The slow magnetisation decay around the QTM step can be ascribed to other magnetic relaxation mechanisms (Raman).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (b) Time dependence of the magnetic field Bext (top) and instantaneous sweep rate (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Note that the sweep rate is not constant around the avoided crossing at Bext = 0, but assumes values in the range 10–20 Oe/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 11 S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' ESTIMATE OF THE INTERNAL FIELDS IN A FROZEN SOLUTION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Dipolar fields In this section we provide an estimate of the internal fields Bint in a disordered ensemble of SMMs, based on field-dependent magnetisation data introduced in Section S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' When a SMM with strongly axial magnetic anisotropy is placed in a strong external magnetic field Bext, it gains a non-zero magnetic dipole moment along its easy axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Once the external field is removed, the SMM partially retains its magnetisation µ = µ ˆµ, which produces a microscopic dipolar field Bdip(r) = µ0µ 4πr3 [3ˆr( ˆµ· ˆr)− ˆµ] (S48) at a point r = rˆr in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This field can then cause a tunnelling gap to open in neighboring SMMs, depending on their relative distance and orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In order to estimate the strength of typical dipolar fields, we need to determine the average distance between SMMs in the sample, and the magnetic dipole moment associated with a single SMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Since we know both volume V and concentration of Dy centres in the sample (see previous section), we can easily obtain the number of SMMs in solution N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The average distance between SMMs can then be obtained simply by taking the cubic root of the volume per particle, as r = �V N �1/3 ≈ 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='4 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S49) The magnetic moment can be obtained from the hysteresis curve shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' S2a, by reading the value of the magnetisation M right before the QTM step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This amounts to an average magnetic moment per molecule ⟨µ∥⟩ = M N ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='07µB (S50) along the direction of the external field Bext, where ⟨·⟩ denotes the average over the ensemble of SMMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Since the orientation of SMMs in a frozen solution is random, the component of the magnetisation µ perpendicular to the applied field averages to zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' ⟨µ⊥⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' However, it still contributes to the formation of the microscopic dipolar field (S48), which depends on µ = µ∥ +µ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Since the sample consists of many randomly oriented SMMs, the average magnetisation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S50) can also be expressed in terms of µ = |µ| via the orientational average ⟨µ∥⟩ = � π/2 0 dθ sinθ µ∥(θ) = µ 2 , (S51) where µ∥(θ) = µ cosθ is the component of the magnetisation of a SMM along the direction of the external field Bext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Thus, the magnetic moment responsible for the microscopic dipolar field is twice as big as the measured value (S50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Based on these estimates, the magnitude of dipolar fields experienced by a Dy atom in the sample is Bdip = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='77 mT× � |3( ˆµ· ˆr)2 −1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S52) The square root averages to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='38 for randomly oriented µ and r and can take values between 1 and 2, represented by the green shaded area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 4 in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Hyperfine coupling Another possible source of microscopic magnetic fields are nuclear spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Among the different isotopes of dysprosium, only 161Dy and 163Dy have non-zero nuclear spin (I = 5/2), making up for approximately 44 % of naturally occurring dysprosium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The nucear spin degrees of freedom are described by the Hamiltonian ˆHnuc = ˆHQ + ˆHHF = ˆI·P· ˆI+ ˆI·A· ˆJ, (S53) where the first term is the quadrupole Hamiltonian ˆHQ = ˆI·P· ˆI, accounting for the zero-field splitting of the nuclear spin states, and the second term ˆHHF = ˆI·A· ˆJ accounts for the hyperfine coupling between nuclear spin ˆI and electronic angular momentum 12 ˆJ operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In analogy with the electronic Zeeman Hamiltonian ˆHZee = µBgJB· ˆJ, we define the effective nuclear magnetic field operator µBgJ ˆBnuc = AT · ˆI, (S54) so that the hyperfine coupling Hamiltonian takes the form of a Zeeman interaction ˆHHF = µBgJ ˆB† nuc · ˆJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' If we consider the nuclear spin to be in a thermal state at temperature T with respect to the quadrupole Hamiltonian ˆHQ, the resulting expectation value of the nuclear magnetic field vanishes, since the nuclear spin is completely unpolarised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' However, the external field Bext will tend to polarise the nuclear spin via the nuclear Zeeman Hamiltonian ˆHnuc, Zee = µNg Bext · ˆI, (S55) where µN is the nuclear magneton and g is the nuclear g-factor of a Dy nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In this case, the nuclear spin is described by the thermal state ρ(th) nuc = e−( ˆHQ+ ˆHnuc, Zee)/kBT Tr � e−( ˆHQ+ ˆHnuc, Zee)/kBT� (S56) and the effective nuclear magnetic field can be calculated as Bnuc = Tr � ˆBnucρ(th) nuc � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S57) To the best of our knowledge, quadrupole and hyperfine coupling tensors for Dy in [Dy(Cpttt)2]+ have not been reported in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' However, ab initio calculations of hyperfine coupling tensors have been performed on DyPc2 [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Although the dysprosium atom in DyPc2 and [Dy(Cpttt)2]+ interacts with different ligands, the crystal field is qualitatively similar for these two complexes, therefore we expect the nuclear spin Hamiltonian to be sufficiently close to the one for [Dy(Cpttt)2]+, at least for the purpose of obtaining an approximate estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Using the quadrupolar and hyperfine tensors determined for DyPc2 [22] and the nuclear g-factors measured for 161Dy and 163Dy [23], we can compute Bnuc = |Bnuc| from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S57) for different orientations of the external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' As shown in Table S1, the effective nuclear magnetic fields at T = 2 K are at least one order of magnitude smaller than the dipolar fields calculated in the previous section, regardless of the orientation of the external field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 161Dy 163Dy Bext//ˆx 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='82×10−8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='34×10−8 Bext//ˆy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='77×10−8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='38×10−8 Bext//ˆz 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='51×10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='08×10−4 TABLE S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Effective Dy nuclear magnetic field Bnuc (T) at T = 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 13 S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' RESULTS FOR A DIFFERENT SOLVENT CONFIGURATION In this section we show that the results presented in the main text are robust against variations of the solvent environment on a qualitative level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In order to show this, we consider a smaller and rounder solvent ball consisting of 111 DCM molecules, and reproduce the results shown in the main text, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' It is worth noting that the vibronic spin-flip probabilities are significantly smaller for the smaller solvent ball, confirming the importance of the low-frequency vibrational modes associated to the solvent for determining QTM behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The general tendency of vibrations to enhance QTM, however, is correctly reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' vibrational DOS a) b) c) d) e) j el j FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Results for a different solvent configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (a) Alternative arrangement of 111 DCM molecules around [Dy(Cpttt)2]+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (b) Spin-phonon coupling strength and vibrational density of states (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (c) Vibronic correction to the energy splitting of the ground Kramers doublet (∆vib 1 − ∆1) for different orientations of the magnetic field (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (d) Ensemble-averaged spin-flip probability for different field sweep rates as a function of the internal field strength (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (e) Orientationally averaged single-mode spin-flip probability ⟨Pj⟩ vs change in magnetic axiality ∆Aj/Ael (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The most evident difference between these results and the ones presented in the main text is the shape of the single-mode axiality distribution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' S3e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In this case, single-mode spin-flip probability ⟨Pj⟩ still correlates to relative single-mode axiality ∆A j/Ael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' However, instead of taking values on a continuous range, the relative axiality seems to cluster around discrete values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In an attempt to clarify the origin of this strange behaviour, we looked at the composition of the vibrational modes belonging to the different clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Vibrational modes belonging to the same cluster were not found to share any evident common feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Rather than in the structure of the vibrational modes, this behaviour seems to originate from the equilibrium electronic g-matrix gel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' This can be seen by computing the single-mode axiality Aj = A(gel +gvib j ) for slightly different choices of gel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In particular, we checked how axiality of the electronic g-matrix affects the mode axiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In order to do that, we considered the singular value decomposition of the electronic g-matrix gel = U·diag(g1,g2,g3)·V†, (S58) △1 A1 (cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='1 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='2 0 元 0 元 2 2(cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='1 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='2 0 一元 2 2A1 VID A1 (cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='1 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='2 元 0 元 2 214 the matrices U and V contain its left and right eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The singular values are g1 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='99, g2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='40 × 10−6, g3 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='98 × 10−6, and the axiality is very close to one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 1 − Ael = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='79 × 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' We artificially change the axiality of gel by rescaling the hard-plane g-values by a factor α and redefining the electronic g-matrix as gel α = U·diag(g1,αg2,αg3)·V†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S59) The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The three different colours distinguish the vibrational modes belonging to the three clusters visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' S3e (corresponding to α = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' When α = 0, the g-matrix has perfect easy-axis anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' In this case, the vibronic correction to the g-matrix is too small to cause significant changes in the magnetic axiality, and all the vibrational modes align around A j ≈ Ael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Increasing α to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='9, clusters begin to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' For α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='3, the single-mode axiality distribution begins to look like the one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 4a in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' The electronic g-matrix obtained for the solvent ball considered in the main text has a lower axiality than the one used throughout this section, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 1−Ael = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content='12×10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Therefore, it makes sense that for α sufficiently larger than 1 we recover the same type of distribution as in the main text, since increasing α corresponds to lowering the electronic axiality A(gel α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Impact of electronic axiality on single-mode axiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Distribution of single-mode spin-flip probability ⟨Pj⟩ and g-matrix axiality A j = A(gelα + gvib j ) relative to the axiality of the modified electronic g-matrix A(gelα) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' (S59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Vibrational modes belonging to different clusters in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' S3e (α = 1) are labelled with different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' 15 [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Svensson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Humbel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Froese, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE5T4oBgHgl3EQfWQ9u/content/2301.05557v1.pdf'} +page_content=' Matsubara, S.' 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b/I9E3T4oBgHgl3EQfuwtu/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7f7111a6e0ce858c87299518bbfb5df00526dc766e50fd95ef630f9efa3e2b4d +size 192480 diff --git a/IdE2T4oBgHgl3EQf_gkq/content/tmp_files/2301.04248v1.pdf.txt b/IdE2T4oBgHgl3EQf_gkq/content/tmp_files/2301.04248v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c95d9df7e049fbe460bc95e63d29524b798ceb13 --- /dev/null +++ b/IdE2T4oBgHgl3EQf_gkq/content/tmp_files/2301.04248v1.pdf.txt @@ -0,0 +1,1024 @@ +Predicting Hateful Discussions on Reddit using +Graph Transformer Networks and Communal +Context +Liam Hebert +University of Waterloo +Waterloo, Canada +liam.hebert@uwaterloo.ca +Lukasz Golab +University of Waterloo +Waterloo, Canada +lgolab@uwaterloo.ca +Robin Cohen +University of Waterloo +Waterloo, Canada +rcohen@uwaterloo.ca +Abstract—We propose a system to predict harmful discussions +on social media platforms. Our solution uses contextual deep +language models and proposes the novel idea of integrating state- +of-the-art Graph Transformer Networks to analyze all conversa- +tions that follow an initial post. This framework also supports +adapting to future comments as the conversation unfolds. In +addition, we study whether a community-specific analysis of hate +speech leads to more effective detection of hateful discussions. +We evaluate our approach on 333,487 Reddit discussions from +various communities. We find that community-specific modeling +improves performance two-fold and that models which capture +wider-discussion context improve accuracy by 28% (35% for the +most hateful content) compared to limited context models. +Index Terms—Hate Speech, Social Media and Social Discourse, +AI for Social Good, Graph Neural Networks, Natural Language +Processing +I. INTRODUCTION +The rise of social media has led to increased harmful +discourse with damaging mental health effects. A recent study +by Gao et al. measured the mental health of over 4800 Chinese +citizens and found a high correlation between increased anx- +iety and depression with high social media usage [1]. These +alarming trends have motivated the usage of automated de- +tection methods to help moderate these platforms and prevent +further harms. +A common approach of methods that tackle this problem is +to detect the hatefulness of individual comments that belong +to a wider discussion [2]–[4]. Two central challenges arise, +however. First, conversational hate speech is deeply contextual. +To properly understand if a comment is hateful, it is important +to understand the context in which it was said. Second, by +only analyzing the hatefulness of individual comments, we +are bound to a reactive kind of moderation, which requires +the damaging comment and the discussion that led to it be +published first. This can lead to the spread of conversations +that discuss harmful or sensitive subjects which can incite +further hateful discourse. As such, it is important to proactively +detect the tone and trend of conversations before they can +delve into explicit hate speech. +Towards the goal of proactive detection, we also believe that +it is important to consider the unique latent communal norms +and behaviours that underpin social groups. Content which +may trigger a hateful reaction from members of one group can +trigger a different reaction from another group. This focus is +inspired by a recent study by Cinellie et al., which found that +online communities are prone to the “echo chamber” effect, +where extreme ideals can be encouraged and amplified [5]. As +such, learning these cultural norms and behaviours is essential +to properly model the direction of a conversation and whether +that direction leads to hate speech. +Taking all these observations into consideration, the ques- +tion is how to design a framework that can perform a rich +analysis of branching social media discussions with a view +of proactive hate speech prediction, and also how best to de- +sign experiments to demonstrate whether community-specific +analysis provides important benefits. +To study the behaviours of diverse communities, we fo- +cus on the social platform Reddit, where discussions take +place in topic-orientated communities called subreddits. These +communities are highly varied, ranging from aww, to discuss +cute animals, and politics, to discuss political affairs. +Beyond their subject differences, prior work by Waller et al. +found that these communities exhibit significant differences +in their social makeup and discussion behaviours [6]. For +example, discussions that take place on The_Donald were +found to have a significant political right leaning bias whereas +politics has an opposing left leaning bias. Within these +communities, conversations take place in branching trees struc- +tures where any user can reply to the comments of other users, +forming a linked chain of replies that can branch off into sub- +discussions. In addition to textual replies, users can explicitly +provide their opinion by up-voting (approving) or down-voting +(disapproving) the comments of other users. +The need for a deeper analysis of discussions with critical +consideration of the relevant context becomes apparent when +examining the excerpt from Reddit provided in Figure 1. +Focusing on the first interaction is insufficient in properly +detecting the tendency towards hate, as these initial comments +are actually quite benign. Indeed, what is perhaps most trou- +bling with this excerpt is the kind of snowball effect of anti- +social sentiment, as the discussion progresses. +arXiv:2301.04248v1 [cs.CL] 10 Jan 2023 + +Fig. 1. Example of a Reddit Discussion +In this work, we propose a novel approach to prevent hate +speech on social platforms which addresses these challenges. +First, it is important to properly encode the complex lexicon +unique to online discussions [7]. To do this, we utilize a fine- +tuned BERT language model that was trained on hateful text +from various social media platforms [8], [9]. This allows our +system to utilize contextual language embeddings of each node +in the discussion graph as initial feature representations. Next, +to capture the structure of conversations, we propose adapta- +tions to Graphormer [10], a state-of-the-art graph transformer +network. We adapt Graphormer away from its original purpose +of detecting global qualities of molecules, a large difference +from our intended use case. Instead, we predict whether +specific nodes in our discussion graph will lead to discussions +that encourage and contain hateful behaviour. By adapting this +model, we benefit from a self-attention aggregation operation +over the entire graph, allowing the model to learn contextual +relationships between relevant comments in the discussion. +Our end-to-end solution utilizes the embeddings generated by +the language model as the representation of each vertex in our +discussion graph, which are then aggregated and transformed +by our Graphormer model to make final node-level predictions. +To train such a system, we assemble a dataset of 333,487 +discussion graphs from different Reddit communities. Each +node in a discussion tree is then algorithmically labeled with +an ordinal value (0-4) according to how prevalent the hate +speech is within the discussion that follows under that node. +By utilizing a ranked representation of hate, we recognize +that encouraged hateful behaviour can be more damaging +than discouraged hateful behaviour, and that not all hateful +behaviour is equivalent. In addition, users can leverage these +scores to specify which level of hateful content they are +comfortable viewing and can be used to help social platforms +prioritize moderation for the most extreme content. We then +investigate the impact of modeling specific communal influ- +ence by measuring performance of our system against variants +that are trained on only one Reddit community. +We conclude this work with a reflection on how our +proactive approach to predicting hate speech can be used +to improve the mental health of users online. This final +discussion will shed more light on the significant value +we derive from considering trees of discussion in social +media, and not simply interpreting whether a single com- +ment is conveying hate speech. Our codebase is available at +github.com/liamhebert/CommunityHateFormer. +II. BACKGROUND AND RELATED WORK +A. Deep Language Models +Our approach draws on recent advancements in deep lan- +guage models and graph neural networks. To encode the +textual content of a comment, we make use of the Transformer +language model architecture proposed by Vaswani et al. [11]. +This architecture has resulted in many performance gains in +various natural language tasks [9], [11]. +Core to the transformer is the scaled dot product self- +attention mechanism. Let Hk = [h0, h1, ..., hn] ∈ Rn×d be +the set of hidden embeddings of each word hi of size d at +layer k. Given three learnable weight matrices, WQ ∈ Rd×d, +WK ∈ Rd×d and WV ∈ Rd×d, the self-attention operation +computes +Q = H(k)WQ, K = H(k)WK, V = H(k)WV , +Attn(H(k+1)) = softmax(QK⊤ +√ +d +)V +where Attn(Hk) represents an attention-weighed representa- +tion of the input. This representation is then processed by a +feedforward neural network to create H(k+1) [10], [11]. The +use of a self-attention mechanism allows the model to encode +the contextual relationship between words in a sentence [11]. +BERT, proposed by Devlin et al., extends this architecture by +adding a special token [CLS] to the initial input and using +the final embedding h[CLS] to encode the entire sentence [9]. +Transformer models can be fined-tuned to perform a variety +of tasks, including hate speech detection on social media. To +utilize this architecture towards hate speech detection, Mathew +et al. curated a corpus of comments from social platforms +Twitter and Gab, labeled as containing either Normal, Hateful +or Offensive content [8]. The authors then fine-tuned a BERT +model on this corpus to classify comments into these three +categories. Their final system, BERT-HateXplain, achieved ≈ +70% classification accuracy according to these three classes + +·3yr.ago 2 +America +How could Flynn have any information that would implicate the president, if the president were +innocent? +↑ 2.3k Give Award Share Report Save +·3 yr.ago 西? +Going further, why would Trump himself attempt or instruct others to attempt the 10 obstructive +acts detailed in the report if he was innocent? +Why would his associates have had to stonewall, destroy evidence, tamper with witnesses, and lie +to investigators if he was innocent? +Why has he claimed executive privilege over the redacted portions of the report if he's innocent +and "totally exonerated?" +It still boggles my mind that conservatives don't seem to possess the basic cognitive function +required to utilize simple logic. +↑ 1.4k Give Award Share Report Save +. 3 yr. ago +Answer: because the trump-drunk crowd doesn't care what he's guilty of--he's the mascot for +their team. +They. Do. Not. Care. +↑ 605 Give Award Share Report Save +. 3 yr. ago +11 more replies +3 yr. ago +Even if we ignore the smoke that suggests fire, obstruction regardless of being guilty or +otherwise, is still illegal and should be an impeachment. +If you are obstructing because you don't want a bad hair day photo go get out to the public +you're worried you'll be mistakenly found guilty, well now you're guilty of obstruction. +↑ 53 Give Award Share Report Save +4 more repliesFig. 2. Aggregate-Combine Operations in a Graph Neural Network +[8]. Our approach utilizes the h[CLS] embeddings created from +a pre-trained BERT-HateXplain model to initially encode each +comment in a discussion graph. +B. Graph Neural Networks +Given a Graph G = (V, E) with vertices V connected by +edges E, Graph Neural Networks (GNNs) compute represen- +tations of the vertices based on their feature embeddings v. +Each layer of a GNN consists of two steps: AGGREGATE and +COMBINE (Figure 2). The AGGREGATE operator combines +the feature embeddings vk of all neighbouring vertices of a +given vertex V i into an aggregated embedding ai. Common +aggregation operators include taking the mean, max and sum +of the feature embeddings of neighbouring vertices [12]–[15]. +Following the AGGREGATE operator, the COMBINE op- +erator creates a new representation of V i by transforming +vi with respect to ai. This new representation can then be +used in subsequent layers for further aggregation. In practice, +COMBINE operations are typically implemented as feed- +forward neural networks, which transform the concatenation +of ai and vi [10]. Key to the expressiveness of GNNs is that +both the COMBINE and AGGREGATE steps are structure- +independent. It is trivial to add vertices and edges to a GNN +input and compute new inferences by simply expanding the +number of vertices in the AGGREGATE operator. This aspect +allows GNNs to be well suited for inputs that can take multiple +shapes, such as branching social media conversations. +Motivated by the performance increases achieved by Trans- +former models, Ying et al. adapted the self-attention mech- +anism to GNNs [10] for use in molecular modeling, which +they name Graphormer. A key feature of this architecture +is the ability to use the self-attention mechanism to encode +relationships between any nodes in the entire graph during +each model layer, regardless of position and structure (Figure +3). To encode graph structure, Ying et al. propose four proxy +variables. Centrality Encoding, representing the in-degree and +out-degree of each node, is given as +H(0) +i += xi + z− +deg−(vi) + z+ +deg+(vi) +Fig. 3. Graphormer Architecture [10] +where H(0) +i +is the initial representation of node vi with +features xi and where z− +deg−(vi) and z+ +deg−(vi) are learnable +embeddings corresponding to the in-degree deg−(vi) and out- +degree deg+(vi) of that node. These features are only added to +the initial representations of each node. Next, Spatial Encoding +is introduced to encode the structure of the graph according +to inter-node distance. This encoding is added during the self +attention mechanism, where the encoding of node vi with +respect to node vj is given as +Aij = (HiWQ)(HjWK)T +√ +d ++ bφ(vi,vj) +where Hi and Hj are the embeddings of nodes vi and vj, WQ +and WK are learnable weight matrices and b is a learnable +embedding representing the shortest distance between vi and +vj in the graph (φ(vi, vj)). The last two features are Edge +Encoding, which encodes features corresponding to edge fea- +tures that connect nodes, and [CLS] nodes, to represent the +entire structure. Following the self-attention mechanism, the +next layer representation H(k+1) +i +of node i is given as +H(k+1) +i += softmax(Ai) × H(k)Wv +where Wv is a learnable weight matrix and softmax(Ai) +is the softmax operation over all Ai embeddings. These Ai +embeddings are created for every node pair in the graph +regardless of position, allowing node embeddings to be created +with respect to the entire graph structure. +C. Prior Work +The urgency of detecting hate speech online has led to a +large research interest in natural language processing tools. +However, preventative hate speech detection is still a difficult +problem to solve. Prior work on this problem by Brassard- +Gourdeau et al. proposed utilizing features from the first two +messages of a linear conversation, in addition to features in +the current message, as useful predictors for future hate speech + +V4 +V5 +V. +2 +V +V2 +4 +V, +V. +1 +3 +V1 +V3 +AGGREGATE +V4' +V5' +V2' +V1' +V3' +COMBINEV1V2U3V4U5 +U1 +MatMul +U2 +SoftMax +U3 +V4 +U5 +U +Scale +Spatial Encoding +U5 +MatMul +U1U2UU4U5 +U1 +02 +U2 +Linear +Linear +Linear +U3 +U4 +Q +K +V +V4 +U3 +U5 +Edge Encoding +Node Feature +Centrality Encoding[4]. Parallel work by Huang et al. proposed methods to pre- +filter comments before being published to a wider audience +[3]. However, both of these systems operate on a limited +context, narrowed to the first few messages of a discussion +or on the next comment to be made. In addition, many of +these systems are trained to generalize to all kinds of social +discourse, which may hinder the detection of nuanced culture- +specific hate speech. A final concern is that the systems are +often designed to handle linear discussions, whereas online +social media typically have branching discussions instead. +GNNs have recently been proposed to capture the im- +portance of context to predict hateful discourse in complex +discussion graphs. Parmentier et al. investigated using GNNs +to detect traits of individual comments by aggregating features +of their immediate neighbours [16]. This included traits such +as predicting the “upvote” and “downvote” score of a comment +and whether the comment contained hateful content [16], [17]. +However, while predicting the behaviour stemming from a +comment, Parmentier et al. only examined immediate node +neighbours. This was a constraint originating from the methods +and computational resources available at the time. +We differ from this prior work in two ways. First, instead +of predicting qualities of individual nodes, we predict whether +specific nodes will lead to future hateful discourse by con- +suming the entire discussion graph during inference. To do +this, we treat predictions of each node in the graph as the +prediction of the future subtree rooted at that node. This allows +our system to be utilized towards preventative detection of hate +speech, rather then analyzing comments after they have been +published. In addition, the self-attention mechanism allows our +model to only pay attention to contextually relevant nodes +when computing new representations, avoiding noise. +Second, we differ from this prior work by incorporating +fine-tuned contextual deep language models rather than using +context-free GloVE language modeling [18]. By utilizing deep +language models, we create embeddings that are fine-tuned +on the lexicon that is present on online social platforms, +rather than relying on embeddings generated from formal +sources, such as Wikipedia. Additionally, self-attention allows +the language model to contextualize to the entire input, as +opposed to GloVE which relies on word co-occurrence ratios +of the training corpus [18]. Notably, the usage of self-attention +mechanisms resulted in much better performance in context- +dependent tasks, such as sentiment analysis [11]. In summary, +the novelty of our approach lies on combining deep language +models with recent innovations in GNNs that can efficiently +capture longer range relationships. +III. METHODOLOGY +A. RedditHate Dataset +To train our model, we collected Reddit posts created be- +tween 2018 and 2019 using the Pushshift API1, focusing on the +The_Donald, IAmA, AmITheAsshole and Politics +communities. This resulted in a dataset of 333,487 posts and +1https://github.com/pushshift/api +6,531,455 comments. These communities were selected due +to their frequent lively discussion around various topics and +known differences in community norms [6], [17]. +Due to the size of the dataset, it is not feasible to label every +subtree by hand. Instead, we rely on several proxy features. +First, we make use of the scores (upvotes and downvotes) that +users assign to comments on Reddit. We utilize an aggregated +tally of these scores to model how inflammatory a given +comment was to a community. We also make use of the +hatefulness prediction of comment text from BERT Hate- +Explain [8] (retaining the definition of hate by the authors) and +the graph structure as further proxy variables. For each vertex +vi in the graph, we compute the hate label of a comment +Lvi by summing over three symbolic terms that estimate +how hateful the conversation is stemming from that node: +Context Cvi, Reaction Rvi and Influence Ivi. These terms are +computed as: +Cvi = 0.25 × hatepi × scorepi +(1) +Rvi = 0.25 × hatevi × scorevi +(2) +Ivi = +� +c∈Sv +0.25 × Lc +(3) +Lvi = Cvi + Rvi + Ivi +(4) +where hatev is the HateXplain predicted score of vertex v +scaled to (−0.7 − 1.5), scorev is the upvote/downvote score, +p is the parent node of v and Sv is the set of child nodes under +v. The Context term Cvi aims to capture the context in which a +comment was made, focusing on the hatefulness of the parent +comment. The Reaction Rvi term then captures the hatefulness +of the instigating comment that led to the conversation that +followed. Lastly, the Influence term Ivi recursively sums the +discounted hatefulness of the conversation that followed as a +result of the current node. Each label is created in a bottom- +up approach starting from the leaves of the discussion tree. +The factor 0.25 in each formula is selected to equally weigh +the contribution of each component. As part of our design +process, we conducted a sensitivity analysis of these weights, +detailed in Section IV-D, and found no significant differences +in performance when perturbed. +In each term comprising our hate label Lvi (Equations 1 +through 3), the multiplication ensures that hateful posts that +are encouraged with a positive score have a similar score to +non-hateful posts that have a negative score. Additionally, the +weights attached to each term allow lively hateful discussions +with many subtrees to decisively influence the label of the +current comment. We then standardize each label into five +ordinal classes, ([< 0], [0 - 5], [6 - 20], [21 - 500], [500 <]), +corresponding to increasing levels of hatefulness. The intuition +behind the use of ordinal values is to enable social media +moderators to prioritize the most hateful content and deploy +different policies for different criteria. The range of values +chosen for each of these classes was selected as a result of +inspection of the overall distribution of hate labels and the +comments contained within them. Our classification of hate +can then be seen as a regression problem. The final label + +TABLE I +COMMUNITY SAMPLE SIZE FOR THE REDDITHATE DATASET +Subreddit +Nodes +Nodes after Filtering +The Donald +6 531 455 +3 005 612 +Politics +6 127 723 +1 457 321 +AmItheAsshole +4 135 270 +1 381 200 +IAmA +183 284 +30 415 +TABLE II +REDDITHATE LABEL DISTRIBUTION +Subreddit / Label +0 +1 +2 +3 +4 +AmItheAsshole +1 103 308 +183 108 +39 815 +45 371 +9598 +IAmA +27 890 +2 240 +216 +68 +1 +The Donald +2 310 601 +441 746 +133 534 +117 091 +2640 +Politics +1 087 286 +264 570 +54 311 +45 563 +5 591 +Total +4 529 085 +891 664 +227 876 +208 093 +17 830 +distribution can be seen in Table II. With each discussion tree +labeled, we then convert each graph into Pytorch Geometric +(PyG) Data Graphs objects for inference [19]. +This approach of using algorithmic labeling follows prior +work for labeling hate speech on social platforms, such as [17] +and [4], among other works. In each of these systems, algo- +rithmic methods such as HateXplain replace human reviewers +to label a sufficiently large dataset. +Next, to analyze the ability of our model to proactively +predict hate speech, we take a subset of each labeled discussion +tree as the model’s input. We do this by taking a horizontal +slice of the discussion tree such that the maximum depth of +the tree is four comments from the root. We then remove +all comments that did not lead to a sub-discussion of at +least two comments. As a result, the leaf nodes of our +trimmed discussion tree are parent nodes of at least two other +comments, requiring the model to predict the hatefulness of an +unseen future discussion. Our trimming strategy also focuses +our model’s input on the most relevant parts of the discussion +[16]. Table I shows the number of nodes before and after +filtering for the four communities we selected. +B. Model Architecture +Our approach modifies the Graphormer model by Ying et +al. [10] with the HateXplain model by Mathew et al. [8]. No- +tably, the original Graphormer architecture was proposed and +geared towards molecular modeling and graph-level predic- +tions (Section II-B). However, with respect to the RedditHate +Dataset (Section III-A), our task is instead to have node- +level predictions regarding the future direction of discussions. +Despite this, our focus on capturing the entire discussion +context makes the Graphormer architecture appealing. +To adapt the Graphormer architecture towards our task, we +first omit the usage of [CLS] nodes and Edge Encoding +since we are not concerned with graph level predictions or +edge features. Second, we adapt the model output to pre- +dict ordinal labels for each node in the graph. These labels +correspond to the predicted hate score defined in Section +III-A. Finally, to encode each comment in the graph, we +TABLE III +MODEL PERFORMANCE (L2 LOSS) OF OUR APPROACH, LABELLED +GRAPHORMER, AND GAT +Method / Label +All +0 +1 +2 +3 +4 +Graphormer +1.066 +0.556 +0.946 +1.275 +2.213 +3.817 +GAT +1.485 +0.473 +1.228 +1.872 +2.661 +5.883 +utilize a pre-trained BERT model trained on the HateXplain +dataset (Section II-A). We use the [CLS] embedding of each +comment as the feature representation in the graph. As a +result, each comment in the discussion graph is encoded as +a 769-dimension embedding. The usage of BERT-HateXplain +allows each contextual comment embedding to be learned from +common social media discourse, rather than formal and clean +text. We also concatenate the upvote/downvote score to the +input vector. In our experiments, we utilize the Graphormer- +BASE variant which consists of ten layers of Graphormer self- +attention, resulting in 144 million total parameters (36 million +excluding BERT-HateXplain). +IV. RESULTS +In our experimental evaluation, we start by investigating +whether global discussion context allows for more accurate de- +tection of future hateful discussions (results in Section IV-A). +This is motivated by contrasting against other attempts at +detecting hate speech, which often rely on just a single +comment or the closest neighbours of a comment [3], [4], +[16]. Second, under the lens of modeling diverse discussion +semantics, we investigate whether models that are trained on +specific communities outperform models that are trained on +all communities (results in Section IV-B). Expanding on these +questions, in Section IV-C we present a qualitative comparison +of our approach against existing approaches, giving examples +of discussions that were predicted more effectively by our +method, compared to others. We then conclude with a sensi- +tivity analysis of our hate labelling function in Section IV-D. +We use the standard Graphormer variant, which consists of +ten transformer layers with a hidden dimension size of 769, as +described in the original work. Additionally, we utilized the +AdamW optimizer with a peak learning rate of 2e-4, which +linearly decayed after a warmup period [20]. We applied an L2 +regression loss for training and as an evaluation metric, which +assigns heavier penalties towards incorrect values. This is +important due to the imbalance in our dataset towards smaller +hate label values. Due to compute constraints, a batch size of +16 was used over ten epochs and only the Graphormer model +was fine-tuned. To evaluate the performance of the model, +we masked 10% of comments for use in validation and 20% +for use in testing, following prior work by Chen et al. [21]. +We utilize the validation set for hyper-parameter tuning in our +experiments. Each model was trained for ≈ 5 hours with 3 +RTX 2080 Ti GPUs. +A. Impact of Deeper Context +We begin by investigating the impact of including the +entire discussion graph during inference. To measure this, we + +TABLE IV +MODEL L2LOSS PERFORMANCE ON DIFFERENT COMMUNITIES +Target Communities +Train +Test +All +1.060 +1.066 +The Donald +0.4913 +0.4911 +Politics +0.4042 +0.4041 +IAmA +0.3687 +0.3444 +AmITheAsshole +0.4388 +0.4397 +selected the Graph Attention Network architecture (GAT) [13] +as a baseline, as implemented in Pytorch Geometric [19]. +This is inspired by previous efforts to utilize graph models +to detect hate speech on social networks, in particular [16], +which utilized this architecture. It is important to note that our +use case differs, where we instead seek to detect comments +that will lead to hateful discussion. Similar to the Graphormer +architecture, the GAT architecture utilizes an attention mech- +anism to aggregate neighbouring node embeddings. However, +an important difference with this architecture is that the +aggregation operation is constrained to the direct neighbouring +comments in each layer. This is contrasted by our proposed +approach, which performs self-attention aggregation over the +entire graph at each layer. +We compare our approach against a two-layer GAT model, +following the recommended configuration by the original +authors and prior work [16]. This implies two iterations of the +Aggregation-Combine operation. Table III shows the results: +our approach (Graphormer) outperforms GAT by an average +L2 loss of 0.419. Our results illustrate that both models +perform well at classifying comments with a low level of +future hateful discussion (0-1). This can be attributed to the +prevalence of these comments in our dataset. However, notable +differences in model performance can be seen when detecting +comments which led to more prevalent hatespeech (2-4), where +our model outperforms GAT by an average L2 loss of 2.066. +We hypothesize that the gains made by Graphormer are +due to its ability to contextualize better to the discourse of +larger discussion graphs, which are often good indicators of +future lively discussions. An example of this would be an +instigating comment in a debate, where a better understanding +of the entire debate could serve as a better predictor of what +could be considered hateful and instigating rebuttal. That being +said, it is still important to note that the performance at the +higher echelon is not optimal, having an average prediction +inaccuracy of ≈ 2 ordinal levels; however, Graphormer still +presents an improvement over GAT. +B. Impact of Community Specific Models +A key question in this work was whether hate could be bet- +ter predicted by modeling the behaviours and cultural norms +of specific communities. To evaluate this, we trained variations +of the model on specific communities present in our dataset. +We compare these results against a general purpose variant +that was trained on data from all four communities (Table +IV). We see that general models trained on all subreddits +perform significantly worse than community-specific models. +When comparing L2 loss, each community-specific model +outperforms general models by a factor of two, for an average +loss of ≈ 0.4. The model for the IAmA subreddit benefited the +most from having a community-specific focus, with an average +L2 loss of 0.3687, followed by the politics subreddit. +We hypothesize that this performance boost came from two +advantages. First, each community we selected is associated +with vastly different topics, with The_Donald created to +discuss alt-right politics and AmITheAsshole discussing +personal advice. As such, it is likely that comments that +would be ignored in The_Donald could be inflammatory +in other communities. A similar trend can be noticed in +politics, which is politically left leaning and would be +more sensitive to hateful political discourse [6]. By modeling +specific communities, we hypothesize that our model captured +these different cultural norms. +Second, some communities discuss in a formal and struc- +tured manner, such as question-answer format. IAmA is an +example of this, a community that focuses on interviewing +celebrities by asking public questions, contrasting other com- +munities which have free-flowing discussion. This difference +was highlighted in our results, where we found that IAmA +benefited the most from a community-specific approach. By +modeling the question and answer format and by sampling +the entire interview, we hypothesize that our model was able +to better understand the topics discussed and the unique +conversational flow. +C. Qualitative Analysis +To better capture the behaviours of our approach, we +conducted a qualitative analysis against GAT and comment- +text only Bert-HateXplain. Since the Bert-HateXplain values +are originally between 0-1, we map them to increasing ordinal +values of [0-0.2], [0.2-0.4], [0.4-0.6], [0.6-0.8] and [0.8- +1.0], for a matching 0-4 scale. We focused on investigating +conversations where predictions between all three models +differed. An example of such a conversation, found in the +politics subreddit, is shown in Table V . The subject of +this conversation is regarding U.S. politics and then-recent +news regarding Michael Flynn, former security advisor to +President Donald Trump. On the politics subreddit, prior +work has discovered a strong left leaning bias, resulting in +heated discussion when talking about topics concerning the +political right [6]. In this discussion, the conversation starts +with negative but still relatively tame comments. However, as +the conversation deepens, it devolves into hate speech against +the Republican party (”If Republicans continue the trend, their +next candidate will only be grunting and throwing feces”). +Investigating the predicted values of all three methods, we +see that Graphormer more accurately predict the direction +the discussion will be heading, even stemming from the +first comment. Our approach predicted an ordinal score of 4 +compared to a label of 3. This is compared to predictions +made from GAT and HateXplain, which fail to capture the +longer dependency (1 and 0 respectively). We also see a +recurring trend with single comment Bert-HateXplain, which + +TABLE V +EXAMPLE CONVERSATION REQUIRING LONG RANGE FORECASTING FROM COMMUNITY CUES (/R/POLITICS) +* INDICATES COMMENTS OUTSIDE OF INITIAL INPUT CONTEXT AND PREDICTED IN SUBSEQUENT ITERATIONS +Depth +Text +Label +Graphormer +GAT +Bert-HateXplain +0 +How could Flynn have any information that would implicate the president, if +the president were innocent? +3 +4 +1 +0 +1 +Because, according to the GOP Hive Mind, the information is a combination of +completely fabricated and not illegal. Trump’s narrative has always been ”Witch +Hunt!” i.e., that there was no basis for the investigation, so any information is +obviously fake. [...] It still boggles my mind that conservatives don’t seem to +possess the basic cognitive function required to utilize simple logic. +3 +3 +2 +1 +1 +Going further, why would Trump himself attempt or instruct others to attempt +the 10 obstructive acts detailed in the report if he was innocent? Why would +his associates have had to stonewall, destroy evidence, tamper with witnesses, +and lie to investigators if he was innocent? +2 +3 +0 +1 +[...] +6 +If Republicans continue the trend, their next candidate will only be grunting +and throwing feces. +4 +4* +4* +4* +7 +I think we can go dumber.. +4 +3* +1* +2* +TABLE VI +MODEL PERFORMANCE ON DIFFERENT LABELING WEIGHTS +Labeling Weight Variant +Train L2Loss +Test L2Loss +Equal +1.060 +1.066 +Influence-Weighted +1.180 +1.200 +Reaction-Weighted +1.072 +1.073 +Context-Weighted +0.985 +0.994 +seems to downplay the intensity of the hateful content early on +compared to the ground truth labels. It is important to note that +the input discussion is cut off after a depth of four comments +(the limit used in [16]), and therefore predictions must be made +from that limited context. +D. Sensitivity of RedditHate Labeling Function +We now investigate the sensitivity of the weights defined in +the HateReddit labeling function, which combines the value of +the Context-term (original node focused), Reaction-term (par- +ent node focused) and Influence-term (aggregated child nodes +focused) (Section III-A). This evaluates the likelihood of our +results to be dependent on the labeling weights chosen. To do +this, we created four variants of the dataset based on different +weighting configurations: Equal, which consists of weighting +each term equally, as well as Influence-Weighted, Reaction- +Weighted and Context-Weighted, which doubles the weight of +each respective term in the labeling function (weighted by 0.5 +instead of 0.25). We conducted each experiment on the entirety +of the HateReddit dataset. +Table VI shows the results. Each variant performs similarly, +with a Loss range of (0.994 - 1.200). As a result, we conclude +that the labeling weights are not sensitive to specific values. +We see that the Context-Weighted variant of the dataset +produced the best performance with a Test L2 loss of 0.994, +whereas the worst performing variant was the Influence- +Weighted variant, with a Test L2 loss of 1.200. It is important +to note that the Influence term of each node is calculated from +the entire discussion tree, despite the input constrained to a +depth of four with each remaining comment being the parent of +at least two others, potentially hidden, comments. As a result, +the Influence-weighted variant represents a more difficult task +compared to the other variants. +V. FUTURE WORK +There are several potential avenues for future work. First, +recall that the RedditHate dataset is founded on algorithmic +deduction that approximates ground truth. Ideally, the Red- +ditHate dataset would be annotated by human evaluators or +more advanced labeling strategies. Potential future work could +also explore creating an expanded and balanced dataset of +hateful and non-hateful discussions. This could be done by +considering a longer time period and then carefully selecting +discussions to fit a desired distribution. In addition, the col- +lection of communities discussed could be further expanded +to consider more diverse and topic orientated communities. +Second, we hypothesize that it would be beneficial to train +the model end-to-end by fine-tuning the BERT-HateXplain and +Graphormer models together. In this work, we circumvented +the need to train the BERT model by using a publicly available +pre-trained variant that was trained on Twitter and Gab data +towards detecting hateful comments. By fine-tuning a model +directly on comments from specific communities and towards +our target task, we can create embeddings that directly reflect +the behaviours of the target community. We believe that this +could allow for better subreddit-specific performance. +Third, future work could investigate the interpretability of +the graph attention weights during inference, following past +work in NLP [22]. This could enable an exploration of the +kinds of comments that enable hateful discussions in specific +communities, and the communal influence that enables them. +Fourth, further analysis could be performed to compare +community-specific models. By analyzing the performance of +one community model on posts from a different community, +it would be possible to derive insights on cultural similarities. +That is, models that perform well on other communities +likely share similar discussion norms. An example of such +a comparison would be evaluating the performance of the +politics model on the_donald subreddit, communities + +that are known to reflect contrasting political biases. This +direction could also expand to applying our model on other +social platforms, such as Twitter, in order to learn whether +the performance of the approach is equally strong in all +environments. We will need to decide how best to model +discussions and communities within a platform such as Twitter, +in order to generate our results. +Lastly, it would be beneficial to conduct additional experi- +ments in order to draw out the real strength of our predictive +approach to hate detection, in comparison with simpler base- +lines which rely on simply tagging existing comments. We +would code a competitor which assumes that the initial post +sets the tone for the overall conversation of each discussion +(i.e., all comments that follow this post are labelled with +the same level of hate detection). We would then examine +the performance of this method for detecting hate speech in +Reddit, both across all subreddits as well as in the specific +communities listed in Table IV, where the performance of +our own approach in terms of L2 loss has already been +measured. We would expect to see, for the baseline, significant +degradation on each subreddit. This would then confirm that +the initial post is often not indicative to predict the hateful +labels of the conversation and thus that our approach of +incorporating attention to predict hate patterns has significant +value. For future work, we can also examine other competitors +that fail to incorporate context, perhaps in other social media +environments, to continue to calibrate the relative benefit of +our solution. +VI. CONCLUSION +Social media have increasingly become the source of many +damaging mental health effects [1]. In this work, we proposed +a system based on state-of-the-art graph transformer models +and deep language models to prevent the spread of harmful +discourse that has perpetuated these damaging effects. To +evaluate our system, we created a dataset that contains a +collection of Reddit posts from various communities. We hope +that our system can be used by social media moderators to +curb the spread of harmful discourse by motivating the usage +of community specific models and the importance of capturing +discourse context. +The scope of this work differs in several ways from previous +work. First, we studied the influence of communities regarding +what content is shared and how that content is reacted to. We +accomplished this by including the entire discussion graph +during inference and creating specific models for individual +communities. This scope is different from traditional hate +speech systems that only examine the text of a comment in +isolation, without considering the discussion context or the +community. As we have shown in our experimental results, +including more context, in terms of the discussion and the +community, can lead to better performance. +Second, the output of the model differed from previous +work by predicting ordinal values from 0 to 4, denoting +how intense and encouraged the hateful discussion will be +following an initial comment. There are two benefits that come +with this objective. First, due to the increasing scope of social +platforms, it becomes important for moderators to prioritize +which content to investigate first and which content should +be immediately quarantined. By including a range of intensity +values, we also support users of these platforms to choose +an intensity level that they are comfortable with. This would +allow users with mental health challenges to self-regulate the +content they want to see without restricting the free speech of +users who are more comfortable seeing intense content. +We also hope that the insights and tools created in this +research can be used to drive future research into enhanced +social media moderation under a proactive lens, especially as +our lives and mental health become increasingly connected +to social platforms. With a deeper exploration of an entire +tree of discussion following a post and with a contextual +attention mechanism that improves the efficiency of predicting +what may be yet to come, our approach has the chance of +detecting when a growing surge of negative expression may +be unleashed. By adapting a more proactive and less reactive +approach to hate speech, the hope is that fragile users may be +better protected from the effects of anti-social behaviour. +ACKNOWLEDGMENT +The authors thank the Natural Sciences and Engineering +Research Council of Canada, the Canada Research Chairs +Program and the University of Waterloo Cheriton Scholarship +for financial support. We are also grateful to the reviewers for +their valued feedback on the paper. +REFERENCES +[1] J. Gao, P. Zheng, Y. Jia, H. Chen, Y. Mao, S. Chen, Y. Wang, H. Fu, +and J. Dai, “Mental health problems and social media exposure during +covid-19 outbreak,” PLOS ONE, no. 4, 2020. +[2] S. Obadinma, H. Guo, and X. Zhu, “Class-wise calibration: A case +study on covid-19 hate speech,” Canadian Conference on Artificial +Intelligence, 6 2021. +[3] Q. Huang, D. Inkpen, J. Zhang, and D. 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Deli´c et al., “Privacy-aware recom- +mender systems challenge on twitter’s home timeline,” arXiv, 2020. +[8] B. Mathew, P. Saha, S. M. Yimam, C. Biemann, P. Goyal, and +A. Mukherjee, “Hatexplain: A benchmark dataset for explainable hate +speech detection,” in AAAI, vol. 35, no. 17, 2021, pp. 14 867–14 875. +[9] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “Bert: Pre-training +of deep bidirectional transformers for language understanding,” arXiv, +2019. +[10] C. Ying, T. Cai, S. Luo, S. Zheng, G. Ke, D. He, Y. Shen, and T.-Y. Liu, +“Do transformers really perform badly for graph representation?” Ad- +vances in Neural Information Processing Systems, vol. 34, pp. 28 877– +28 888, 2021. +[11] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, +L. Kaiser, and I. Polosukhin, “Attention is all you need,” 2017. + +[12] W. Hamilton, Z. Ying, and J. Leskovec, “Inductive representation +learning on large graphs,” NeurIPS, vol. 30, 2017. +[13] P. Veliˇckovi´c, G. Cucurull, A. Casanova, A. Romero, P. Lio, and +Y. Bengio, “Graph attention networks,” arXiv, 2017. +[14] M. Welling and T. N. Kipf, “Semi-supervised classification with graph +convolutional networks,” in ICLR, 2016. +[15] K. Xu, W. Hu, J. Leskovec, and S. Jegelka, “How powerful are graph +neural networks?” in ICLR, 2019. +[16] A. Parmentier, J. P’ng, X. Tan, and R. Cohen, “Learning reddit user +reputation using graphical attention networks,” in Future Technologies +Conference (FTC) 2020, Volume 1, 2021, pp. 777–789. +[17] A. Parmentier and R. Cohen, “Learning user reputation on reddit,” in +Web Intelligence (WI), 2019, pp. 242–247. +[18] J. Pennington, R. Socher, and C. D. Manning, “Glove: Global vectors +for word representation,” in EMNLP, 2014, pp. 1532–1543. +[19] M. Fey and J. E. Lenssen, “Fast graph representation learning with +pytorch geometric,” arXiv, 2019. +[20] I. Loshchilov and F. Hutter, “Decoupled weight decay regularization,” +in ICLR, 2018. +[21] J. Chen, T. Ma, and C. 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Faruqui, “Attention +interpretability across nlp tasks,” arXiv, 2019. + diff --git a/IdE2T4oBgHgl3EQf_gkq/content/tmp_files/load_file.txt b/IdE2T4oBgHgl3EQf_gkq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b991cdae2e4bda89036f02251a680db1a605a86a --- /dev/null +++ b/IdE2T4oBgHgl3EQf_gkq/content/tmp_files/load_file.txt @@ -0,0 +1,608 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf,len=607 +page_content='Predicting Hateful Discussions on Reddit using Graph Transformer Networks and Communal Context Liam Hebert University of Waterloo Waterloo, Canada liam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='hebert@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='ca Lukasz Golab University of Waterloo Waterloo, Canada lgolab@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='ca Robin Cohen University of Waterloo Waterloo, Canada rcohen@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='ca Abstract—We propose a system to predict harmful discussions on social media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Our solution uses contextual deep language models and proposes the novel idea of integrating state- of-the-art Graph Transformer Networks to analyze all conversa- tions that follow an initial post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This framework also supports adapting to future comments as the conversation unfolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' In addition, we study whether a community-specific analysis of hate speech leads to more effective detection of hateful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We evaluate our approach on 333,487 Reddit discussions from various communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We find that community-specific modeling improves performance two-fold and that models which capture wider-discussion context improve accuracy by 28% (35% for the most hateful content) compared to limited context models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Index Terms—Hate Speech, Social Media and Social Discourse, AI for Social Good, Graph Neural Networks, Natural Language Processing I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' INTRODUCTION The rise of social media has led to increased harmful discourse with damaging mental health effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' A recent study by Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' measured the mental health of over 4800 Chinese citizens and found a high correlation between increased anx- iety and depression with high social media usage [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' These alarming trends have motivated the usage of automated de- tection methods to help moderate these platforms and prevent further harms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' A common approach of methods that tackle this problem is to detect the hatefulness of individual comments that belong to a wider discussion [2]–[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Two central challenges arise, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' First, conversational hate speech is deeply contextual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' To properly understand if a comment is hateful, it is important to understand the context in which it was said.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Second, by only analyzing the hatefulness of individual comments, we are bound to a reactive kind of moderation, which requires the damaging comment and the discussion that led to it be published first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This can lead to the spread of conversations that discuss harmful or sensitive subjects which can incite further hateful discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' As such, it is important to proactively detect the tone and trend of conversations before they can delve into explicit hate speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Towards the goal of proactive detection, we also believe that it is important to consider the unique latent communal norms and behaviours that underpin social groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Content which may trigger a hateful reaction from members of one group can trigger a different reaction from another group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This focus is inspired by a recent study by Cinellie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=', which found that online communities are prone to the “echo chamber” effect, where extreme ideals can be encouraged and amplified [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' As such, learning these cultural norms and behaviours is essential to properly model the direction of a conversation and whether that direction leads to hate speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Taking all these observations into consideration, the ques- tion is how to design a framework that can perform a rich analysis of branching social media discussions with a view of proactive hate speech prediction, and also how best to de- sign experiments to demonstrate whether community-specific analysis provides important benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' To study the behaviours of diverse communities, we fo- cus on the social platform Reddit, where discussions take place in topic-orientated communities called subreddits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' These communities are highly varied, ranging from aww, to discuss cute animals, and politics, to discuss political affairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Beyond their subject differences, prior work by Waller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' found that these communities exhibit significant differences in their social makeup and discussion behaviours [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' For example, discussions that take place on The_Donald were found to have a significant political right leaning bias whereas politics has an opposing left leaning bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Within these communities, conversations take place in branching trees struc- tures where any user can reply to the comments of other users, forming a linked chain of replies that can branch off into sub- discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' In addition to textual replies, users can explicitly provide their opinion by up-voting (approving) or down-voting (disapproving) the comments of other users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' The need for a deeper analysis of discussions with critical consideration of the relevant context becomes apparent when examining the excerpt from Reddit provided in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Focusing on the first interaction is insufficient in properly detecting the tendency towards hate, as these initial comments are actually quite benign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Indeed, what is perhaps most trou- bling with this excerpt is the kind of snowball effect of anti- social sentiment, as the discussion progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='04248v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='CL] 10 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Example of a Reddit Discussion In this work, we propose a novel approach to prevent hate speech on social platforms which addresses these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' First, it is important to properly encode the complex lexicon unique to online discussions [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' To do this, we utilize a fine- tuned BERT language model that was trained on hateful text from various social media platforms [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This allows our system to utilize contextual language embeddings of each node in the discussion graph as initial feature representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Next, to capture the structure of conversations, we propose adapta- tions to Graphormer [10], a state-of-the-art graph transformer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We adapt Graphormer away from its original purpose of detecting global qualities of molecules, a large difference from our intended use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Instead, we predict whether specific nodes in our discussion graph will lead to discussions that encourage and contain hateful behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' By adapting this model, we benefit from a self-attention aggregation operation over the entire graph, allowing the model to learn contextual relationships between relevant comments in the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Our end-to-end solution utilizes the embeddings generated by the language model as the representation of each vertex in our discussion graph, which are then aggregated and transformed by our Graphormer model to make final node-level predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' To train such a system, we assemble a dataset of 333,487 discussion graphs from different Reddit communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Each node in a discussion tree is then algorithmically labeled with an ordinal value (0-4) according to how prevalent the hate speech is within the discussion that follows under that node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' By utilizing a ranked representation of hate, we recognize that encouraged hateful behaviour can be more damaging than discouraged hateful behaviour, and that not all hateful behaviour is equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' In addition, users can leverage these scores to specify which level of hateful content they are comfortable viewing and can be used to help social platforms prioritize moderation for the most extreme content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We then investigate the impact of modeling specific communal influ- ence by measuring performance of our system against variants that are trained on only one Reddit community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We conclude this work with a reflection on how our proactive approach to predicting hate speech can be used to improve the mental health of users online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This final discussion will shed more light on the significant value we derive from considering trees of discussion in social media, and not simply interpreting whether a single com- ment is conveying hate speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Our codebase is available at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='com/liamhebert/CommunityHateFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' BACKGROUND AND RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Deep Language Models Our approach draws on recent advancements in deep lan- guage models and graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' To encode the textual content of a comment, we make use of the Transformer language model architecture proposed by Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This architecture has resulted in many performance gains in various natural language tasks [9], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Core to the transformer is the scaled dot product self- attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Let Hk = [h0, h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=', hn] ∈ Rn×d be the set of hidden embeddings of each word hi of size d at layer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Given three learnable weight matrices, WQ ∈ Rd×d, WK ∈ Rd×d and WV ∈ Rd×d, the self-attention operation computes Q = H(k)WQ, K = H(k)WK, V = H(k)WV , Attn(H(k+1)) = softmax(QK⊤ √ d )V where Attn(Hk) represents an attention-weighed representa- tion of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This representation is then processed by a feedforward neural network to create H(k+1) [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' The use of a self-attention mechanism allows the model to encode the contextual relationship between words in a sentence [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' BERT, proposed by Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=', extends this architecture by adding a special token [CLS] to the initial input and using the final embedding h[CLS] to encode the entire sentence [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Transformer models can be fined-tuned to perform a variety of tasks, including hate speech detection on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' To utilize this architecture towards hate speech detection, Mathew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' curated a corpus of comments from social platforms Twitter and Gab, labeled as containing either Normal, Hateful or Offensive content [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' The authors then fine-tuned a BERT model on this corpus to classify comments into these three categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Their final system, BERT-HateXplain, achieved ≈ 70% classification accuracy according to these three classes 3yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='ago 2 America How could Flynn have any information that would implicate the president, if the president were innocent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' ↑ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='3k Give Award Share Report Save 3 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='ago 西?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Going further, why would Trump himself attempt or instruct others to attempt the 10 obstructive acts detailed in the report if he was innocent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Why would his associates have had to stonewall, destroy evidence, tamper with witnesses, and lie to investigators if he was innocent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Why has he claimed executive privilege over the redacted portions of the report if he\'s innocent and "totally exonerated?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='" It still boggles my mind that conservatives don\'t seem to possess the basic cognitive function required to utilize simple logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' ↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='4k Give Award Share Report Save .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' 3 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=" ago Answer: because the trump-drunk crowd doesn't care what he's guilty of--he's the mascot for their team." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' They.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' ↑ 605 Give Award Share Report Save .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' 3 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' ago 11 more replies 3 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' ago Even if we ignore the smoke that suggests fire, obstruction regardless of being guilty or otherwise, is still illegal and should be an impeachment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=" If you are obstructing because you don't want a bad hair day photo go get out to the public you're worried you'll be mistakenly found guilty, well now you're guilty of obstruction." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' ↑ 53 Give Award Share Report Save 4 more repliesFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Aggregate-Combine Operations in a Graph Neural Network [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Our approach utilizes the h[CLS] embeddings created from a pre-trained BERT-HateXplain model to initially encode each comment in a discussion graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Graph Neural Networks Given a Graph G = (V, E) with vertices V connected by edges E, Graph Neural Networks (GNNs) compute represen- tations of the vertices based on their feature embeddings v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Each layer of a GNN consists of two steps: AGGREGATE and COMBINE (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' The AGGREGATE operator combines the feature embeddings vk of all neighbouring vertices of a given vertex V i into an aggregated embedding ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Common aggregation operators include taking the mean, max and sum of the feature embeddings of neighbouring vertices [12]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Following the AGGREGATE operator, the COMBINE op- erator creates a new representation of V i by transforming vi with respect to ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This new representation can then be used in subsequent layers for further aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' In practice, COMBINE operations are typically implemented as feed- forward neural networks, which transform the concatenation of ai and vi [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Key to the expressiveness of GNNs is that both the COMBINE and AGGREGATE steps are structure- independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' It is trivial to add vertices and edges to a GNN input and compute new inferences by simply expanding the number of vertices in the AGGREGATE operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This aspect allows GNNs to be well suited for inputs that can take multiple shapes, such as branching social media conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Motivated by the performance increases achieved by Trans- former models, Ying et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' adapted the self-attention mech- anism to GNNs [10] for use in molecular modeling, which they name Graphormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' A key feature of this architecture is the ability to use the self-attention mechanism to encode relationships between any nodes in the entire graph during each model layer, regardless of position and structure (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' To encode graph structure, Ying et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' propose four proxy variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Centrality Encoding, representing the in-degree and out-degree of each node, is given as H(0) i = xi + z− deg−(vi) + z+ deg+(vi) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Graphormer Architecture [10] where H(0) i is the initial representation of node vi with features xi and where z− deg−(vi) and z+ deg−(vi) are learnable embeddings corresponding to the in-degree deg−(vi) and out- degree deg+(vi) of that node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' These features are only added to the initial representations of each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Next, Spatial Encoding is introduced to encode the structure of the graph according to inter-node distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This encoding is added during the self attention mechanism, where the encoding of node vi with respect to node vj is given as Aij = (HiWQ)(HjWK)T √ d + bφ(vi,vj) where Hi and Hj are the embeddings of nodes vi and vj, WQ and WK are learnable weight matrices and b is a learnable embedding representing the shortest distance between vi and vj in the graph (φ(vi, vj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' The last two features are Edge Encoding, which encodes features corresponding to edge fea- tures that connect nodes, and [CLS] nodes, to represent the entire structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Following the self-attention mechanism, the next layer representation H(k+1) i of node i is given as H(k+1) i = softmax(Ai) × H(k)Wv where Wv is a learnable weight matrix and softmax(Ai) is the softmax operation over all Ai embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' These Ai embeddings are created for every node pair in the graph regardless of position, allowing node embeddings to be created with respect to the entire graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Prior Work The urgency of detecting hate speech online has led to a large research interest in natural language processing tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' However, preventative hate speech detection is still a difficult problem to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Prior work on this problem by Brassard- Gourdeau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' proposed utilizing features from the first two messages of a linear conversation, in addition to features in the current message, as useful predictors for future hate speech V4 V5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' 2 V V2 4 V, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=" 1 3 V1 V3 AGGREGATE V4' V5' V2' V1' V3' COMBINEV1V2U3V4U5 U1 MatMul U2 SoftMax U3 V4 U5 U Scale Spatial Encoding U5 MatMul U1U2UU4U5 U1 02 U2 Linear Linear Linear U3 U4 Q K V V4 U3 U5 Edge Encoding Node Feature Centrality Encoding[4]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Parallel work by Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' proposed methods to pre- filter comments before being published to a wider audience [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' However, both of these systems operate on a limited context, narrowed to the first few messages of a discussion or on the next comment to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' In addition, many of these systems are trained to generalize to all kinds of social discourse, which may hinder the detection of nuanced culture- specific hate speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' A final concern is that the systems are often designed to handle linear discussions, whereas online social media typically have branching discussions instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' GNNs have recently been proposed to capture the im- portance of context to predict hateful discourse in complex discussion graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Parmentier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' investigated using GNNs to detect traits of individual comments by aggregating features of their immediate neighbours [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This included traits such as predicting the “upvote” and “downvote” score of a comment and whether the comment contained hateful content [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' However, while predicting the behaviour stemming from a comment, Parmentier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' only examined immediate node neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This was a constraint originating from the methods and computational resources available at the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We differ from this prior work in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' First, instead of predicting qualities of individual nodes, we predict whether specific nodes will lead to future hateful discourse by con- suming the entire discussion graph during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' To do this, we treat predictions of each node in the graph as the prediction of the future subtree rooted at that node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This allows our system to be utilized towards preventative detection of hate speech, rather then analyzing comments after they have been published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' In addition, the self-attention mechanism allows our model to only pay attention to contextually relevant nodes when computing new representations, avoiding noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Second, we differ from this prior work by incorporating fine-tuned contextual deep language models rather than using context-free GloVE language modeling [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' By utilizing deep language models, we create embeddings that are fine-tuned on the lexicon that is present on online social platforms, rather than relying on embeddings generated from formal sources, such as Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Additionally, self-attention allows the language model to contextualize to the entire input, as opposed to GloVE which relies on word co-occurrence ratios of the training corpus [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Notably, the usage of self-attention mechanisms resulted in much better performance in context- dependent tasks, such as sentiment analysis [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' In summary, the novelty of our approach lies on combining deep language models with recent innovations in GNNs that can efficiently capture longer range relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' METHODOLOGY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' RedditHate Dataset To train our model, we collected Reddit posts created be- tween 2018 and 2019 using the Pushshift API1, focusing on the The_Donald, IAmA, AmITheAsshole and Politics communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This resulted in a dataset of 333,487 posts and 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='com/pushshift/api 6,531,455 comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' These communities were selected due to their frequent lively discussion around various topics and known differences in community norms [6], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Due to the size of the dataset, it is not feasible to label every subtree by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Instead, we rely on several proxy features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' First, we make use of the scores (upvotes and downvotes) that users assign to comments on Reddit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We utilize an aggregated tally of these scores to model how inflammatory a given comment was to a community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We also make use of the hatefulness prediction of comment text from BERT Hate- Explain [8] (retaining the definition of hate by the authors) and the graph structure as further proxy variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' For each vertex vi in the graph, we compute the hate label of a comment Lvi by summing over three symbolic terms that estimate how hateful the conversation is stemming from that node: Context Cvi, Reaction Rvi and Influence Ivi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' These terms are computed as: Cvi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='25 × hatepi × scorepi (1) Rvi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='25 × hatevi × scorevi (2) Ivi = � c∈Sv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='25 × Lc (3) Lvi = Cvi + Rvi + Ivi (4) where hatev is the HateXplain predicted score of vertex v scaled to (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='7 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='5), scorev is the upvote/downvote score, p is the parent node of v and Sv is the set of child nodes under v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' The Context term Cvi aims to capture the context in which a comment was made, focusing on the hatefulness of the parent comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' The Reaction Rvi term then captures the hatefulness of the instigating comment that led to the conversation that followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Lastly, the Influence term Ivi recursively sums the discounted hatefulness of the conversation that followed as a result of the current node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Each label is created in a bottom- up approach starting from the leaves of the discussion tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' The factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='25 in each formula is selected to equally weigh the contribution of each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' As part of our design process, we conducted a sensitivity analysis of these weights, detailed in Section IV-D, and found no significant differences in performance when perturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' In each term comprising our hate label Lvi (Equations 1 through 3), the multiplication ensures that hateful posts that are encouraged with a positive score have a similar score to non-hateful posts that have a negative score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Additionally, the weights attached to each term allow lively hateful discussions with many subtrees to decisively influence the label of the current comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We then standardize each label into five ordinal classes, ([< 0], [0 - 5], [6 - 20], [21 - 500], [500 <]), corresponding to increasing levels of hatefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' The intuition behind the use of ordinal values is to enable social media moderators to prioritize the most hateful content and deploy different policies for different criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' The range of values chosen for each of these classes was selected as a result of inspection of the overall distribution of hate labels and the comments contained within them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Our classification of hate can then be seen as a regression problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' The final label ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='TABLE I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='COMMUNITY SAMPLE SIZE FOR THE REDDITHATE DATASET ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='Subreddit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='Nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='Nodes after Filtering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='The Donald ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='6 531 455 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='3 005 612 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='Politics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='6 127 723 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='1 457 321 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='AmItheAsshole ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='4 135 270 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='1 381 200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='IAmA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='183 284 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='30 415 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='TABLE II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='REDDITHATE LABEL DISTRIBUTION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='Subreddit / Label ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='AmItheAsshole ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='1 103 308 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='183 108 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='39 815 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='45 371 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='9598 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='IAmA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='27 890 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='2 240 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='216 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='68 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='The Donald ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='2 310 601 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='441 746 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='133 534 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='117 091 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='2640 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='Politics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='1 087 286 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='264 570 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='54 311 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='45 563 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='5 591 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='Total ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='4 529 085 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='891 664 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='227 876 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='208 093 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='17 830 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='distribution can be seen in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' With each discussion tree labeled, we then convert each graph into Pytorch Geometric (PyG) Data Graphs objects for inference [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This approach of using algorithmic labeling follows prior work for labeling hate speech on social platforms, such as [17] and [4], among other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' In each of these systems, algo- rithmic methods such as HateXplain replace human reviewers to label a sufficiently large dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Next, to analyze the ability of our model to proactively predict hate speech, we take a subset of each labeled discussion tree as the model’s input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We do this by taking a horizontal slice of the discussion tree such that the maximum depth of the tree is four comments from the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We then remove all comments that did not lead to a sub-discussion of at least two comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' As a result, the leaf nodes of our trimmed discussion tree are parent nodes of at least two other comments, requiring the model to predict the hatefulness of an unseen future discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Our trimming strategy also focuses our model’s input on the most relevant parts of the discussion [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Table I shows the number of nodes before and after filtering for the four communities we selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Model Architecture Our approach modifies the Graphormer model by Ying et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' [10] with the HateXplain model by Mathew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' No- tably, the original Graphormer architecture was proposed and geared towards molecular modeling and graph-level predic- tions (Section II-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' However, with respect to the RedditHate Dataset (Section III-A), our task is instead to have node- level predictions regarding the future direction of discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Despite this, our focus on capturing the entire discussion context makes the Graphormer architecture appealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' To adapt the Graphormer architecture towards our task, we first omit the usage of [CLS] nodes and Edge Encoding since we are not concerned with graph level predictions or edge features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Second, we adapt the model output to pre- dict ordinal labels for each node in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' These labels correspond to the predicted hate score defined in Section III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Finally, to encode each comment in the graph, we TABLE III MODEL PERFORMANCE (L2 LOSS) OF OUR APPROACH, LABELLED GRAPHORMER, AND GAT Method / Label All 0 1 2 3 4 Graphormer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='066 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='556 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='946 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='275 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='213 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='817 GAT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='485 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='473 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='228 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='872 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='661 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='883 utilize a pre-trained BERT model trained on the HateXplain dataset (Section II-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We use the [CLS] embedding of each comment as the feature representation in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' As a result, each comment in the discussion graph is encoded as a 769-dimension embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' The usage of BERT-HateXplain allows each contextual comment embedding to be learned from common social media discourse, rather than formal and clean text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We also concatenate the upvote/downvote score to the input vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' In our experiments, we utilize the Graphormer- BASE variant which consists of ten layers of Graphormer self- attention, resulting in 144 million total parameters (36 million excluding BERT-HateXplain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' RESULTS In our experimental evaluation, we start by investigating whether global discussion context allows for more accurate de- tection of future hateful discussions (results in Section IV-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This is motivated by contrasting against other attempts at detecting hate speech, which often rely on just a single comment or the closest neighbours of a comment [3], [4], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Second, under the lens of modeling diverse discussion semantics, we investigate whether models that are trained on specific communities outperform models that are trained on all communities (results in Section IV-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Expanding on these questions, in Section IV-C we present a qualitative comparison of our approach against existing approaches, giving examples of discussions that were predicted more effectively by our method, compared to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We then conclude with a sensi- tivity analysis of our hate labelling function in Section IV-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We use the standard Graphormer variant, which consists of ten transformer layers with a hidden dimension size of 769, as described in the original work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Additionally, we utilized the AdamW optimizer with a peak learning rate of 2e-4, which linearly decayed after a warmup period [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We applied an L2 regression loss for training and as an evaluation metric, which assigns heavier penalties towards incorrect values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This is important due to the imbalance in our dataset towards smaller hate label values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Due to compute constraints, a batch size of 16 was used over ten epochs and only the Graphormer model was fine-tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' To evaluate the performance of the model, we masked 10% of comments for use in validation and 20% for use in testing, following prior work by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We utilize the validation set for hyper-parameter tuning in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Each model was trained for ≈ 5 hours with 3 RTX 2080 Ti GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Impact of Deeper Context We begin by investigating the impact of including the entire discussion graph during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' To measure this, we TABLE IV MODEL L2LOSS PERFORMANCE ON DIFFERENT COMMUNITIES Target Communities Train Test All 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='060 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='066 The Donald 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='4913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='4911 Politics 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='4042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='4041 IAmA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='3687 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='3444 AmITheAsshole 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='4388 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='4397 selected the Graph Attention Network architecture (GAT) [13] as a baseline, as implemented in Pytorch Geometric [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This is inspired by previous efforts to utilize graph models to detect hate speech on social networks, in particular [16], which utilized this architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' It is important to note that our use case differs, where we instead seek to detect comments that will lead to hateful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Similar to the Graphormer architecture, the GAT architecture utilizes an attention mech- anism to aggregate neighbouring node embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' However, an important difference with this architecture is that the aggregation operation is constrained to the direct neighbouring comments in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This is contrasted by our proposed approach, which performs self-attention aggregation over the entire graph at each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We compare our approach against a two-layer GAT model, following the recommended configuration by the original authors and prior work [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This implies two iterations of the Aggregation-Combine operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Table III shows the results: our approach (Graphormer) outperforms GAT by an average L2 loss of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Our results illustrate that both models perform well at classifying comments with a low level of future hateful discussion (0-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This can be attributed to the prevalence of these comments in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' However, notable differences in model performance can be seen when detecting comments which led to more prevalent hatespeech (2-4), where our model outperforms GAT by an average L2 loss of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We hypothesize that the gains made by Graphormer are due to its ability to contextualize better to the discourse of larger discussion graphs, which are often good indicators of future lively discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' An example of this would be an instigating comment in a debate, where a better understanding of the entire debate could serve as a better predictor of what could be considered hateful and instigating rebuttal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' That being said, it is still important to note that the performance at the higher echelon is not optimal, having an average prediction inaccuracy of ≈ 2 ordinal levels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' however, Graphormer still presents an improvement over GAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Impact of Community Specific Models A key question in this work was whether hate could be bet- ter predicted by modeling the behaviours and cultural norms of specific communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' To evaluate this, we trained variations of the model on specific communities present in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We compare these results against a general purpose variant that was trained on data from all four communities (Table IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We see that general models trained on all subreddits perform significantly worse than community-specific models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' When comparing L2 loss, each community-specific model outperforms general models by a factor of two, for an average loss of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' The model for the IAmA subreddit benefited the most from having a community-specific focus, with an average L2 loss of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='3687, followed by the politics subreddit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We hypothesize that this performance boost came from two advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' First, each community we selected is associated with vastly different topics, with The_Donald created to discuss alt-right politics and AmITheAsshole discussing personal advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' As such, it is likely that comments that would be ignored in The_Donald could be inflammatory in other communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' A similar trend can be noticed in politics, which is politically left leaning and would be more sensitive to hateful political discourse [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' By modeling specific communities, we hypothesize that our model captured these different cultural norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Second, some communities discuss in a formal and struc- tured manner, such as question-answer format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' IAmA is an example of this, a community that focuses on interviewing celebrities by asking public questions, contrasting other com- munities which have free-flowing discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This difference was highlighted in our results, where we found that IAmA benefited the most from a community-specific approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' By modeling the question and answer format and by sampling the entire interview, we hypothesize that our model was able to better understand the topics discussed and the unique conversational flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Qualitative Analysis To better capture the behaviours of our approach, we conducted a qualitative analysis against GAT and comment- text only Bert-HateXplain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Since the Bert-HateXplain values are originally between 0-1, we map them to increasing ordinal values of [0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='2], [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='4], [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='6], [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='6-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='8] and [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='8- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='0], for a matching 0-4 scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We focused on investigating conversations where predictions between all three models differed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' An example of such a conversation, found in the politics subreddit, is shown in Table V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' The subject of this conversation is regarding U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' politics and then-recent news regarding Michael Flynn, former security advisor to President Donald Trump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' On the politics subreddit, prior work has discovered a strong left leaning bias, resulting in heated discussion when talking about topics concerning the political right [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' In this discussion, the conversation starts with negative but still relatively tame comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' However, as the conversation deepens, it devolves into hate speech against the Republican party (”If Republicans continue the trend, their next candidate will only be grunting and throwing feces”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Investigating the predicted values of all three methods, we see that Graphormer more accurately predict the direction the discussion will be heading, even stemming from the first comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Our approach predicted an ordinal score of 4 compared to a label of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This is compared to predictions made from GAT and HateXplain, which fail to capture the longer dependency (1 and 0 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We also see a recurring trend with single comment Bert-HateXplain, which TABLE V EXAMPLE CONVERSATION REQUIRING LONG RANGE FORECASTING FROM COMMUNITY CUES (/R/POLITICS) INDICATES COMMENTS OUTSIDE OF INITIAL INPUT CONTEXT AND PREDICTED IN SUBSEQUENT ITERATIONS Depth Text Label Graphormer GAT Bert-HateXplain 0 How could Flynn have any information that would implicate the president, if the president were innocent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' 3 4 1 0 1 Because, according to the GOP Hive Mind, the information is a combination of completely fabricated and not illegal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Trump’s narrative has always been ”Witch Hunt!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=', that there was no basis for the investigation, so any information is obviously fake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='] It still boggles my mind that conservatives don’t seem to possess the basic cognitive function required to utilize simple logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' 3 3 2 1 1 Going further, why would Trump himself attempt or instruct others to attempt the 10 obstructive acts detailed in the report if he was innocent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Why would his associates have had to stonewall, destroy evidence, tamper with witnesses, and lie to investigators if he was innocent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' 2 3 0 1 [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='] 6 If Republicans continue the trend, their next candidate will only be grunting and throwing feces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' 4 4* 4* 4* 7 I think we can go dumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='. 4 3* 1* 2* TABLE VI MODEL PERFORMANCE ON DIFFERENT LABELING WEIGHTS Labeling Weight Variant Train L2Loss Test L2Loss Equal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='060 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='066 Influence-Weighted 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='180 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='200 Reaction-Weighted 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='072 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='073 Context-Weighted 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='994 seems to downplay the intensity of the hateful content early on compared to the ground truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' It is important to note that the input discussion is cut off after a depth of four comments (the limit used in [16]), and therefore predictions must be made from that limited context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Sensitivity of RedditHate Labeling Function We now investigate the sensitivity of the weights defined in the HateReddit labeling function, which combines the value of the Context-term (original node focused), Reaction-term (par- ent node focused) and Influence-term (aggregated child nodes focused) (Section III-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This evaluates the likelihood of our results to be dependent on the labeling weights chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' To do this, we created four variants of the dataset based on different weighting configurations: Equal, which consists of weighting each term equally, as well as Influence-Weighted, Reaction- Weighted and Context-Weighted, which doubles the weight of each respective term in the labeling function (weighted by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='5 instead of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We conducted each experiment on the entirety of the HateReddit dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Table VI shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Each variant performs similarly, with a Loss range of (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='994 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' As a result, we conclude that the labeling weights are not sensitive to specific values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We see that the Context-Weighted variant of the dataset produced the best performance with a Test L2 loss of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='994, whereas the worst performing variant was the Influence- Weighted variant, with a Test L2 loss of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' It is important to note that the Influence term of each node is calculated from the entire discussion tree, despite the input constrained to a depth of four with each remaining comment being the parent of at least two others, potentially hidden, comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' As a result, the Influence-weighted variant represents a more difficult task compared to the other variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' FUTURE WORK There are several potential avenues for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' First, recall that the RedditHate dataset is founded on algorithmic deduction that approximates ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Ideally, the Red- ditHate dataset would be annotated by human evaluators or more advanced labeling strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Potential future work could also explore creating an expanded and balanced dataset of hateful and non-hateful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This could be done by considering a longer time period and then carefully selecting discussions to fit a desired distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' In addition, the col- lection of communities discussed could be further expanded to consider more diverse and topic orientated communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Second, we hypothesize that it would be beneficial to train the model end-to-end by fine-tuning the BERT-HateXplain and Graphormer models together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' In this work, we circumvented the need to train the BERT model by using a publicly available pre-trained variant that was trained on Twitter and Gab data towards detecting hateful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' By fine-tuning a model directly on comments from specific communities and towards our target task, we can create embeddings that directly reflect the behaviours of the target community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We believe that this could allow for better subreddit-specific performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Third, future work could investigate the interpretability of the graph attention weights during inference, following past work in NLP [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This could enable an exploration of the kinds of comments that enable hateful discussions in specific communities, and the communal influence that enables them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Fourth, further analysis could be performed to compare community-specific models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' By analyzing the performance of one community model on posts from a different community, it would be possible to derive insights on cultural similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' That is, models that perform well on other communities likely share similar discussion norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' An example of such a comparison would be evaluating the performance of the politics model on the_donald subreddit, communities that are known to reflect contrasting political biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This direction could also expand to applying our model on other social platforms, such as Twitter, in order to learn whether the performance of the approach is equally strong in all environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We will need to decide how best to model discussions and communities within a platform such as Twitter, in order to generate our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Lastly, it would be beneficial to conduct additional experi- ments in order to draw out the real strength of our predictive approach to hate detection, in comparison with simpler base- lines which rely on simply tagging existing comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We would code a competitor which assumes that the initial post sets the tone for the overall conversation of each discussion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=', all comments that follow this post are labelled with the same level of hate detection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We would then examine the performance of this method for detecting hate speech in Reddit, both across all subreddits as well as in the specific communities listed in Table IV, where the performance of our own approach in terms of L2 loss has already been measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We would expect to see, for the baseline, significant degradation on each subreddit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This would then confirm that the initial post is often not indicative to predict the hateful labels of the conversation and thus that our approach of incorporating attention to predict hate patterns has significant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' For future work, we can also examine other competitors that fail to incorporate context, perhaps in other social media environments, to continue to calibrate the relative benefit of our solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' CONCLUSION Social media have increasingly become the source of many damaging mental health effects [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' In this work, we proposed a system based on state-of-the-art graph transformer models and deep language models to prevent the spread of harmful discourse that has perpetuated these damaging effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' To evaluate our system, we created a dataset that contains a collection of Reddit posts from various communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We hope that our system can be used by social media moderators to curb the spread of harmful discourse by motivating the usage of community specific models and the importance of capturing discourse context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' The scope of this work differs in several ways from previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' First, we studied the influence of communities regarding what content is shared and how that content is reacted to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We accomplished this by including the entire discussion graph during inference and creating specific models for individual communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This scope is different from traditional hate speech systems that only examine the text of a comment in isolation, without considering the discussion context or the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' As we have shown in our experimental results, including more context, in terms of the discussion and the community, can lead to better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Second, the output of the model differed from previous work by predicting ordinal values from 0 to 4, denoting how intense and encouraged the hateful discussion will be following an initial comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' There are two benefits that come with this objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' First, due to the increasing scope of social platforms, it becomes important for moderators to prioritize which content to investigate first and which content should be immediately quarantined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' By including a range of intensity values, we also support users of these platforms to choose an intensity level that they are comfortable with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' This would allow users with mental health challenges to self-regulate the content they want to see without restricting the free speech of users who are more comfortable seeing intense content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We also hope that the insights and tools created in this research can be used to drive future research into enhanced social media moderation under a proactive lens, especially as our lives and mental health become increasingly connected to social platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' With a deeper exploration of an entire tree of discussion following a post and with a contextual attention mechanism that improves the efficiency of predicting what may be yet to come, our approach has the chance of detecting when a growing surge of negative expression may be unleashed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' By adapting a more proactive and less reactive approach to hate speech, the hope is that fragile users may be better protected from the effects of anti-social behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' ACKNOWLEDGMENT The authors thank the Natural Sciences and Engineering Research Council of Canada, the Canada Research Chairs Program and the University of Waterloo Cheriton Scholarship for financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' We are also grateful to the reviewers for their valued feedback on the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' REFERENCES [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQf_gkq/content/2301.04248v1.pdf'} +page_content=' Gao, P.' metadata={'source': 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a/JdE2T4oBgHgl3EQf_wnL/content/tmp_files/2301.04252v1.pdf.txt b/JdE2T4oBgHgl3EQf_wnL/content/tmp_files/2301.04252v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ec807f8d0c1c67321c4905a9e53f920fce68a2e8 --- /dev/null +++ b/JdE2T4oBgHgl3EQf_wnL/content/tmp_files/2301.04252v1.pdf.txt @@ -0,0 +1,3532 @@ +arXiv:2301.04252v1 [math.GR] 11 Jan 2023 +Conjugacy in Semigroups: the Partition and Brauer Diagram +Monoids, Conjugacy Growth, and Partial Inner Automorphisms +Jo˜ao Ara´ujo, Wolfram Bentz, Michael Kinyon, Janusz Konieczny, +Ant´onio Malheiro, and Valentin Mercier +Abstract +The conjugacy relation plays an important role in group theory. If a and b are elements of a group G, +a is conjugate to b if g−1ag = b for some g ∈ G. Group conjugacy extends to inverse semigroups in a +natural way: for a and b in an inverse semigroup S, a is conjugate to b if g−1ag = b and gbg−1 = a +for some g ∈ S. The fourth author has recently defined a conjugacy for an arbitrary semigroup S that +coincides with inverse semigroup conjugacy if S is an inverse semigroup, and is included in all existing +semigroup conjugacy relations. We will call it the natural conjugacy for semigroups, and denote it by +∼n. +The first purpose of this paper is to study ∼n in various contexts, chiefly the partition monoid and +some of its friends (Brauer and partial Brauer monoids), and also to characterize ∼n in several important +classes of semigroups, transformation semigroups and in the polycyclic monoids. +The second purpose of this paper is to show how the notion of natural conjugacy leads to the definition +of the inverse semigroup of partial automorphisms of an arbitrary semigroup (in the same way conjugation +in groups induces the notion of inner automorphism). Attached to the majority of mathematical objects +there is a notion of morphism and hence notions of automorphism and endomorphism that often encode +relevant information about the original object. Our approach allows to attach to the endomorphisms +of a mathematical object an inverse semigroup that hopefully will bring the deep results on inverse +semigroups to help the study of the original object. +Finally we extend the notion of conjugacy growth from groups to semigroups and give closed formulas +for the conjugacy growth series of the polycyclic monoid, for ∼n and two other semigroup conjugacies. +The paper ends with some open problems. +2020 Mathematics Subject Classification. 20M10, 20M20, 20M15, 05C20. +Keywords: Conjugacy; partial inner automorphisms; transformation semigroups; partition monoids; poly- +cyclic monoids; conjugacy growth series. +1 +Introduction +In a semigroup S, define a relation ∼n, which we will call natural conjugacy, as follows: for all a, b ∈ S, +a ∼n b ⇐⇒ ∃g,h∈S1 ( ag = gb, bh = ha, hag = b, and gbh = a ) . +(∼n) +The main goals of this paper are the following: +1. Describe the natural conjugacy classes in the partition monoid and some of its friends; these monoids +(Partition, Brauer, Jones, Kauffman, Martin, Temperley and Lieb, etc.) belong to the general family +of diagram monoids and (with the associated algebras and categories) arise in many areas of mathe- +matics such as invariant theory, classical groups, representation theory, logic, knot theory or statistical +mechanics (e.g. [7,30,32,33,40,41,59]; for an excellent overview on the literature and interconnections +of these areas please see the introduction of [21]). Given the importance of these objects, about one +third of the paper is dedicated to the description of the conjugacy classes in the partition monoid, the +1 + +Brauer monoid and the partial Brauer monoid. We describe the classes for ∼n and for several other +notions of conjugacy. +2. As conjugation in groups induces in a natural way the group of inner automorphisms (a → g−1ag), +the notion ∼n induces on every semigroup the inverse semigroup of partial automorphisms; when +the semigroup is a group, then this object is the group of inner automorphisms with a zero adjoined. +Computing this object for a given semigroup will be challenging in general; here we computed it for the +full transformation monoid, the symmetric inverse semigroup and for a completely simple semigroup. +3. Extend to monoids the group theory notion of conjugacy growth. As a proof of concept, investigate the +conjugacy growth in the polycyclic monoids (a natural family of finitely generated infinite monoids). +4. Prove for the natural conjugacy results similar to the ones proved in [4] for other notions of conjugacy. +In addition to these general goals, this paper explores many other paths as we now explain. +Let a and b be conjugate elements of a group G, that is, g−1ag = b for some g ∈ G. There are equivalent +formulations that avoid inverses, for example, ag = gb for some g ∈ G or a = uv and b = vu for some +u, v ∈ G. The latter formulations have been used to define relations ∼l (left conjugate) [51,60,61] and ∼p +(primary conjugate) [43] on an arbitrary semigroup S: +a ∼l b ⇐⇒ ∃g∈S1 ag = gb, +(1.1) +a ∼p b ⇐⇒ ∃u,v∈S1 a = uv and b = vu, +(1.2) +where S1 is S with an identity adjoined. In a general semigroup S, the relation ∼l is reflexive and transitive, +but not symmetric; while ∼p is reflexive and symmetric, but not transitive. However, these relations can +serve as a conjugacy in the class of free semigroups: if S is a free semigroup, then ∼l and ∼p are equivalence +relations, and they coincide [44]. +The relation ∼l has been restricted to ∼o [51], and ∼p has been extended to ∼∗ +p [42,43], in such a way +that the modified relations are equivalences on an arbitrary semigroup S: +a ∼o b ⇐⇒ ∃g,h∈S1 ag = gb and bh = ha, +(1.3) +∼∗ +p = the transitive closure of ∼p . +(1.4) +The relation ∼o reduces to S × S for any semigroup S with zero. This deficiency has been remedied in [5], +where the following relation has been defined on an arbitrary semigroup S: +a ∼c b ⇐⇒ ∃g∈P(a)∃h∈P(b) ag = gb and bh = ha, +(1.5) +where for a ̸= 0, P(a) = {g ∈ S1 : ∀m∈S1 (ma ̸= 0 ⇒ (ma)g ̸= 0)}, and P(0) = {1}. (See [5, Section 2] for a +motivation of this definition.) The relation ∼c is an equivalence, it does not reduce to S × S if S has a zero, +and it is equal to ∼o if S does not have a zero. +The relations ∼o, ∼∗ +p, and ∼c are not satisfactory as conjugacies when applied to inverse semigroups. +Let S be an inverse semigroup. Then the following relation ∼i on S is a natural extension of the group +conjugacy [2]: +a ∼i b ⇐⇒ ∃g∈S1 g−1ag = b and gbg−1 = a. +(1.6) +However, none of the relations ∼o, ∼∗ +p, or ∼c reduces to ∼i when S is an inverse semigroup. +In 2018, the fourth author [38] defined a conjugacy ∼n on any semigroup S by (∼n) above, that is, +a ∼n b ⇐⇒ ∃g,h∈S1 ( ag = gb, bh = ha, hag = b, and gbh = a ) . +(1.7) +The relation ∼n is an equivalence relation on any semigroup S, it does not reduce to S × S if S has a zero, +and it coincides with ∼i if S is an inverse semigroup. In fact, it is the smallest of all conjugacies defined up +to this point for general semigroups. For these reasons, we will call ∼n the natural conjugacy for semigroups. +2 + +Note that each of the relations (1.1)–(1.7) reduces to group conjugacy when S is a group. However, +assuming we require conjugacy to be an equivalence relation on general semigroups, only ∼∗ +p, ∼o, ∼c, and +∼n can provide possible definitions of conjugacy. +There are equivalence relations, however, that can serve as conjugacies for special classes of semigroups. +For example, as we have already mentioned, each of ∼l and ∼p can serve as a conjugacy in the class of +free semigroups (in which they coincide). Another such relation, called trace conjugacy, originally defined +for finite monoids, defines a notion of conjugacy in the class of epigroups [4]. A semigroup S is called an +epigroup if for every a ∈ S, there exists a positive integer n such that an belongs to a subgroup of S, that +is, the H-class H = Han of an is a group (see §2.4 for more details). We denote by aω the identity in the +group H [54, §2], and we set aω+1 = aωa (which is also an element of H). Every finite semigroup, or more +generally, every periodic semigroup S is an epigroup, and in this case, aω itself is a power of a. We define +the relation ∼tr on any epigroup S as follows [4]: +a ∼tr b ⇐⇒ ∃g,h∈S1 ghg = g, hgh = h, gh = aω, hg = bω, and haω+1g = bω+1. +(1.8) +The relation ∼tr, called trace conjugacy, is an equivalence relation on any epigroup. Its definition was inspired +by the representation theory of finite monoids (see [55] for details). +In any semigroup, we have +∼n ⊆ ∼∗ +p ⊆ ∼o and ∼n ⊆ ∼c ⊆ ∼o, +and, with respect to inclusion, ∼∗ +p and ∼c are not comparable [38, Prop. 2.3]. For detailed comparison and +analysis in various classes of semigroups, of the conjugacies ∼∗ +p, ∼o, ∼c, and ∼tr, see [4]. +As noted above, the aim of this paper is to study conjugacy ∼n in various classes of semigroups. In +§2.1, we provide various alternative definitions of ∼n, which we will use throughout the paper. It was stated +in [4] that “. . . in general, Green’s relations and the conjugacies under consideration are not comparable with +respect to inclusion.” However, in §2.2, we will show a very nice feature of ∼n, namely that in any semigroup, +∼n is included in Green’s relation D, and that ∼n and D coincide when restricted to idempotents. In §2.3– +2.4, we study ∼n in inverse and stable semigroups, and in epigroups and completely regular semigroups. In +§2.5, we characterize ∼n in several well-known semigroups of transformations. The definition of ∼n was not +available during the work that led to [4], so this section can be viewed as an extension of [4] that includes the +investigation of properties of ∼n. In particular, it seems clear that ∼n has very nice features, when compared +with the notions treated in [4]. +The next three sections contain the most important results of this paper. In §3, we show how the notion +of the natural conjugacy ∼n leads to the definition of partial inner automorphisms of an arbitrary semigroup +(in analogy with the inner automorphisms of an arbitrary group). +Therefore, we are able to assign to +each semigroup (linear, topological, or any other kind) a natural inverse semigroup that in many cases will +encode important information about the original semigroup and will hopefully be tractable using techniques +of inverse semigroup theory. In particular, we describe this inverse semigroup for the full transformation +monoid and for a Rees matrix semigroup. Section §4 characterizes ∼n in several finite partition monoids, +namely the partition monoid itself, the Brauer monoid and the partial Brauer monoid. We also characterize +the other notions of conjugation (∼tr, ∼∗ +p, ∼o, and ∼c) in these monoids. Finally, in §5, we characterize ∼n +in the finite polycyclic monoids, and give closed formulas for the conjugacy growth series of the polycyclic +monoid for ∼n, ∼∗ +p, and ∼o. +2 +General results on ∼n +The goal of this section is to study ∼n in a manner analogous to what was carried out for the other notions +in [4]. +2.1 +Alternative definitions of ∼n +For a semigroup S, a, b ∈ S and g, h ∈ S1, consider the following equations. +3 + +(i) +ag = gb +(ii) +bh = ha +(iii) +hag = b +(iv) +gbh = a +(v) +hg · b = b +(vi) +gh · a = a +(vii) +b · hg = b +(viii) +a · gh = a +Our definition of ∼n is based on the set {(i),(ii),(iii),(iv)}. We now give some alternative characterizations +which will be useful later. In particular, we could have defined ∼n less symmetrically. +Lemma 2.1. Let S be a semigroup, and let a, b ∈ S and g, h ∈ S1. Then: +(a) +(i) =⇒ ( (iii) ⇐⇒ (v) ); +(b) +(i) =⇒ ( (iv) ⇐⇒ (viii) ); +(c) +(ii) =⇒ ( (iv) ⇐⇒ (vi) ); +(d) +(ii) =⇒ ( (iii) ⇐⇒ (vii) ); +(e) +{(iii),(vi)} =⇒ {(i),(v)}; +(f) +{(iv),(v)} =⇒ {(ii),(vi)}; +(g) +{(iv),(vii)} =⇒ {(i),(viii)}; +(h) +{(iii),(viii)} =⇒ {(ii),(vii)}. +Proof. If (i) holds, then hg · b = hag and a · gh = gbh. The first of these implies (a), the second implies (b). +If (ii) holds, then gh · a = gbh and b · hg = hag. The first of these implies (c), the second implies (d). +For (e), ag = ghag = gb and then (v) follows from (a). For (f), bh = hgbh = ha and then (vi) follows +from (c). For (g), gb = gbhg = ag and then (viii) follows from (b). For (h), ha = hagh = bh and then (vii) +follows from (d). +Proposition 2.2. Let S be a semigroup, and let a, b ∈ S and g, h ∈ S1. Each of the following sets of +equations implies all of (i)–(viii), and thus a ∼n b. +(1) +{(i),(iii),(iv)} +(2) +{(i),(iii),(viii)} +(3) +{(i),(v),(viii)} +(4) +{(ii),(iii),(iv)} +(5) +{(ii),(iii),(vi)} +(6) +{(ii),(iv),(vii)} +(7) +{(iii),(vi),(viii)} +(8) +{(iv),(v),(vii)} +Proof. Each case follows from tracking implications in Lemma 2.1. We prove case (1) and leave the rest to +the reader. Thus assume (i),(iii),(iv) hold. Then (v) and (viii) hold by parts (a) and (b) of Lemma 2.1. +Then (ii) holds by part (f), and so (vi) and (vii) hold by parts (c) and (d). +It turns out that any subset of {(i),. . . ,(viii)} which is sufficient to prove all eight equations must contain +one of the subsets listed in Proposition 2.2. We omit the unenlightening list of counterexamples necessary +to establish this claim. +For a semigroup S, if a, b ∈ S satisfy a ∼n b, then there exist g, h ∈ S1 satisfying all of the conditions +(i)–(viii). For brevity, we will say that g, h are conjugators for a, b. We shall also use (i)–(viii) freely in +calculations. +As already noted, we refer to ∼n as natural conjugacy or just n-conjugacy, for short. For a ∈ S we write +[a]n = {b ∈ S : b ∼n a} for the conjugacy class of a relative to ∼n. +Remark 2.3. Note that in any semigroup with a zero, [0]n = {0}, and in any monoid M, [1]n = {gh ∈ M : +hg = 1}. +4 + +2.2 +Conjugacy ∼n and Green’s relations +If S is a semigroup and a, b ∈ S, we say that a L b if S1a = S1b, a R b if aS1 = bS1, and a J b if S1aS1 = +S1bS1. We define H as the intersection of L and R, and D as the join of L and R, that is, the smallest +equivalence relation on S containing both L and R. These five equivalence relations are known as Green’s +relations [35, p. 45]. The relations L and R commute [35, Proposition 2.1.3], and consequently D = L ◦ R = +R ◦ L. If S is finite, then D = J [35, Proposition 2.1.4]. Green’s relations are one of the most important +tools in studying semigroups. +Because D = R ◦ L, we may express D equationally as follows: +a D b ⇐⇒ ∃g1,g2,h1,h2∈S1( ag1 = g2b, +ag1h1 = a, +h2g2b = b ) . +Comparing this observation with Proposition 2.2, we immediately have the following. +Proposition 2.4. In a semigroup, ∼n ⊆ D. +Example 2.5. From Proposition 2.4 and [38, Prop. 2.3], we have ∼n ⊆ D ∩ ∼p ∩ ∼c. (Although the cited +reference states ∼n ⊆ ∼∗ +p, it actually proves the stronger result ∼n ⊆ ∼p.) This inclusion is strict in general. +Consider the monoid S defined by the Cayley table +· +0 +1 +2 +3 +4 +5 +6 +7 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +1 +2 +3 +4 +5 +6 +7 +2 +0 +2 +6 +6 +3 +2 +6 +2 +3 +0 +3 +6 +6 +3 +2 +6 +2 +4 +0 +4 +6 +6 +4 +5 +6 +5 +5 +0 +5 +6 +6 +4 +5 +6 +5 +6 +0 +6 +6 +6 +6 +6 +6 +6 +7 +0 +7 +2 +3 +4 +5 +6 +7 +We have 2 = 3 · 7 and 3 = 7 · 3, so 2 ∼p 3. Next, 2 · 4 = 3 and 3 · 5 = 2, and so 2 R 3 (and thus certainly +2 D 3). Finally, for all x, y ∈ S\{0}, xy ̸= 0, and thus x ∼c y in S if and only if x ∼o y in S\{0}. In the +latter semigroup, ∼o is the universal relation because 6 is a zero, and so 2 ∼c 3. However, 2 ≁n 3 because, +as can be checked, there are no suitable conjugators. +Next we consider how n-conjugacy interacts with idempotents. First we note that if an n-conjugacy class +contains an idempotent, then it consists only of idempotents. +Proposition 2.6. Let S be a semigroup, let e, a ∈ S, and assume e is an idempotent. If e ∼n a, then a is +also an idempotent. +Proof. Let g, h ∈ S1 be conjugators for a and e. Then aa = aagh = ageh = geeh = geh = agh = a. +Restricted to idempotents, n-conjugacy and the D-relation turn out to coincide. A pair g, h of elements +of a semigroup S are said to be mutually inverse if ghg = g and hgh = h. +Theorem 2.7. Let S be a semigroup and let e, f ∈ S be idempotents. Then e ∼n f if and only if e D f. +When this is the case, there exist mutually inverse conjugators g, h of e, f in the same D-class as e, f. +Proof. One direction is covered by Proposition 2.4, so assume e D f. We just follow the proof of [35, Thm. +2.3.4], noting that the construction therein gives mutually inverse conjugators. Indeed, by assumption, there +exist g, h1, h2 ∈ S1 such that eg = g = gf, gh1 = e and h2g = f. +(Here we are using the fact that +an idempotent e is a left identity element for the R-class Re and a right identity element for the L-class +Le [35, Prop. 2.3.3].) Set h = fh1e and check that gh = gfh1e = gh1e = ee = e and hg = fh1eg = fh1g = +h2gh1g = h2eg = h2g = f. Since eg = gf, egh = e and hgf = f, it follows from Proposition 2.2 that e ∼n f +with g, h as conjugators. Finally ghg = eg = g and hgh = fh = h. +5 + +Recall that a band is a semigroup in which every element is an idempotent. +Corollary 2.8. In any band, ∼n = D. +We conclude this section with a brief discussion of the two extreme cases: where n-conjugacy is the +universal relation, that is, ∼n= S × S, and where ∼n is the equality relation. In neither case will we arrive +at a complete characterization, but each case still entails interesting necessary conditions. +A semigroup is bisimple if D is the universal relation. A rectangular band is an idempotent semigroup +satisfying xyx = x; every rectangular band is isomorphic to one of the form I × J for sets I, J with +multiplication (i, j) · (k, ℓ) = (i, ℓ). +Proposition 2.9. If S is a semigroup in which ∼n is universal, then S is bisimple. If, in addition, S has +an idempotent, then S is a rectangular band. +Proof. The first assertion follows from Proposition 2.4 and the second follows from Proposition 2.6. +At the other extreme, we have the following. +Proposition 2.10. Let S be a semigroup in which ∼n is the equality relation. Then each D-class has at +most one idempotent, and each regular D-class is an H-class. +Proof. The first assertion follows from Theorem 2.7. For the second, assume e is an idempotent and c D e. +Then c is regular and hence there exists an idempotent f such that c L f. But then f D e and so by assumption +e = f, that is, c L e. By a similar argument, c R e and so c H e. +As noted in the introduction, in ( [4], §3), it was shown that Green’s relations and the four notions of +conjugation considered are not particularly well related. The results of this subsection show that ∼n tells a +completely different story. (See also Theorem 3.4 and Corollary 3.6 below.) +2.3 +Conjugacy ∼n in inverse and stable semigroups +As we pointed out in §1, of the known conjugacy relations for general semigroups, ∼n is the only one that +coincides with the conjugacy ∼i (1.6) in inverse semigroups. This was first proved in [38, Thm. 2.6] using the +Wagner-Preston representation of inverse semigroups as semigroups of partial injective transformations [35, +Thm. 5.1.7]. Here we present a purely equational proof. +Proposition 2.11. In inverse semigroups, ∼n = ∼i. +Proof. Let S be an inverse semigroup. The inclusion ∼i ⊆ ∼n follows from [2, Prop. 1.3], but we give a +brief proof here to keep the discussion self-contained. Suppose a ∼i b for some a, b ∈ S. Then g−1ag = b +and gbg−1 = a for some g ∈ S1. We have a · gg−1 = gbg−1gg−1 = gbg−1 = a and gg−1 · a = gg−1gbg−1 = +gbg−1 = a. Now condition (7) of Proposition 2.2 holds with h = g−1 and so a ∼n b. +Now suppose a ∼n b for some a, b ∈ S, and let g, h ∈ S1 be conjugators. Then +g−1 · ag +���� = g−1g · b +(by (i)) += g−1g · bb−1 +� +�� +� ·b += +b +���� b−1 · g−1g · b +(since idempotents commute) += hg · bb−1 · g−1g· +� +�� +� b +(by (v)) += h · gg−1g +� �� � · bb−1b +� �� � +(since idempotents commute) += hg · b += b +(by (v)) +The equality gbg−1 = a is proved similarly, and so a ∼i b. +6 + +The natural partial order (or Mitsch order) ≤ in a semigroup S is defined as follows: +a ≤ b ⇐⇒ ∃s,t∈S1 sa = a = sb and at = a = bt ; +see [49]. We now consider how natural conjugacy and the natural partial order interact. +A semigroup S is left stable if, for all a, b ∈ S, S1a ⊆ S1ab implies S1a = S1ab, that is, a L ab. This can +be equivalently formulated as a ∈ S1ab implies ab ∈ S1a for all a, b ∈ S. Right stability is defined dually, and +a semigroup is said to be stable if it is both left and right stable [15, Vol. I, p. 31]. Every periodic semigroup, +and in particular every finite semigroup, is stable. +Theorem 2.12. Let S be a stable semigroup. Then ∼n ∩ ≤ is the identity relation. +Proof. Assume a ∼n b and a ≤ b for some a, b ∈ S. Let g, h ∈ S1 be conjugators for a, b and let s, t ∈ S1 +witness a ≤ b, that is, sa = a = sb and at = a = bt. We have a = sb = shag. By stability, there exists u ∈ S1 +such that ag = ua. Thus ua = uat = agt = gbt = ga, hence ag = ga. Now a = bt = hgbt = hga = hag = b, +as claimed. +2.4 +Conjugacy ∼n in epigroups and completely regular semigroups +An element a of a semigroup S is an epigroup element (or a group-bound element) if there exists a positive +integer n such that an is contained in a subgroup of S. The smallest n for which this is satisfied is the index +of a, and for all k ≥ n, ak is contained in the group H-class of an. The set of all epigroup elements of S is +denoted by Epi(S) and the subset consisting of elements of index no more than n is denoted by Epin(S). +We have Epim(S) ⊆ Epin(S) for m ≤ n and Epi(S) = � +n≥1 Epin(S). The elements of Epi1(S) are called +completely regular (or group elements); thus Epi1(S) is the union of all group H-classes of S. +For a ∈ Epin(S), let e denote the identity element of the group H-class H of an. Then ae = ea is in H. +The pseudo-inverse a′ of a is a′ = (ae)−1, the inverse of ae in the group H [54, (2.1)]. We have the following +characterization: a ∈ Epi(S) if and only if there exists a positive integer n and a (unique) a′ ∈ S such that +the following hold [54, §2]: +a′aa′ = a′ , +aa′ = a′a , +an+1a′ = an, +(2.9) +where the smallest n such that an+1a′ = an is the index of a. If a is an epigroup element, then so is a′ +with a′′ = aa′a. The element a′′ is always completely regular and a′′′ = a′. We set aω = aa′. We also +have aω = a′′a′ = a′a′′, (a′)ω = (a′′)ω = aω, and more generally aω = (aa′)m = (a′)mam = am(a′)m, for all +m > 0. For finite semigroups, aω is usually called the idempotent power of a. +A semigroup S is said to be an epigroup if Epi(S) = S. If Epi1(S) = S (that is, if S is a union of groups), +then S is called a completely regular semigroup. For n > 0, the class En consists of all epigroups S such that +S = Epin(S); thus E1 is the class of completely regular semigroups. +We will need the following lemma. +Lemma 2.13. ([4, Lem. 4.1]) Let S be a semigroup and suppose that uv, vu ∈ Epi(S) for some u, v ∈ S. +Then +(uv)′u = u(vu)′ . +(2.10) +As a relation on the set Epi1(S) of completely regular elements of a semigroup S (that is, as the restriction +to Epi1(S) × Epi1(S)), ∼p is transitive (that is, ∼p = ∼∗ +p) and coincides with ∼tr [4, Cor. 4.9]. We extend +this result to ∼n. +Theorem 2.14. Let S be a semigroup. Then on Epi1(S), ∼n = ∼p. +Proof. The inclusion ∼n ⊆ ∼p holds in all semigroups [38]. For the converse, suppose a ∼p b, where a, b ∈ +Epi1(S). Then a = uv and b = vu for some u, v ∈ S1. Set g = u and h = v(uv)−1. Then ag = uvu = gb, +bh = vuv(uv)−1 = v(uv)−1uv = ha and hag = v(uv)−1uvu = vu(vu)−1vu = vu = b, using Lemma 2.13. +Thus a ∼n b by Proposition 2.2. +7 + +Corollary 2.15. In a completely regular semigroup, ∼n = ∼p. +Example 2.16. An epigroup in which ∼n = ∼p need not be completely regular. For example, a null semi- +group S (S has a zero and ab = 0 for all a, b ∈ S) of order greater than 1 is not completely regular, but ∼p, +and hence ∼n, are both identity relations in S. +Theorem 2.17. Let S be a regular epigroup. Then S is completely simple if and only if ∼n = ∼o. +Proof. From [4, Thm. 4.22], we know that a regular epigroup is completely simple if and only if ∼p = ∼o. This +is stated in the cited reference with the additional assumption that the epigroup does not have a zero, and +we now take the opportunity to point out that this assumption was never used in the proof of [4, Thm. 4.22]. +Suppose that S is completely simple. Then S is completely regular [35, Prop. 4.1.2], and so ∼n = ∼p, +by Corollary 2.15, and ∼p = ∼o, by [4, Thm. 4.22], so ∼n = ∼o. Conversely, suppose that ∼n = ∼o. Then +∼p = ∼o since ∼n ⊆ ∼p ⊆ ∼o in any semigroup, and so S is completely simple by [4, Thm. 4.22]. +Theorem 2.18. Let S be a semigroup in which ∼n = ∼p and let c be a regular epigroup element. Then c is +completely regular. +Proof. Let c∗ denote an inverse of c, that is, cc∗c = c and c∗cc∗ = c∗. Let c′ denote the epigroup pseudoinverse +of c, so cn+1c′ = cn for some n > 1. We will prove that cnc′ = cn−1. It will then follow by induction that +c ∈ Epi1(S), that is, c is completely regular. +Since c∗c · c ∼p c · c∗c = c and ∼n = ∼p, it follows that c∗c2 ∼n c. Thus there exist conjugators g, h ∈ S1 +for c∗c2, c. By Corollary 3.3, g, h are also conjugators for (c∗c2)k, ck for any positive integer k. Note that +(c∗c2)k = c∗ck+1. Thus gck = c∗ck+1g, which we will use multiple times in the calculation that follows. We +have +gcnc′ = c∗cn+1gc′ += c∗c · cngc′ += c∗c · c′cn+1gc′ += c∗c′ · cn+2gc′ += c∗c′ · c c∗cn+2g +� +�� +� c′ = c∗c′cg cn+1c′ +� �� � += c∗c′cgcn += c∗c′ cc∗cn+1 +� �� � g += c∗ c′cn+1 +� �� � g += c∗cng = gcn−1 . +Thus cnc′ = hgcnc′ = hgcn−1 = cn−1, as claimed. +Combining Theorem 2.18 with Corollary 2.15, we obtain the following result. +Corollary 2.19. Let S be a regular epigroup. Then S is completely regular if and only if ∼n = ∼p. +Form the previous result and [4, Theorem 4.21] we get the following. +Corollary 2.20. Let S be a completely simple semigroup. Then ∼n = ∼p = ∼∗ +p = ∼tr = ∼o. +For an element a in a completely regular semigroup S, it is customary to denote the unique idempotent +aω in the H-class of a by a0, that is, a0 = aa−1 = a−1a. +We know by Theorems 2.7 and 3.4 that group H-classes He and Hf, where e and f are idempotents, are +isomorphic via mutually inverse conjugators of e, f in the D-class of e and f. The next result shows that we +may select those conjugators to be the same as those for a, b for any a ∈ He and b ∈ Hf such that a ∼n b. +Proposition 2.21. Let a, b be completely regular elements of a semigroup S such that a ∼n b. Then there +exist mutually inverse conjugators in the D-class of a and b. +Proof. Let e = a0, f = b0, and let g, h ∈ S1 be conjugators of a, b. By Theorem 3.4, φg,h is an isomorphism +of Ha onto Hb. In particular, e ∼n f with the same conjugators g, h, so eg = gf, fh = he, heg = f, and +gfh = e. Set ¯g = eg and ¯h = fh. Then a¯g = aeg = ag = gb = gfb = ¯gb, a¯g¯h = aegfh = aee = e, and ¯h¯gb = +fhegb = ffb = b. Thus ¯g, ¯h are conjugators of a, b. Finally, ¯g¯h¯g = egfheg = egff = egf = eeg = eg = ¯g +and ¯h¯g¯h = fhegfh = fffh = fh = ¯h. +8 + +We also have a characterization of ∼n in a completely regular semigroup S in terms of a single conjugator +g ∈ S1 instead of a pair g, h ∈ S1. First we need a bit of notation and a lemma. Note that for positive +integers m, (am)−1 = (a−1)m, and so we may denote this by a−m unambiguously. +Lemma 2.22. Let S be a completely regular semigroup and suppose a, b ∈ S, g ∈ S1 satisfy ag = gb. Then +for all integers m, amg = gbm. +Proof. We first verify the case m = 0: +a0g = a−1 ag +���� = a−1gb = a−1 gb +���� b0 = a0gb0 = a0 gb +���� b−1 = a0agb−1 = ag +���� b−1 = gbb−1 = gb0 . +Next we check m = −1: +a−1g = a−1a0g = a−1gb0 = a−1 gb +���� b−1 = a−1agb−1 = a0gb−1 = gb0b−1 = gb−1 . +The remaining cases follow by an easy induction. +Theorem 2.23. Let S be a completely regular semigroup. Then, for all a, b ∈ S, +a ∼n b ⇐⇒ ∃g ∈ S1 ( ag = gb, g0a = a, bg0 = b ). +Proof. Fix a, b ∈ S, assume a ∼n b and let g, h ∈ S1 be conjugators. Then +g0a = g0gha = gha = a +and +bg0 = bhgg0 = bhg = b , +using (vi) and (vii). +For the converse, assume that there exists g ∈ S1 such that ag = gb, g0a = a and bg0 = b. +Set +h = bg−1a−1. We use Lemma 2.22 (with m = −1) in the following: +hg = bg−1 a−1g +� �� � = bg−1gb−1 = bg0 +���� b−1 = bb−1 = b0 +and +gh = gb +���� g−1a−1 = agg−1a−1 = ag0 a−1a +� �� � a−1 = a g0a +���� a−1a−1 = aaa−1a−1 = a0 . +Thus hg · b = b and a · gh = a, and so condition (3) of Proposition 2.2 is satisfied. Therefore a ∼n b. +We have already seen that n-conjugacy is equivalent to i-conjugacy in inverse semigroups. Now we discuss +the analog of i-conjugacy for completely regular semigroups, this time using the commuting inverse. For a +completely regular semigroup S, we define ∼i by: +a ∼i b ⇐⇒ ∃g ∈ S1( g−1ag = b and gbg−1 = a ) . +The relation ∼i cannot be regarded as a conjugacy in the class of completely regular semigroups because it +is not, in general, transitive in this class. +Example 2.24. The following multiplication table defines a smallest example of a completely regular semi- +group in which ∼i is not transitive: +· +0 +1 +2 +3 +4 +5 +6 +0 +0 +0 +0 +0 +0 +0 +0 +1 +1 +1 +1 +1 +1 +1 +1 +2 +2 +2 +2 +2 +2 +2 +2 +3 +0 +1 +0 +3 +3 +5 +5 +4 +2 +1 +2 +4 +4 +6 +6 +5 +1 +0 +1 +5 +5 +3 +3 +6 +1 +2 +1 +6 +6 +4 +4 +9 + +The commuting inverse is just the identity map: x−1 = x. Set a = 0, b = 1, c = 2, g = 5, and h = 6. We +have g−1ag = 5 · 0 · 5 = 1 = b and gbg−1 = 5 · 1 · 5 = 0 = a, and so a ∼i b. Also h−1bh = 6 · 1 · 6 = 2 = c and +hch−1 = 6 · 2 · 6 = 1 = b, and so b ∼i c. Suppose, however, that x−1ax = c and xcx−1 = a. Then, we must +have x = 2 or x = 4, but 2c2 = 2 · 2 · 2 = 2 ̸= 0 = a and 4c4 = 4 · 2 · 4 = 2 ̸= 0 = a, so a ̸∼i c. +However, one can check that in the variety of completely regular semigroups defined by the identity +xx(yxx)−1 = x(yx)−1 (which includes Clifford semigroups), the relation ∼i is transitive. In this class, ∼i is +strictly included in ∼n. +We conclude this subsection by characterizing n-conjugacy in 0-Rees matrix semigroups. +Theorem 2.25. Let Γ be a group, I and Λ two nonempty sets, and P a Λ×I matrix with entries in Γ∪{0}. +Let M0(G; I, Λ; P) be the 0-Rees matrix semigroup induced by Γ, I, Λ and P. Let (A, a, α), (B, b, β) ∈ +M0(G; I, Λ; P) \ {0}. Then +(A, a, α) ∼n (B, b, β) iff pβB ̸= 0 ̸= pαA & ∃g∈Γ pβBb = g−1apαAg. +Proof. We start by proving the direct implication. By definition, (A, a, α) ∼n (B, b, β) implies that there +exist (G, g, γ), (H, h, η) ∈ M0(G; I, Λ; P) such that +(A, a, α)(G, g, γ) = (G, g, γ)(B, b, β) +(B, b, β) = (H, h, η)(A, a, α)(G, g, γ) +(A, a, α) = (G, g, γ)(B, b, β)(H, h, η) . +From the first equality we get G = A and γ = β, from the second we get H = B, and from the third we get +η = α. Therefore, +(A, apαAg, β) = (A, a, α)(A, g, β) = (A, g, β)(B, b, β) += (A, gpβBb, β) +(B, b, β) = (B, h, α)(A, a, α)(A, g, β) = (B, hpαAapαAg, β) +(A, a, α) = (A, g, β)(B, b, β)(B, h, α) = (A, gpβBbpβBh, α) . +The second line of equalities implies that pαA ̸= 0 (otherwise (B, b, β) would equal 0 in M0(G; I, Λ; P), +contrary to our assumptions). Similarly, the third line implies that pβB ̸= 0. The first line implies that +apαAg = gpβBb, that is, g−1apαAg = pβBb as claimed. +Conversely, let (A, a, α), (B, b, β) ∈ M0(G; I, Λ; P) such that pβB ̸= 0 ̸= pαA and there exists g ∈ Γ such +that pβBb = g−1apαAg. Consider the elements (A, g, β), (B, p−1 +βBg−1p−1 +αA, α) ∈ M0(G; I, Λ; P). Then +(A, a, α)(A, g, β) = (A, apαAg, β) +apαAg=gpβBb += +(A, gpβBb, β) = (A, g, β)(B, b, β). +On the other hand, +(B, p−1 +βBg−1p−1 +αA, α)(A, a, α)(A, g, β) = (B, p−1 +βBg−1p−1 +αApαAapαAg, β) = (B, p−1 +βBg−1apαAg, β) = (B, b, β) . +Similarly, +(A, g, β)(B, b, β)(B, p−1 +βBg−1p−1 +αA, α) = (A, gpβBbpβBp−1 +βBg−1p−1 +αA, α) = (A, gpβBbg−1p−1 +αA, α) = (A, a, α) . +The result follows. +2.5 +Conjugacy ∼n in semigroups of transformations +Let X be a non-empty set. In [38], n-conjugacy was characterized in the semigroup P(X) of partial transfor- +mations on X, the semigroup T (X) of full transformations on X, the symmetric inverse semigroup I(X) of +partial injective transformations on X, and the semigroup J (X) of full injective transformation on X. In this +10 + +section, we describe ∼n for other basic transformation semigroups. As in [38], we will use the representation +of transformations by directed graphs. +A directed graph (or a digraph) is a pair Γ = (A, E) where A is a set (not necessarily finite and possibly +empty) and E is a binary relation on A. Any element x ∈ A is called a vertex of Γ, and any pair (x, y) ∈ E +is called an edge of Γ. A vertex x of Γ is called initial if there is no vertex y such that (y, x) ∈ E; x is called +terminal if there is no vertex y such that (x, y) ∈ E. Let Γ = (A, E) and Υ = (B, F) be digraphs. A function +φ : A → B is called a homomorphism from Γ to Υ if for all x, y ∈ A, (x, y) ∈ E implies (xφ, yφ) ∈ F. A +bijection φ : A → B is called an isomorphism from Γ to Υ if for all x, y ∈ A, (x, y) ∈ E if and only if +(xφ, yφ) ∈ F. We will say that Γ and Υ are isomorphic, written Γ ∼= Υ, if there exists an isomorphism from +Γ to Υ. +Let α ∈ P(X). We denote by dom(α) and im(α) the domain and image of α, respectively. We define +the span of α, written span(α), to be dom(α) ∪ im(α). Any α ∈ P(X) can be represented by the digraph +Γ(α) = (A, E), where A = span(α) and for all x, y ∈ A, (x, y) ∈ E if and only if x ∈ dom(α) and xα = y. +(We apply transformations on the right and compose from left to right: x(αβ) = (xα)β.) Any digraph +Γ = (A, E) such that Γ = Γ(α) for some α ∈ P(X), where A ⊆ X, is called a functional digraph. For the +structure of functional graphs, see [5]. +The following definitions and theorem are fundamental to studying n-conjugacy in semigroups of trans- +formations. +Definition 2.26. Let Γ = (A, E) be a digraph. An initial vertex x of Γ will be called bottom initial if for +all vertices y, z of Γ, if (x, y) ∈ E and (z, y) ∈ E, then z is initial. +Let α ∈ P(X), x be a bottom initial vertex of Γ(α) = (A, E), and y be a unique vertex in Γ(α) such that +(x, y) ∈ E (y = xα). We will call the set yα−1 = {z ∈ A : (z, y) ∈ E} the initial bundle in Γ(α) containing x. +Note that every vertex in an initial bundle in Γ(α) is bottom initial. +For example, the functional digraph presented in Figure 2.1 on the left has four initial bundles. +Definition 2.27. ([38, Def. 3.1]) Let Γ = (A, E) and Υ = (B, F) be digraphs. A homomorphism φ : A → +B is called a restricted homomorphism (or an r-homomorphism) from Γ to Υ if: +(1) for every terminal vertex x of Γ, xφ is a terminal vertex of Υ; +(2) for every bottom initial vertex x of Γ, xφ is an initial vertex of Υ. +Definition 2.28. ([38, Def. 3.4]) Let S be a subsemigroup of P(X). We will say that S is closed under +restrictions to spans if for all α, β ∈ S such that span(α) ⊆ dom(β), β|span(α) ∈ S. +Note that every semigroup of full transformations on X is closed under restrictions to spans. +Theorem 2.29. ([38, Thm. 3.5]) Let S be a subsemigroup of P(X) that is closed under restrictions to +spans, and let α, β ∈ S. Then α ∼n β in S if and only if there are φ, ψ ∈ S1 such that φ is an r-homomorphism +from Γ(α) to Γ(β), ψ is an r-homomorphism from Γ(β) to Γ(α), y(φψ) = y for every non-initial vertex y of +Γ(α), and v(ψφ) = v for every non-initial vertex v of Γ(β). +Conjugacy ∼n in P(X) and T (X) was characterized in [38] in terms of a trim of a functional digraph. +Definition 2.30. ([38, Def. 4.3]) For α ∈ P(X), we define a trim of Γ(α) as a digraph obtained from Γ(α) +by removing all initial vertices except that we retain exactly one vertex from each initial bundle. Any two +trims of Γ(α) are isomorphic. We denote by Γt(α) any trim of Γ(α). +In the semigroups P(X) and T (X), α ∼n β if and only if Γt(α) ∼= Γt(β) [38, Thms. 4.8 and 4.11]. The +concept of a trim of Γ(α), where α ∈ P(X), can be replaced by a simpler concept of the prune of Γ(α). +Definition 2.31. Let α ∈ P(X). The digraph Γp(α) obtained from Γ(α) by removing all initial vertices of +Γ(α) will be called the prune of Γ(α). +11 + +• +• +• +• +• +• +• +• +• +• • • • • +... +• +• +• +• +• +• +• +• +� +� +�⑧ +⑧ +⑧⑧ +⑧⑧ +� �❄❄❄❄❄❄ +�✞✞✞✞✞ +� �✼✼✼✼✼ +�✞✞✞✞✞ +�✗✗✗✗ +�✬✬✬✬ +�✼✼✼✼✼ +�⑧ +⑧ +⑧⑧ +⑧⑧ +�❄❄❄❄❄❄ +� +� +� +� +�✿✿✿✿ +�❄❄❄ �❄❄❄ +• +• +• +• +• +• +• +... +• +• +• +• +• +• +• +� +� +�⑧ +⑧⑧ +⑧⑧ +⑧ +�❄❄❄❄❄❄ +�✞✞✞✞✞ +�✼✼✼✼✼ +�⑧ +⑧⑧ +⑧⑧ +⑧ +� +� +� +� +�❄❄❄ �❄❄❄ +• +• +• +• +• +... +• +• +• +• +• +� +� +�⑧⑧ +⑧⑧ +⑧ +⑧ +�❄❄❄❄❄❄ +�⑧⑧ +⑧⑧ +⑧ +⑧ +� +� +� +�❄❄❄ +Figure 2.1: A functional digraph (left), its trim (middle), and its prune (right). +The prune of Γ(α), where α ∈ P(X), is a subgraph of a trim of Γ(α) since in the latter some initial vertices +of Γ(α) may be preserved. Note that the prune of Γ(α) is unique (not just unique up to isomorphism). +Figure 2.1 presents an example of a functional digraph, its trim, and its prune. +For a function f : A → B and A1 ⊆ A, denote by f|A1 the restriction of f to A1. +Proposition 2.32. For all α, β ∈ P(X), Γt(α) ∼= Γt(β) if and only if Γp(α) ∼= Γp(β). +Proof. Let α, β ∈ P(X) with Γt(α) = (At, Et), Γp(α) = (Ap, Ep), Γt(β) = (Bt, Ft), and Γp(β) = (Bp, Fp). +Suppose Γt(α) ∼= Γt(β) and let σ : At → Bt be an isomorphism from Γt(α) to Γt(β). The set Ap consists +of the non-initial vertices of Γt(α), and the subgraph of Γt(α) induced by Ap is equal to Γp(α). +The +corresponding statement is true for β. Since σ maps the set of non-initial vertices of Γt(α) onto the set of +non-initial vertices of Γt(β), it follows that σ|Ap is an isomorphism from Γp(α) to Γp(β). +Conversely, suppose Γp(α) ∼= Γp(β) and let δ : Ap → Bp be an isomorphism from Γp(α) to Γp(β). Let +y1, . . . , yk, where k ≥ 0, be the initial vertices of Γp(α). Then v1, . . . , vk, where vi = yiδ for each i, are +the initial vertices of Γp(β). By the definitions of a trim and the prune of a functional graph, for every +i ∈ {1, . . . , k}, there is a unique initial vertex xi of Γt(α) such that (xi, yi) ∈ E, and x1, . . . , xk are the only +initial vertices of Γ(α). Similarly, for every i ∈ {1, . . . , k}, there is a unique initial vertex ui of Γt(β) such +that (ui, vi) ∈ E, and u1, . . . , uk are the only initial vertices of Γ(β). Hence σ : At → Bt that extends δ in +such a way that xiσ = ui, for every i ∈ {1, . . . , k}, is an isomorphism from Γt(α) to Γt(β). +The following theorem follows immediately from Proposition 2.32 and the characterizations of ∼n in +P(X) and T (X) (stated above) obtained in [38] in terms of trims. +Theorem 2.33. In the semigroups P(X) and T (X), α ∼n β if and only if Γp(α) ∼= Γp(β). +We are now ready to characterize ∼n in some transformation semigroups not considered in [38]. We will +begin with the semigroups of transformations whose image is restricted by a prescribed set. Such semigroups +have been studied extensively; see, for example, [48,50,56–58]. Let X be an arbitrary set and ∅ ̸= Y ⊆ X. +Then T (X, Y ) = {α ∈ T (X) : im(α) ⊆ Y } is a subsemigroup of T (X), consisting of transformations whose +image is restricted by Y . We will now describe n-conjugacy in T (X, Y ). +Lemma 2.34. Let S be a subsemigroup of P(X) and let α, β ∈ S. Suppose φ, ψ ∈ S1 are r-homomorphisms +as in Theorem 2.29. Let Ap and Bp be the sets of vertices of Γp(α) and Γp(β), respectively. Then φ|Ap is +an isomorphism from Γp(α) to Γp(β) and (φ|Ap)−1 = ψ|Bb. +Proof. By [38, Lem. 4.6], for every non-initial vertex y of Γ(α), yφ is not initial in Γ(β), and an analogous +statement is true for ψ. Thus, φ|Ap is a homomorphism from Γp(α) to Γp(β), and ψ|Bp is a homomorphism +from Γp(β) to Γp(α). +Moreover, φ|Ap and ψ|Bb are inverses of each other, which implies that they are +isomorphisms. +12 + +Theorem 2.35. Let X and Y be sets such that ∅ ̸= Y ⊆ X, and let α, β ∈ T (X, Y ). Then α ∼n β in +T (X, Y ) if and only if Γp(α) ∼= Γp(β), and if Z is an initial bundle in Γ(α) or in Γ(β), then Z ∩ Y ̸= ∅. +Proof. Let Γ(α) = (X, E), Γ(β) = (X, F), Γp(α) = (A, Ep), and Γp(β) = (B, Fp). Suppose α ∼n β in +T (X, Y ). Let φ, ψ ∈ T (X, Y ) be r-homomorphisms as in Theorem 2.29, where S = T (X, Y ). By Lemma 2.34, +Γp(α) ∼= Γp(β). Let Z be an initial bundle in Γ(β). Then Z = vβ−1 for some initial vertex v in Γp(β). Let +y = vψ. Then y is an initial vertex in Γp(α) (since, by Lemma 2.34, ψ|B is an isomorphism form Γp(β) to +Γp(α)), and yα−1 is an initial bundle in Γ(α) (by [38, Lem. 4.6]). Let x ∈ yα−1. Since φ is a homomorphism +and (x, y) ∈ E, we have (xφ, v) = (xφ, v(ψφ)) = (xφ, yφ) ∈ F. Thus xφ ∈ Z, and so Z ∩Y ̸= ∅ since xφ ∈ Y . +By symmetry, we have Z ∩ Y ̸= ∅ for every initial bundle Z in Γ(α). +Conversely, suppose that Γp(α) ∼= Γp(β), and if Z is an initial bundle in Γ(α) or in Γ(β), then Z ∩Y ̸= ∅. +Let δ: A → B be an isomorphism from Γp(α) to Γp(β). Let v ∈ B. If v is not initial in Γp(β), then fix +v∗ ∈ B such that (v∗, v) ∈ F. If v is initial in Γp(β), then fix v∗ ∈ Y such that (v∗, v) ∈ F (possible since +Z = {u ∈ X : (u, v) ∈ F} is an initial bundle in Γ(α), and so Z ∩ Y ̸= ∅). Define φ : X → X by +xφ = +� +xδ +if x ∈ A, +(yδ)∗ +if x is initial in Γ(α) and (x, y) ∈ E. +It is straightforward to check that φ ∈ T (X, Y ) and φ is an r-homomorphism from Γ(α) to Γ(β). Symmet- +rically, we can define ψ ∈ T (X, Y ) such that ψ is an r-homomorphism from Γ(β) to Γ(α) with vψ = vδ−1 +for every v ∈ B. Then α ∼n β in T (X, Y ) by Theorem 2.29. +Next, we consider the semigroup of full order-preserving transformations on a chain with n elements, +where n ≥ 1, say Xn = {1 < . . . < n}. Viewing Xn as a set, we denote by Tn the semigroup T (Xn). Let On +be the subset of Tn consisting of full order-preserving transformations, that is, +On = {α ∈ Tn : ∀x,y∈Xn(x ≤ y ⇒ xα ≤ yα)}. +The semigroup On has been studied in numerous papers since the 1960s (see [29, 14.5.1]). We will now +describe n-conjugacy in On. +Notation 2.36. Let α, β ∈ P(Xn). Suppose Γ′(α) = (A′, E′) and Γ′(β) = (B′, F ′) are subgraphs of Γ(α) and +Γ(β), respectively, where A′ = {x1 < . . . < xk} and B′ = {y1 < . . . < yk} (k ≥ 0). We denote by Γ′ +β(α) the +digraph obtained from Γ′(α) by replacing every vertex xi with yi. +Theorem 2.37. Let α, β ∈ On, with Γ(α) = (X, E), Γ(β) = (X, F), Γp(α) = (A, Ep), and Γp(β) = (B, Fp), +where A = {x1 < . . . < xk} and B = {y1 < . . . < ym} (k, m ≥ 0). Then α ∼n β in On if and only if k = m +and Γp +β(α) = Γp(β). +Proof. Suppose α ∼n β in On. Let φ, ψ ∈ On be r-homomorphisms as in Theorem 2.29. By Lemma 2.34, +φp = φ|A is an isomorphism from Γp(α) to Γp(β), ψp = ψ|B is an isomorphism from Γp(β) to Γp(α), and +ψp = φ−1 +p . This gives k = m. Further, Γp +β(α) = (B, E0), where (yi, yj) ∈ E0 if and only if (xi, xj) ∈ Ep. +It remains to show that E0 = Fp. Since φp preserves order, we have x1φp < . . . < xkφp, which implies +xiφp = yi for every i. The equality E0 = Fp follows since for all i, j, (xi, xj) ∈ Ep if and only if (yi, yj) = +(xiφp, xjφp) ∈ Fp. Hence Γp +β(α) = Γp(β). +Conversely, suppose that k = m and Γp +β(α) = Γp(β). Let i ∈ {1, . . . , k}. Fix y∗ +i ∈ X such that (y∗ +i , yi) ∈ F +(such a y∗ +i exists since yi is not initial in Γ(β)). Let Ai = {xj : (xj, xi) ∈ E}. Let x be an initial vertex in +Γ(α). Then xα = xi (so (x, xi) ∈ E) for some i. Note that x is bottom initial in Γ(α) if and only if Ai = ∅. +Suppose Ai ̸= ∅. Write Ai = {xj1 < . . . < xjw}, where w ≥ 1, and define mx ∈ {j1, . . . , jw} as follows: +mx = j1 if x < xj1, mx = jw if xw < x, and mx = js if xjs < x < xjs+1. Now, define φ : X → X by +xφ = + + + +yi +if x = xi, +y∗ +i +if x is bottom initial in Γ(α) (so Ai = ∅) and (x, xi) ∈ E, +ymx +if x is initial, but not bottom initial, in Γ(α) (so Ai ̸= ∅) and (x, xi) ∈ E. +13 + +Note that xiφ = yi for every i. First, we will prove that φ is an r-homomorphism from Γ(α) to Γ(β). Since +Γp +β(α) = Γp(β), (xi, xj) ∈ E if and only if (yi, yj) ∈ F, for all i and j. Moreover, for every i, (y∗ +i , yi) ∈ F +and if x is initial, but not bottom initial, in Γ(α) with xα = xi, then (ymx, yi) ∈ F (since (xmx, xi) ∈ E). It +follows that φ is a homomorphism. Since Γ(α) does not have any terminal vertices, (1) of Definition 2.27 is +vacuously satisfied. Let x be a bottom initial vertex of Γ(α) and let xi = xα (so (x, xi) ∈ E). Suppose to +the contrary that xφ is not initial in Γ(β). Then xφ = yj, for some j, and (yj, yi) = (xφ, xiφ) ∈ F. Thus +(xj, xi) ∈ E, which is a contradiction since (x, xi) ∈ E and x is bottom initial. Hence xφ is initial in Γ(β). +Therefore, φ is an r-homomorphism from Γ(α) to Γ(β). +Next, we will prove that φ ∈ On. Let x, z ∈ X with x < z, and let xi = xα and xj = zα (so (x, xi) ∈ E +and (z, xj ∈ E). Since α ∈ On, we have xi ≤ xj. We want to prove that xφ ≤ zφ. Consider three possible +cases. +Case 1. x and z are not initial in Γ(α). +Then x = xs and z = xt, for some s and t. Thus xs < xt, and so xφ = xsφ = ys < yt = xtφ = zφ. +Case 2. x or z is initial in Γ(α), and i ̸= j. +Then xi < xj, and so yi < yj. Since φ is a homomorphism from Γ(α) to Γ(β), we have (xφ, yi) = +(xφ, xiφ) ∈ F and (zφ, yj) = (zφ, xjφ) ∈ F, that is, (xφ)β = yi and (zφ)β = yj. Since β ∈ On, zφ ≤ xφ +would imply yj ≤ yi, which would contradict yi < yj. Hence xφ < zφ. +Case 3. x or z is initial in Γ(α), and i = j. +If Ai = ∅, then both x and z are bottom initial in Γ(α), and so xφ = y∗ +i = zφ. Let Ai = {xj1 < . . . < +xjw} ̸= ∅. Suppose x is initial in Γ(α). Then xφ = ymx. Suppose z is not initial in Γ(α). Then z = xjq for +some q. Since x < z = xjq, we have xmx ≤ xjq (by the definition of mx), and so xφ = ymx ≤ yjq = xjqφ = zφ. +Suppose z is initial in Γ(α). Then zφ = ymz. Since x < z, xmx ≤ xmz, and so xφ = ymx ≤ ymz = zφ. If z is +initial in Γ(α), then we obtain xφ ≤ zφ by a similar argument. +Hence, in all cases, xφ ≤ zφ, that is, φ ∈ On. By symmetry, there exists an r-homomorphism ψ from +Γ(β) to Γ(α) such that yiψ = xi for all i, and ψ ∈ On. Then for every i, xi(φψ) = xi and yi(ψφ) = yi. +Hence φ and ψ are as in Theorem 2.29, and so α ∼n β in On. +Example 2.38. Consider α, β, δ ∈ O6 whose digraphs are given in Figure 2.2. The prunes of the digraphs +are presented in Figure 2.3, with the orderings of vertices: 4 < 5 < 6 in Γp(α), 3 < 4 < 5 in Γp(β), and +2 < 4 < 5 in Γp(δ). Replacing the vertices in Γp(α) according to these orderings, we obtain Γp +β(α) and Γp +δ(α) +as in Figure 2.4. We can see that Γp +β(α) = Γp(β), but Γp +δ(α) ̸= Γp(δ). Thus, by Theorem 2.37, α and β are +n-conjugate in O6, but α and δ are not. +• +• +� +• +• +• +• +�✄✄✄✄✄ +� +� +�❀❀❀❀❀ +� +5 +3 +2 +1 +6 +4 +• +• +�• +• +� +• +• +• +• +�⑧⑧ +⑧⑧ +⑧⑧ +� +� +�� +4 +3 +2 +1 +5 +6 +• +• +� +• +• +• +• +�⑧⑧ +⑧⑧ +⑧⑧ +� +� +�� +5 +4 +3 +6 +2 +1 +Figure 2.2: Γ(α) (left), Γ(β) (middle), and Γ(δ) (right). +In the semigroups I(X) and J (X) of injective transformations on X (partial and full, respectively), +α ∼n β if and only if Γ(α) ∼= Γ(β) [38, Cor. 5.2 and Thm. 5.3]. +The latter result is also true for the semigroup Ω(X) of surjective transformations on X, which was studied +in [39]. We actually have a stronger result for Ω(X). Let Sym(X) be the symmetric group of permutations +on X. Let S be any subsemigroup of P(X) such that Sym(X) ⊆ S. For α, β ∈ S, we say that α is conjugate +to β by permutation if β = σ−1ασ for some σ ∈ Sym(X). +Note that the conjugacy-by-permutation is +included in ∼n in any such semigroup S. +14 + +• +• +� +• +� +� +5 +6 +4 +• +• +� +• +� +� +4 +3 +5 +• +• +� +• +� +� +5 +4 +2 +Figure 2.3: Γp(α) (left), Γp(β) (middle), and Γp(δ) (right). +• +• +� +• +� +� +4 +5 +3 +• +• +� +• +� +� +4 +2 +5 +Figure 2.4: Γp +β(α) (left) and Γp +δ(α) (right). +Theorem 2.39. For all α, β ∈ Ω(X), the following conditions are equivalent: +(a) α and β are n-conjugate in Ω(X); +(b) the digraphs Γ(α) and Γ(β) are isomorphic; +(c) α and β are conjugate by permutation. +Proof. Let α, β ∈ Ω(X). Suppose that α ∼n β in Ω(X). By Theorem 2.29 and Lemma 2.34, Γp(α) ∼= Γp(β). +Since the digraph of any surjective transformation does not have any initial vertices, Γp(α) = Γ(α) and +Γp(β) = Γ(β), and so Γ(α) ∼= Γ(β). Hence (a) implies (b). +Suppose that Γ(α) ∼= Γ(β), and let σ be an isomorphism from Γ(α) = (X, E) to Γ(β) = (X, F). Then +clearly σ ∈ Sym(X). Let u ∈ X and v = uβ. Then (u, v) ∈ F, and so (uσ−1, vσ−1) ∈ E. Thus (uσ−1)α = +vσ−1 = (uβ)σ−1, which implies u(σ−1ασ) = u(βσ−1σ) = uβ. Hence β = σ−1ασ. We have proved that (b) +implies (c). Finally, (c) implies (a) since the conjugacy-by-permutation is included in ∼n. +The same result is true for the semigroup J (X) of full injective transformations on X [38, Thm. 5.3], +and for the finite symmetric inverse semigroup I(X). However, for an infinite set X, the conjugacy-by- +permutation in I(X) is strictly included in n-conjugacy in I(X) [38]. +Recall that for an integer n ≥ 1, Xn = {1 < . . . < n}. Viewing Xn as a set, we denote by In the +symmetric inverse semigroup I(Xn). +Let OIn be the subset of In consisting of partial injective order- +preserving transformations, that is, +OIn = {α ∈ In : ∀x,y∈Xn(x < y ⇒ xα < yα)}. +Then OIn is an inverse semigroup [25,26]. We will now describe n-conjugacy in OIn. +Let Γ be a digraph and let v0, v1, . . . , vk, k ≥ 1, be pairwise distinct vertices of Γ. Suppose that +v0 → v1 → · · · → vk−1 → v0, +(2.1) +v0 → v1 → · · · → vk−1 → vk +(2.2) +are sub-digraphs of Γ. +We call (2.1) and (2.2), respectively, a cycle of length k (or a k-cycle), writ- +ten (v0 v1 . . . vk−1), and a chain of length k (or a k-chain), written [v0 v1 . . . vk], in Γ. +We can view +(v0 v1 . . . vk−1) and [v0 v1 . . . vk] as partial injective transformations on the set of vertices of Γ, both with +domain {v0, v1, . . . , vk−1}, and the values calculated according to (2.1) and (2.2). +15 + +Definition 2.40. Let α ∈ P(X), where X is any set, and let x ∈ span(α). The subgraph of Γ(α) induced +by the set +{y ∈ span(α) : αk(y) = αm(x) for some integers k, m ≥ 0} +is called the component of Γ(α) containing x. The components of Γ(α) correspond to the connected compo- +nents of the underlying undirected graph of Γ(α). +If α ∈ In, then each component of Γ(α) is either a cycle or a chain, that is, Γ(α) is a disjoint union of +cycles and chains. We will use the language “a cycle [chain] in α” to mean “a component in Γ(α) that is a +cycle [chain].” If α ∈ OIn, then each cycle in α has length 1, and if [v0 v1 . . . vm] is a chain in α, then either +v0 < v1 < . . . < vm or v0 > v1 > . . . > vm. +Recall that for α ∈ P(X), span(α) = dom(α) ∪ im(α) and that span(α) is the set of vertices of Γ(α). For +the meaning of Γβ(α), which appears in the following theorem, see Notation 2.36. +Theorem 2.41. Let α, β ∈ OIn with span(α) = {x1 < . . . < xk} and span(β) = {y1 < . . . < ym}. Then +α ∼n β in OIn if and only if k = m and Γβ(α) = Γ(β). +Proof. Suppose α ∼n β in OIn. Since OIn is closed under restrictions to spans, there is φ ∈ OIn such +that φ is an isomorphism from Γ(α) to Γ(β) (by [38, Thm. 5.1]). Thus k = m. Let Γ(α) = (A, E) and +Γ(β) = (B, F). We have Γβ(α) = (B, E0), where (yi, yj) ∈ E0 if and only if (xi, xj) ∈ E. It remains to +show that E0 = F. Since φ preserves order, we have x1φ < . . . < xkφ, which implies xiφ = yi for every i. +The equality E0 = F follows since for all i, j, (xi, xj) ∈ E if and only if (yi, yj) = (xiφ, xjφ) ∈ F. Hence +Γβ(α) = Γ(β). +Conversely, suppose that k = m and Γβ(α) = Γ(β). Define φ : A → B by xiφ = yi for every i. Then +φ ∈ OIn and for all i, j, (xi, xj) ∈ E ⇔ (yi, yj) ∈ E0 ⇔ (yi, yj) ∈ F ⇔ (xiφ, xjφ) ∈ F. Thus, φ is an +isomorphism from Γ(α) to Γ(β), and so α ∼n β in OIn by [38, Thm. 5.1]. +Let α ∈ OIn with span(α) = {x1 < . . . < xk}, k ≥ 1. Using Theorem 2.41, we can construct the +n-conjugacy class [α]n as follows: +(a) begin with [α]n = ∅ and Yk = the set of all subchains {y1 < . . . < yk} of Xn; +(b) select a subchain {y1 < . . . < yk} from Yk; +(c) replace each xi in Γ(α) with yi; +(d) add β to [α]n, where β is the transformation represented by the digraph obtained in (c); +(e) remove the subchain {y1 < . . . < yk} selected in (b) from Yk; +(f) if Yk ̸= ∅, return to (b); otherwise STOP. +By the above algorithm and the fact that [0]n = {0} in any semigroup with zero, we have +if α ∈ OIn with | span(α)| = k, then |[α]n| = +�n +k +� +for every k ∈ {0, 1, . . ., n}. +Let ∅ ̸= α ∈ OIn. If Γ(α) has s + t components, where σ1, . . . , σs are 1-cycles and τ1, . . . , τt are chains, +then we will write α = σ1 ⊔ · · · ⊔ σs ⊔ τ1 ⊔ · · · ⊔ τt, where each σi and τj is viewed as an element of OIn, and +“⊔” (called the join) is the union of functions viewed as sets. +Example 2.42. Consider α = (1) ⊔ (4) ⊔ [3 5 7] ⊔ [10 9 8] ∈ OI11, and note that we have +span(α) = {1 < 3 < 4 < 5 < 7 < 8 < 9 < 10} +and | span(α)| = 8. Select any subchain of X11 with 8 elements, say {2 < 3 < 5 < 6 < 7 < 8 < 10 < 11}. +Now, replace each x in α, written as above, with the corresponding (according to the orderings) y from that +subchain. Then, β = (2) ⊔ (5) ⊔ [3 6 7] ⊔ [11 10 8] is n-conjugate to α. +16 + +3 +Conjugacy ∼n and partial inner automorphisms +If G is a group, then any g ∈ G defines an inner automorphism of G by a �→ g−1ag. The notion of natural +conjugacy ∼n leads us to a definition of a partial inner automorphism of an arbitrary semigroup. +Let S be a semigroup, fix g, h ∈ S1, and define +Dg,h = {a ∈ S | gh · a = a · gh = a} . +Note that for all a, b ∈ S, a ∼n b with conjugators g and h if and only if a ∈ Dg,h and b = hag (see +Proposition 2.2). +Let ⪯ be a preorder on a set A (that is, ⪯ is a binary relation on A that is reflexive and transitive). We +say that a subset B of A is downward directed in ⪯ if for all a ∈ A and b ∈ B, a ⪯ b implies a ∈ B. +Let S be a semigroup. Then the relation ⪯H on S defined by a ⪯H b if sb = a = bt for some s, t ∈ S1 is +a preorder on S. Note that if a ⪯H b and b ⪯H a, then a H b. +Lemma 3.1. Let S be a semigroup and let g, h ∈ S1. Then: +(1) Dg,h is a subsemigroup of S; +(2) Dg,h is downward directed in the H-preorder ⪯H; +(3) Dg,h is downward directed in the natural partial order ≤; +(4) if a ∈ Dg,h, then Ha ⊆ Dg,h, where Ha denotes the H-class of a in S. +Proof. (1) is clear. For (2), assume a ∈ Dg,h and c ⪯H a. Then there exist s, t ∈ S1 such that sa = c = at. +We have c · gh = s a · gh +� �� � = sa = c and gh · c = gh · a +� �� � t = at = c, and so c ∈ Dg,h, as claimed. Now (3) follows +from (2) since the natural partial order ≤ refines the H-preorder ⪯H. Finally, (4) also follows from (2). +Now we define a mapping by +φg,h : Dg,h → S; a �→ hag . +Note that for all a, b ∈ S, a ∼n b with conjugators g and h if and only if aφg,h = b. +Theorem 3.2. φg,h is a partial automorphism of S, specifically, it is an isomorphism of Dg,h onto Dh,g. +Proof. For a ∈ Dg,h, set b = aφg,h = hag. By Proposition 2.2, a ∼n b with g, h as conjugators. Thus we also +have hg·b = b·hg = b, that is, b ∈ Dh,g. In addition, gbh = a, that is, bφh,g = a. Since aφg,hφh,g = ghagh = a +and bφh,gφg,h = b, we have φg,h is a bijection from Dg,h to Dh,g. +Finally we show that φg,h is a homomorphism. Let a1, a2 ∈ Dg,h be given and set bi = haig for i = 1, 2. +Since ai ∼n bi, we have (a1a2)φg,h = ha1 a2g +���� = ha1g +���� b2 = b1b2, which establishes the claim. +Corollary 3.3. Let S be a semigroup and suppose a, b ∈ S satisfy a ∼n b. Then ak ∼n bk for all positive +integers k, and if g, h ∈ S1 are conjugators for a, b, then g, h are also conjugators for ak, bk. +Theorem 3.4. The bijection φg,h : Dg,h → Dh,g restricts to bijections between H-classes, that is, for +a ∈ Dg,h and b = aφg,h, the restriction of φg,h to Ha is a bijection onto Hb. Further, if Ha is a group +H-class then φg,h is a group isomorphism. +Proof. Fix c ∈ Ha and let d = cφg,h = hcg. There exist s1, s2, t1, t2 ∈ S1 such that s1a = c, s2c = a, at1 = c, +ct2 = a. Set ¯si = hsig and ¯ti = htig for i = 1, 2. Then +¯s1b = hs1 gb +���� = h s1a +���� g = hcg = d , +¯s2d = hs2 ghc +���� g = hs2cg = hag = b , +b¯t1 = bh +���� t1g = h at1 +���� g = hcg = d +and +d¯t2 = h cgh +���� t2g = h ct2 +���� g = hag = b . +17 + +This proves d H b. Thus (Ha)φg,h ⊆ Hb and by symmetry, (Hb)φh,g ⊆ Ha. Finally Hb = (Hb)φh,gφg,h ⊆ +(Ha)φg,h ⊆ Hb, so that φg,h is a bijection of Ha onto Hb. The remaining assertion follows from Theorem +3.2. +Remark 3.5. It is a basic result in semigroup theory that any two group H-classes in the same D-class of a +semigroup are isomorphic [35, Prop. 2.3.6]. We have actually reproved this; it follows from Theorem 2.7 and +Theorem 3.4. Our proofs are certainly more involved but better highlight the role of n-conjugacy. +Corollary 3.6. H ◦ ∼n = ∼n ◦ H. +Proof. Say c H a ∼n b and let g, h ∈ S1 be conjugators for a, b. Set d = (c)φg,h. By Theorem 3.4, we have +b H d ∼n c. The other inclusion is similarly proved. +Now we consider the composition of partial automorphisms. +Proposition 3.7. For gi, hi ∈ S1, i = 1, 2, we have +φg1,h1φg2,h2 ⊆ φg1g2,h2h1 . +(3.1) +Proof. The domain of φg1,h1φg2,h2 is +C = {a ∈ Dg1,h1 | h1ag1 ∈ Dg2,h2} . +If a ∈ C, then +g1g2h2h1 · a = g1 g2h2h1ag1 +� +�� +� h1 = g1h1ag1h1 = a +and +a · g1g2h2h1 = g1 h1ag1g2h2 +� +�� +� h1 = g1h1ag1h1 = a . +Thus a ∈ Dg1g2,h2h1. Clearly aφg1,h1φg2,h2 = aφg1g2,h2h1 for a ∈ C. +Example 3.8. In general, the inclusion (3.1) is proper. For instance, in the group Z2 written additively, +the map φ0,1 is the empty map and thus so is φ0,1φ0,1. However, φ0+0,1+1 = φ0,0 is the identity map. +Let Inn(S) denote the inverse monoid of partial automorphisms generated by the φg,h’s. We will call +Inn(S) the partial inner automorphism monoid of S. +This is a natural generalization to semigroups of the inner automorphism group of a group. Indeed, +suppose S is a group. For g, h ∈ S, if Dg,h ̸= ∅, then gh · a = a for some a, so gh = 1, that is, h = g−1. But +then Dg,g−1 = S and φg,g−1 is the usual inner automorphism of conjugacy by g. Thus if S is a nontrivial +group, our Inn(S) is a zero group, the union of the usual inner automorphism group of S and the empty +mapping. +Remark 3.9. The case where S is an inverse semigroup is studied in detail in [37]. It turns out that for any +g, h ∈ S1, Dg,h ⊆ Dg,g−1. In that case, we may just work with the partial inner automorphisms φg,g−1 and +for those, the inclusion (3.1) is an equality. We then get a homomorphism Φ : S → Inn(S); g �→ φg,g−1, whose +kernel is precisely the central congruence of S. In particular, if S is the symmetric inverse semigroup of partial +injective transformations on a set X, then the homomorphism Φ is an isomorphism, and so S ∼= Inn(S). +Example 3.10. It is well known that nonisomorphic groups can have isomorphic automorphism groups +(eg, Q8 and S4 both have automorphism groups isomorphic to S4). The same happens with partial inner +automorphisms. The cyclic groups of order 2 and 3, both have the 2-chain as the semigroup of partial inner +automorphisms (and the 2-chain is isomorphic to its semigroup of partial inner automorphisms). +18 + +Example 3.11. An elementary observation in group theory is that if two elements a, b are conjugate, then +every element of the centralizer Ca of a is conjugate to some element of the centralizer Cb of b. This is not +true for ∼n, even in highly structured semigroups. Consider the semigroup defined by this table: +· +e +r1 +r2 +s1 +s2 +s3 +f +c +e +e +r1 +r2 +s1 +s2 +s3 +e +s1 +r1 +r1 +r2 +e +s3 +s1 +s2 +r1 +s3 +r2 +r2 +e +r1 +s2 +s3 +s1 +r2 +s2 +s1 +s1 +s2 +s3 +e +r1 +r2 +s1 +e +s2 +s2 +s3 +s1 +r2 +e +r1 +s2 +r2 +s3 +s3 +s1 +s2 +r1 +r2 +e +s3 +r1 +f +e +r1 +r2 +s1 +s2 +s3 +f +c +c +s1 +s2 +s3 +e +r1 +r2 +c +f +This is a Clifford semigroup, that is, an inverse semigroup in which the idempotents (in this case, e and f) +commute with all elements. We see that this semigroup is a union (in fact, semilattice) of the subgroups +A = {e, r1, r2, s1, s2, s3} and B = {e, c}. Since s2 +3 = e, the identity element of A, we have that A ⊆ Ds3,s3. +Now (s1)φs3,s3 = s3s1s3 = s2, and thus s1 ∼n s2. +We see from the table that Cs1 = {e, f, s1, c} and +Cs2 = {e, f, s2}. If gh · c = c = c · gh, then from the table, gh = f, and so g = h = f or g = h = c. +We compute cφf,f = c and cφc,c = c. Therefore the n-conjugacy class of c is [c]n = {c}, and so c is not +n-conjugate to any element of Cs2. +We can use the machinery above to show that in epigroups, we can impose additional restrictions on +conjugators without loss of generality. Recall that elements g, h of a semigroup S are mutually inverse if +ghg = g and hgh = h. +Theorem 3.12. Let S be an epigroup. Then for all g, h ∈ S1, there exist mutually inverse ¯g, ¯h ∈ S1 such +that φg,h ⊆ φ¯g,¯h. +Proof. Let g, h ∈ S1. Setting +¯g = (gh)ωg +and +¯h = h(gh)′, +(3.2) +we obtain: +¯g¯h = (gh)ωgh(gh)′ = (gh)ω, +(3.3) +¯h¯g = h(gh)′(gh)ωg = h(gh)′g +(2.10) += +hg(hg)′ = (hg)ω, +(3.4) +¯g¯h¯g = (gh)ω(gh)ωg = (gh)ωg = ¯g, +¯h¯g¯h = h(gh)′(gh)ω = h(gh)′ = ¯h . +Therefore ¯g, ¯h are mutually inverse. +Now assume aφg,h = b, that is, a ∼n b with g, h as conjugators. We will now show that +(gh)ωa = a = a(gh)ω +and +(hg)ωb = b = b(hg)ω . +(3.5) +Indeed, choose n such that (gh)n(gh)ω = (gh)n+1(gh)′ = (gh)n. Then a(gh)ω = a(gh)n·(gh)ω = a(gh)n = a. +The other three equations in (3.5) are proved similarly. +Now we use (3.2), (3.3), (3.4), and (3.5) in the following calculations: +a¯g = a(gh)ωg = ag = gb = g(hg)ωb = (gh)ωgb = ¯gb , +¯h¯g · b = (hg)ωb = b , and +a · ¯g¯h = a(gh)ω = a . +By Proposition 2.2, ¯g, ¯h are conjugators for a, b, and thus aφ¯g,¯h = b. This completes the proof. +19 + +Example 3.13. In general, the conclusion of Theorem 3.12 is a strict inclusion. For example, consider the +semigroup defined by the multiplication table +· +1 +2 +3 +4 +1 +1 +1 +4 +4 +2 +2 +2 +3 +3 +3 +3 +3 +2 +2 +4 +4 +4 +1 +1 +Set g = 1 and h = 3. Then ¯g = 1 and ¯h = 2. For a = 1, b = 2, we have a¯g = 1 = ¯gb, a¯g¯h = 1 = a, +¯h¯gb = 2 = b. Thus a ∼n b with ¯g, ¯h as conjugators, so aφ¯g,¯h = b. However, agh = 3 ̸= a and so a ̸∈ Dg,h. +Corollary 3.14. If a ∼n b in an epigroup S, then there exist mutually inverse conjugators for a, b. +3.1 +The partial inner automorphism monoid of T(X) +Computing the partial inner automorphisms of a given semigroup is a challenge in itself. We already observed +that the symmetric inverse semigroup is isomorphic to its inverse semigroup of partial inner automorphisms. +In this subsection, we describe the partial inner automorphism monoid S = Inn(T (X)), for the full transfor- +mation monoid of a set X. It turns out that the structure of S is essentially isomorphic to the combination +of two components, one of which is the symmetric inverse semigroup on X. The other component consists +of bijections between partitions of X with the same number of parts. In the same way that the partial +composition operation of the symmetric inverse semigroup is based on the intersection of an image and a +domain, the operation of the second component is based on the join ∨ of two partitions. +In the above description, we write “essentially” for two reasons. The two components are not entirely +independent, but are required to be compatible which each other in a natural way. In addition, further small +adjustments are needed. The number of elements of Inn(T (X)) that are affected by these adjustments are +small relative to the size of S. +Throughout this subsection, we will blur the distinction between partitions and their corresponding +equivalence relations. +Theorem 3.15. Let g, h ∈ T (X) and Dg,h be as defined above, that is, +Dg,h = {x ∈ T (X) : ghx = xgh = x} . +Then there exists a partition P of X, and a partial section I of P, such that Dg,h consists of all transfor- +mations t with im t ⊆ I and P ⊆ ker t. Moreover, I, P can be chosen so that every singleton part S of P +satisfies S ⊆ I. +I is uniquely determined by Dg,h, and if Dg,h contains more than one transformation, then P is uniquely +determined by Dg,h as well. +Conversely, suppose that P is a partition of X and I is a partial section of P such that all singleton parts +of P intersect I. Then there exist g, h ∈ T (X) such that Dg,h consists of all transformations t ∈ T (X) with +im t ⊆ I and P ⊆ ker t. +In the above cases, if |I| ≥ 2, then I, P uniquely determine Dg,h, while if |I| ≤ 1, then I uniquely +determines Dg,h. +Proof. Assume first that g, h ∈ T (X), and let D = Dg,h. Clearly D only depends on the product p = gh. +Let I ⊆ X be the set of points fixed by p, and let P be the collection of connected components of the +function graph of p. In each part of P, there is at most a single point x with xp = x, and so I is a partial +section of P. If for some x ∈ X, {x} is a singleton part of P, then xp = x, and so {x} ⊆ I. +Let t ∈ Dg,h. Because tp = t, p acts as the identity on the image of t and so t maps into I. Because +pt = t, if xp = y, then yt = x(pt) = xt, and so (x, y) ∈ ker t. It follows that the connected component of x +in the function graph of p is contained in the kernel of t. Hence P ⊆ ker t. +20 + +Conversely, if t ∈ T (X) maps into I and P ⊆ ker t, it is straightforward to check that pt = tp = t, and so +t ∈ D. It follows that D consists of all t with im t ⊆ I and P ⊆ ker t. +Now, let I and P be any set and partition that characterize D in this way. Then I is the union of +all images of transformations in D, and hence is uniquely determined by D. If |D| ≥ 2, then |I| ≥ 2 and +|P| ≥ 2, the latter because I is a partial section of P. Suppose that P ∈ {P1, P2}, where P1, P2 are two +distinct partitions of X, each with at least two parts. Then w.l.og. P1 is a refinement of a 2-partition P ′ of +X that does not contain P2. Because |I| ≥ 2, there exists a t ∈ T (X) with im t ⊆ I and ker t = P ′ ⊇ P1, +but P2 ̸⊆ P ′ = ker t. It follows that P is uniquely determined by D. +Now suppose that P is a partition of X and I is a partial section of P such that all singleton parts of P +are contained in I. +Let g ∈ T (X) be the identity, and define h ∈ T (X) as follows: if x ∈ X is in a part B of P intersecting +I, then let xh = y were y is the unique element of B ∩ I. If B is a part of P not intersecting I then |B| ≥ 2. +Pick b1 ̸= b2 ∈ B, and let b1h = b2, xh = b1 for x ∈ B \ {b1}. Applying the construction in the first part of +the proof to Dg,h, it is straightforward to verify that we recover the sets I and P. Hence Dg,h contains all +transformations t with im t ⊆ I and P ⊆ ker t. +The final uniqueness result now also follows from the first part for |I| ≥ 2, and is trivial for |I| ≤ 1. +For any X-partition P and I ⊆ X, we will use the notation DP,I to refer to the set of t ∈ T (X) with +im t ⊆ I, P ⊆ ker t, where we also include such I, P in which I is not a partial section of P, or for which not +all singleton parts of P intersect I. +Lemma 3.16. Let Dg,h = DP,I and Dh,g = DP ′,I′. Then g|I : I → I′ , h|I′ : I′ → I are inverse bijections. +Proof. The result is clear if I = ∅. Otherwise, pick i ∈ I, and define t ∈ T (X) by [j]P t = j for j ∈ I, xt = i +otherwise. Clearly, t ∈ Dg,h and im t = I. Because ght = t, im(ht) = I, and because htg ∈ DP ′,I′, we see +that g|I maps into I′. Dually, hI′ maps into I. +Because t ∈ Dg,h, tgh = t, and so gh acts as the identity on the image I. Applying the argument to a +correspondingly constructed element t′ ∈ Dh,g, we get that hg is the identity on I′. The result follows. +Lemma 3.17. Let Dg,h = DP,I, Dh,g = DP ′,I′ with |I| ≥ 2 (and therefore |I′| ≥ 2). +Then ˆg : P → P ′, given by [p]P ˆg = [pg]P ′, and ˆh : P ′ → P, given by [p′]P ′ˆh = [p′h]P , are well-defined +inverse bijections. +Moreover, for all B ∈ P, B′ ∈ P ′, we get B ∩ I = ∅ ⇔ Bˆg ∩ I′ = ∅ and B′ ∩ I′ = ∅ ⇔ B′ˆh ∩ I = ∅. +Proof. Pick distinct i, j ∈ I, and [p] ∈ P. +Define t ∈ T (X) by [p]P t = j, xt = i otherwise. +Clearly, +t ∈ Dg,h = DP,I. +Because j = pt = p(ght) we see that p(gh) ∈ [p]P , and therefore [p]P (gh) ⊆ [p]P . +Suppose that p1, p2 ∈ [p]P are such that [p1g]P ′ ̸= [p2g]P ′. Let t′ ∈ Dh,g be a transformation that maps +[p1g]P ′, [p2g]P ′ to distinct elements i′ +1, i′ +2 ∈ I′ (such t′ clearly exists). Then gt′h ∈ Dg,h = DP,I, and therefore +i′ +1h = p1gt′h = p2gt′h = i′ +2h, which contradicts the injectivity of h|I′. It follows that ˆg is well-defined. A +dual argument shows the corresponding claim for ˆh. +We already have seen that p(gh) ∈ [p]P , and so [p]P ˆgˆh = [p]P . As [p]P was arbitrary, we see that ˆgˆh acts +as the identity on ¯P. An analogous argument shows that ˆhˆg is the identity on P ′, and hence ˆg and ˆh are +inverse bijections. +The last claim follows from Lemma 3.16. +We now can derive a classification theorem for the generating elements φg,h of the partial inner automor- +phism monoid. +Theorem 3.18. The partial inner automorphisms of T (X) having the form φg,h, and acting on more than +one transformation are in bijective correspondence with the tuples (P, P ′, I, I′, α, β), where +• P and P ′ are partitions of X, with |P| = |P ′|; +21 + +• I and I′ are partial sections, of P and P ′, respectively, with |I| = |I′| ≥ 2, and intersecting all singleton +sets of P, P ′, respectively; +• α : I → I′ is a bijection; +• β : P → P ′ is a bijection extending the partial bijection between P and P ′ induced by α; +such that +• The domain of φg,h consists of all transformations t ∈ T (X) with im t ⊆ I, P ⊆ ker t; +• The image of φg,h consists of all transformations t ∈ T (X) with im t ⊆ I′, P ′ ⊆ ker t; +• Given t in the domain of φg,h, and x ∈ X, we have (x)(tφg,h) = iα, where i ∈ I is the unique element +in (([x]P ′)β−1)t. +The partial inner automorphisms of T (X) having the form φg,h and acting on at most one transformation +consist of all functions mapping one constant transformation on X to another, and (for |X| ̸= 1), the empty +mapping. +Proof. We first consider the case of the partial inner automorphisms φg,h whose domain contains more +than one transformation. By Theorem 3.15, P, I, P ′, I′ exist, have the stated properties and are uniquely +determined by Dg,h and Dh,g. Set α = g|I, and β = ˆg, where ˆg is defined as in Lemma 3.17. By Lemmas 3.16 +and 3.17, α and β are bijections, and by its definition, β extends the partial function on P induced by α. +Let t ∈ dom φg,h = DP,I, and x ∈ X. By Lemma 3.17, β−1 = ˆh. Therefore [x]P ′β−1 ∈ P. As t ∈ DP,I, +(([x]P ′)β−1)t contains a single element i ∈ I. +We now have that x(ht) ∈ ([x]P ′ˆh)t = {i}, and so x(htg) = (x(ht))g = ig = ig|I = iα, as required. +Now for any i ∈ I, let ci ∈ DP,I be the constant function with image i. It follows from the above that +ciφg,h = ciα, and hence α is uniquely determined by φg,h. +Finally suppose that β, β′ : P → P ′ are two bijections, that, together with some φg,h, α, P, I, P ′, I′ +satisfy the conditions of the theorem. Pick two distinct elements i, j ∈ I, and for each B ∈ P, let tB be +the transformation with Bt = {i}, xt = j for x /∈ B. Let x ∈ Bβ, then x(tBφg,h) = iα, as ([x]P ′β−1)tB = +{i}. +Because α is injective, it follows that ([x]P ′β′−1)tB = {i}. +From the definition of tB this implies +([x]P ′β′−1) = ([x]P ′β−1), and so β−1 and β′−1 agree on Bβ. As B was arbitrary, we get β = β′. +The final claim about φg,h with |Dg,h| ≤ 1 easily follows from Theorem 3.15. +We will now turn our attention to general elements of Inn(T (X)). +Definition 3.19. Let P, P ′ be partitions of X, and γ : P → P ′ a bijection. If ¯P = {Bi} is a partition that +refines to P, we define ¯γ on ¯P by (∪Bi)¯γ = ∪((Bi)γ). +It is clear that ¯γ is well-defined, and that its image is a partition that refines to P ′. +Theorem 3.20. Let φ ∈ Inn(T (X)). Then there exist +• partitions P, P ′ of X; +• I, I′ ⊆ X; +• bijections α : I → I′, β : P → P ′ satisfying [i]P β = [iα]P ′ for all i ∈ I; +such that +• The domain of φ consists of all transformations t ∈ T (X) with im t ⊆ I, P ⊆ ker t; +• The image of φ consists of all transformations t ∈ T (X) with im t ⊆ I′, P ′ ⊆ ker t; +• Given t in the domain of φ, and x ∈ X, we have (x)(tφ) = iα, where i ∈ I is the unique element in +(([x]P ′)β−1)t. +22 + +Moreover, if φ1, φ2 ∈ Inn(T (X)) have corresponding parameters +(P1, I1, P ′ +1, I′ +1, α1, β1) and (P2, I2, P ′ +2, I′ +2, α2, β2) +then φ1φ2 corresponds to +((P ′ +1 ∨ P2)¯β−1 +1 , (I′ +1 ∩ I2)α−1 +1 , (P ′ +1 ∨ P2)¯β2, (I′ +1 ∩ I2)α2, α1α2, ¯β1 ¯β2) , +where α1α2 refers to the partial composition α1|(I′ +1∩I2)α−1 +1 α2. +Proof. We will show the assertions by structural induction over the involved elements φ, φ1, φ2. The beginning +of the induction corresponds to those φ of the form φg,h, and follows from Theorem 3.18 (in the cases with +|Dg,h| ≤ 1, we can chose P = P ′ = {X}, β = id{{X}}). +Suppose the theorem holds for φ1, φ2 ∈ Inn(T (X)). Then L := im φ1 ∩ dom φ2 consists of all transforma- +tions t with im t ⊆ I′ +1 ∩ I2 and P ′ +1 ∨ P2 ⊆ ker t. It is now straightforward to check that +Lφ−1 +1 += D(P ′ +1∨P2) ¯β−1 +1 +,(I′ +1∩I2)α−1 +1 +and Lφ2 = D(P ′ +1∨P2) ¯β2,(I′ +1∩I2)α2 +and hence these parameters define the domain and image of φ1φ2. +Let i ∈ (I′ +1 ∩ I2)α−1 +1 +⊆ I, then +[i](P ′ +1∨P2) ¯β−1 +1 +¯β1 ⊇ [i]P1β1 = [iα1]P ′ , +and so +[i](P ′ +1∨P2) ¯β−1 +1 +¯β1 = [iα1]P ′ +1∨P2 ⊃ [iα1]P2 . +Because iα1 ∈ I′ +1 ∩ I2 ⊆ I2, we get that +[i](P ′ +1∨P2) ¯β−1 +1 +¯β1 ¯β2 ⊃ [iα1]P2β2 = [iα1α2]P ′ +2 . +Hence we get +[i](P ′ +1∨P2) ¯β−1 +1 +¯β1 ¯β2 = [iα1α2](P ′ +1∨P2) ¯β2 , +as required. +Let t ∈ Lφ−1 +1 , and x ∈ X. Pick an element y ∈ [x](P ′ +1∨P2) ¯β2 ¯β−1 +2 . Because ¯β−1 +2 +is injective, we have +[x](P ′ +1∨P2) ¯β2 ¯β−1 +2 += [y]P ′ +1∨P 2. It follows that +([x](P ′ +1∨P2) ¯β2(¯β1 ¯β2)−1)t = ([x](P ′ +1∨P2) ¯β2 ¯β−1 +2 +¯β−1 +1 )t = ([y]P ′ +1∨P2 ¯β−1 +1 )t = ([y]P ′ +1β−1 +1 )t , +where the last equality holds because the kernel of t contains (P ′ +1 ∨ P2)¯β−1 +1 . By induction, this set contains +a unique element i such that y(tφ1) = iα1. +Also by induction, x((tφ1)φ2) = jα2, where j is the unique element in +([x](P ′ +1∨P2) ¯β2 ¯β−1 +2 )(tφ1) = ([y]P ′ +1∨P 2)(tφ1) = {y(tφ1)} = {iα1} . +Hence x((tφ1)φ2) = (iα1)α2. Because i ∈ ([x](P ′ +1∨P2) ¯β2(¯β1 ¯β2)−1), the result follows. +We can now obtain results about the structure of Inn(T (X)). For a set X, let A(X), B(X) be the +set of all bijections between subsets of X, and bijections on partitions of X, respectively. We say that +α ∈ A(X), α : I → I′ and β ∈ B(X), β : P → P ′ are compatible, written α ≈ β, if [i]P β = [iα]P ′ for all +i ∈ I. +Let V (X) = {(α, β) : α ∈ A(X), β ∈ B(X), α ≈ β}. On V (X) we define a binary operation +(α1, β1)(α2, β2) = (α1α2, ¯β1 ¯β2) , +23 + +where ¯βi is as in Theorem 3.20, and where we fix the domain of α1α2 [of ¯β1 ¯β2] as the largest subset of X +[finest partition on X] for which these expressions are well-defined. It is easy to check that domains and +images of α1α2 and ¯β1 ¯β2 are given by the expressions from Theorem 3.20. +It will follow from our results below that V (X) with this operation is an inverse monoid. Because for +every partial bijection α on X, there is a compatible β, the projection of V (X) to its the first component is +essentially the symmetric inverse monoid on X. +On V (X), define a binary relation +θ = ∆V (X) ∪ {((α, β1), (α, β2)) : α ∈ A(X), | dom α| ≤ 1, β1, β2 ∈ B(X)} . +Clearly, θ is an equivalence relation, and because {(α, β) : | dom α| ≤ 1} is an ideal of V (X), θ is compatible +with the operation on V (X). We set W(X) = V (X)/θ. For [(α, β)]θ ∈ W(X) we will also use the short +notation [α, β]. +Theorem 3.21. Let X be any set. For φ ∈ Inn(T (X)), let αφ, βφ be the bijectionss associated with φ by +Theorem 3.20. Then ϕ : Inn(T (X)) → W(X), given by ϕ(φ) = [(αφ, βφ)]θ is an embedding. +In particular Inn(T (X)) is isomorphic to the substructure of W(X) generated by all elements of W(X) +that can be represented as [(α, β)]θ such that dom α is a partial section of dom β, and all singleton parts of +dom β intersect dom α. +Proof. Our construction guarantees that ϕ is a homomorphism, provided it is well defined. +Hence let φ ∈ Inn(T (X)), and α, β be the bijections associated with φ. Because dom α and im α are the +maximal images of all transformations in dom φ and im φ, respectively, they are uniquely determined by φ. +For each i ∈ dom α, let ci be the constant function with image i. Then ci ∈ dom φ, and ciφ = ciα. It +follows that α is uniquely determined by φ. +If | dom α| ≤ 1, then one θ-class contains (α, β) for all choices of β. So assume otherwise, say i, j ∈ dom α. +Let B ∈ dom β. Because dom φ contains the transformation tB that maps B to i and X \ B to j, it +follows that the parts of dom β are determined by all minimal kernel classes of transformations in dom φ. +Hence dom β is unique, and similarly, we see that im β is unique. +Finally, because tBφ maps exactly Bβ to iα, we see that β itself is uniquely determined. It follows that +ϕ is well-defined, and hence a homomorphism. +Moreover, for every t ∈ dom φ, and x ∈ X, we have (x)(tφ) = iα, where i ∈ I is the unique element in +(([x]P ′)β−1)t. Therefore tφ is uniquely determined by α, β, and hence ϕ is injective. +The final assertion follows from the description of the generators φg,h of Inn(T (X)) in Theorem 3.18, +noting that in the case of | dom α| ≤ 1, we may always choose β = id{{X}}, in which case the representation +[α, β] is as claimed. +For a complete classification, it remains to determine the image of the embedding ϕ. We will have to +distinguish between finite and infinite X. In the following, by the term “generator”, we will mean an element +of the form φg,hϕ. +Theorem 3.22. Let X be infinite. Then Inn(T (X)) is isomorphic to W(X), and the embedding ϕ from +Theorem 3.21 is an isomorphism. +Proof. By Theorem 3.21, it suffices to show that W(X) is indeed generated by all generators. +Let I ⊆ X, and P be a partition X. Clearly, idI ≈ idP . We first show that [(idI, idP )]θ is in the image +of ϕ. +Chose a bijection σ : X → X2. Let P1 be the singleton partition on X, P ′ +1 = {({x}×X)σ−1 : x ∈ X}, and +define α1 : X → (∆Y )σ−1, β1 : P1 → P ′ +1 by xα1 = (x, x)σ−1, {x}β1 = ({x} × X)σ−1. It is straightforward +to check that [α1, β1] is a generator. +Next let α2 and β2 be the identities on {(x, x)σ−1 : x ∈ I} and P ′ +1, respectively. Because P ′ +1 does not +contain any singleton blocks, [α2, β2] is once again a generator. +Let β3 be the identity on the partition P3 consisting of all sets of the form {(x, y), (y, x)}σ−1 for x, y ∈ X +with [x]P = [y]P , and singletons otherwise. Moreover, let I3 be the union of all singleton sets in P3 and +α3 = idI3. Once again, (α3, β3) is a generator. +24 + +Finally, let α4 = α−1 +1 , β4 = β−1 +1 . We claim [(idI, idP )]θ = Π4 +i=1[(αi, βi)]θ. +Let x ∈ I, then +xα1α2α3α4 = ((x, x)σ−1)α2α3α4 = ((x, x)σ−1)α3α4 = ((x, x)σ−1)α4 = x . +If x /∈ I, then α2 is undefined at xα1 = ((x, x)σ−1). Hence α1α2α3α4 = idI. +Let B ∈ P, and C ⊆ B. Then +C ¯β1 ¯β2 ¯β3 ¯β4 = ((C × X)σ−1)¯β2 ¯β3 ¯β4 = ((C × X)σ−1)¯β3 ¯β4 = ((B × X)σ−1)¯β4 = B . +From this it follows that the domain of ¯β1 ¯β2 ¯β3 ¯β4 is indeed P (as opposed to a refinement), and that ¯β1 ¯β2 ¯β3 ¯β4 +acts as the identity. Hence [(idI, idP )]θ is in the image of ϕ, as claimed. +For the general case, let [α, β]θ ∈ W(X) be arbitrary. +Construct [α′, β′] as follows: If Bi ∈ dom β +intersects dom α, choose a partition PBi of Bi that contains exactly one element of dom α in each part, and +let dom β′ be the union of the PBi, together with all B ∈ dom β not intersecting dom α. Note that dom β′ +is a refinement of dom β. Let im β′ be the refinement obtained from im β in the same way. If B′ +i ∈ dom β′ +contains a (unique) element i ∈ dom α, then let B′ +iβ′ = [iα]im β′, otherwise, set B′ +iβ′ = B′ +iβ. If Bi ∈ dom β +does not intersect dom α, choose an element bi ∈ Bi. Let dom α′ be obtained from dom α by adjoining all +the elements bi. Similarly enlarge im α to im α′ by choosing one element from each Bi ∈ im β that does not +intersect im α. Now let xα′ be the unique element in im α′ ∩ [x]dom β′β′. +Then [α′, β′] is a generator. Since [iddom α, iddom β] ∈ im ϕ, this also holds for [iddom α, iddom β][α′, β′]. A +straightforward check shows that this product is [α, β], and the result follows. +Theorem 3.23. Let X be finite, and [α, β]θ ∈ W(X). If | dom α| ≥ 2, then [α, β]θ ∈ im ϕ if and only if one +of the following holds: +1. dom α = X and dom β is the partition of X into singletons; +2. there exists B ∈ dom β with |B| ≥ 2, B ̸⊆ dom α. +If | dom α| ≤ 1, then [α, β]θ ∈ im ϕ, unless |X| = 1 and dom α = ∅. +Proof. Suppose first that | dom α| ≥ 2. If [α, β] satisfies condition 1, then it is a generator, and hence in the +image of ϕ (in fact its preimage will be a unit of T (X)). +So assume that there exists a set B ∈ dom β with |B| ≥ 2, B ̸⊆ dom α. Let I = dom α, P = dom β. As +in the infinite case, we first show that [(idI, idP )]θ is in the image of ϕ. +Enumerate X as x1, x2, . . . , xm, such that the parts of P correspond to consecutive index ranges in +{1, . . . , m}, with xm ∈ B \ I. We will use three different types of generators to obtain [idI, idP ]. +For J ⊆ I \ {xm}, let QJ be the partition with part J ∪ {xm}, and singletons otherwise. If J = {xj}, we +will just write Qxj. We set kj = [idI\{xm}, idQxj ], and lJ = [idI\J, idQJ ]. Moreover, let βj : Qj → Qj+1 be +defined by {xj, xm}βj = {xj}, {xj+1}βj = {xj+1, xm}, and the identity otherwise. Set sj = [idI\{xm}, βj]. +It is easy to check that all kj, lJ, and sj are generators. +Let C1, . . . , Cr = B be the parts of P, in the order of their index ranges. For each Ci = {xdi, . . . , xei}, +i = 1, . . . , r−1, let Ji = Ci \I, and set pi = kdikdi+1 . . . keilJisei. For Cr = B = {xdr, . . . , xm}, let Jr = B\I +and set pr = kdrkdr+1 . . . km−1lJr. +We leave it up to the reader to confirm that [idI, idP ] = p1 · · · pr. We now can show that im ϕ contains +any [α, β] with dom α = I, dom β = P exactly as in the infinite case in Theorem 3.22. +For the converse, suppose that a = [α, β]θ ∈ im ϕ, say a = g1 · · · gn for some generators gi = [αi, βi]. +If dom α = X, then by finiteness, dom αi = X for all i, and hence (as the gi are generators), dom βi is the +partition into singletons. From this, we get that dom α = X and dom β is the partition of X into singletons, +as well. +Let dom α ̸= X. We may assume that the number of generators n is the smallest possible. If dom α1 = X, +then it is easy to see that g1g2 is a generator as well (note that this requires finiteness, which forces g1ϕ−1 +to be a unit of T (X)). +25 + +Hence by minimality, dom α1 ̸= X. As g1 is a generator, it follows that dom β1 contains a set B′, |B′| ≥ 2 +with B′ ̸⊆ dom α1. But then dom β contains a set B with B′ ⊆ B and dom α ∩ B′ ⊆ dom α1. It follows that +B satisfies the criteria in condition 2. +If | dom α| = 1 then [α, β]θ = [α, id{X}]θ, which is a generator. If | dom α| = 0 and |X| ̸= 1, then [α, β], +which is the empty mapping, is the generator [∅, id{X}]. Conversely, if |X| = 1, then Inn(T (X)) only contains +the trivial full automorphism. The result follows. +3.2 +The partial inner automorphism monoid of a completely simple semigroup +Every completely simple semigroup is isomorphic to a Rees matrix semigroup and hence we assume at the +outset of this subsection that our semigroups have this form. +Lemma 3.24. Let Γ be a group, I and Λ two nonempty sets, and P a Λ × I matrix with entries in +Γ. Let M(G; I, Λ; P) be the Rees matrix semigroup induced by Γ, I, Λ and P. Let (G, g, γ), (H, h, η) ∈ +M(G; I, Λ; P). Then +D(G,g,γ),(H,h,η) ̸= ∅ +⇐⇒ +h = (pη,G g pγ,H)−1 +and +D(G,g,γ),(H,(pηG g pγ,H)−1,η) = {G} × Γ × {η}. +Proof. Regarding the equivalence, we start by proving the direct implication and the second equality. Let +(A, a, α) ∈ M(G; I, Λ; P) such that +(G, g, γ)(H, h, η)(A, a, α) = (A, a, α) = (A, a, α)(G, g, γ)(H, h, η). +Then A = G and α = η so that +D(G,g,γ),(H,h,η) ⊆ {G} × Γ × {η} +and hence the two sets are equal (by Lemma 3.1(4)). This proves the last equality in the statement of the +lemma. +Now, from (G, g, γ)(H, h, η)(G, a, η) = (G, a, η), we get g pγ,H h pη,G a = a, that is, h = (pη,G g pγ,H)−1. +The direct implication is proved. +For the converse implication, let h = (pη,G g pγ,H)−1 and (G, a, η) ∈ M(G; I, Λ; P). Then +(G, g, γ)(H, p−1 +γ,Hg−1p−1 +η,G, η)(G, a, η) = (G, a, η) +and similarly +(G, a, η)(G, g, γ)(H, p−1 +γ,Hg−1p−1 +η,G, η) = (G, a, η). +It is proved that D(G,g,γ),(H,h,η) ̸= ∅ and the lemma follows. +Now we can state the main result of this subsection. +Theorem 3.25. Let Γ be a group, I and Λ two nonempty sets, and P a Λ × I matrix with entries in +Γ. +Let M(G; I, Λ; P) be the Rees matrix semigroup induced by Γ, I, Λ and P. +Then the semigroup +Inn(M(G; I, Λ; P)) is generated by the following maps and corresponding inverses: +φ(G,g,γ),(H,(pη,G g pγ,H)−1,η) : +{G} × Γ × {η} +→ +{H} × Γ × {γ} +(G, a, η) +�→ +(H, (gpγ,H)−1 a (pη,G g), γ), +for g ∈ Γ, G, H ∈ I and γ, η ∈ Λ. +26 + +4 +Conjugacies ∼n, ∼tr, ∼∗ +p, ∼o, and ∼c in finite partition monoids +The partition monoid PX on a set X has the set of all partitions of X ∪ X′ as its underlying set, where +X′ is a disjoint copy of X. +These monoids originally arose in the study of partition algebras (see, for +example, [32,47]) and subsequently attracted the attention of mathematicians working in semigroup theory +(see, for example, [20,22,28]. One reason for the attention is that PX contains some important semigroups +as subsemigroups, such as T (X) and I(X) (see §2.5), as well as the symmetric group Sym(X) on X [22]. +In this section, we will be interested in the finite partition monoid Pn on a set with n elements, and in the +submonoids BPn and Bn of Pn, which are called partial Brauer monoids and Brauer monoids, respectively. +Our goal is to characterize the conjugacies ∼n, ∼tr, ∼p, ∼o, and ∼c in these monoids. (See §1 for the +definitions of all these conjugacy relations.) +From now on, we will identify an equivalence relation R on a set Y with the partition of Y induced by R. +It will always be clear from the context how we view R. +Using the notation from [20], we let n = {1, . . . , n} and n′ = {1′, . . . , n′}. Symbols x, y, z, , k, l, m . . . will +always refer to elements in n, and x′, y′, z′, k′, l′, m′ . . . to the corresponding elements in n′. If A ⊆ n, then +A′ = {x′ : x ∈ A} ⊆ n′. +As customary, we represent an element a ∈ Pn (a partition of n ∪ n′) as a simple graph with vertices +1, . . . , n in a row, vertices 1′, . . . , n′ directly below, and edges drawn in such a way that the connected +components of the graph correspond to the blocks of the partition a. Such a graph is not unique, so we +identify two graphs that have the same connected components. For example, the graph +1 +2 +3 +4 +5 +• +• +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +• +• +❧❧❧❧❧❧❧ +• +• +• +• +• +• +represents the element a ∈ P5 whose blocks are: {1, 3}, {2, 4′}, {1′, 2′}, {3′, 4, 5}, {5′}. For x ∈ n, [x]a will +denote the block of a containing x. Similarly, we write [x′]a for the block containing x′ ∈ n′. +We multiply elements of Pn as follows. If a is as above and b is represented by the graph +• +• +• +• +• +• +♠ +♠ +♠ +♠ +♠ +♠ +♠ +• +♠ +♠ +♠ +♠ +♠ +♠ +♠ +• +♠ +♠ +♠ +♠ +♠ +♠ +♠ +• +• , +then to obtain the product ab, we first draw a over b: +• +• +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +• +• +❧❧❧❧❧❧❧ +• +• +• +• +• +• +• +• +• +• +• +• +♠ +♠ +♠ +♠ +♠ +♠ +♠ +• +♠ +♠ +♠ +♠ +♠ +♠ +♠ +• +♠ +♠ +♠ +♠ +♠ +♠ +♠ +• +• , +then we glue two middle rows: +• +• +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +❳ +• +• +❧❧❧❧❧❧❧ +• +• +• +• +• +• +• +♠ +♠ +♠ +♠ +♠ +♠ +♠ +• +♠ +♠ +♠ +♠ +♠ +♠ +♠ +• +♠ +♠ +♠ +♠ +♠ +♠ +♠ +• +• , +and finally we remove the middle row, keeping in the same block the elements of X ∪ X′ such that there is +a path between these elements in the graph with the middle row: +• +• +❘ +❘ +❘ +❘ +❘ +❘ +❘ +• +• +❢❢❢❢❢❢❢❢❢❢❢❢❢❢ +• +• +• +• +• +• . +27 + +(See [22, §4.1].) +Let a ∈ Pn. Throughout this section, we will need the following definitions: +ker a = {[x]a ∩ [n] : x ∈ [n]}, +coker a = {[x′]a ∩ [n′] : x′ ∈ [n′]}, +dom(a) = {x ∈ X : x belongs to a transversal block of a}, +codom∧(a) = {x ∈ X : x′ belongs to a transversal block of a}, +coker∧(a) = {A ⊆ [n] : A′ ∈ coker(a)}, +rank(a) = the number of transversal blocks of a. +(We follow [19, §2] and [22, §4.2], with some changes in names and notation to make our arguments clearer.) +We will also need the restriction of ker(a) and coker∧(a) to dom(a) and codom∧(a), respectively. For a ∈ Pn, +we define +kert(a) = {A ∈ ker(a) : A ⊆ dom(a)} and cokert(a) = {B ∈ coker∧(a) : B ⊆ codom∧(a)}. +(4.6) +Note that for every A ∈ kert(a), there exists a unique B ∈ cokert(a) such that A ∪ B′ is a transversal block +of a; and that rank(a) = | kert(a)| = | cokert(a)|. +We now define the following subsets of Pn: +BPn = {a ∈ Pn : each block of a has size at most 2}, +Bn = {a ∈ Pn : each block of a has size 2}. +The subsets BPn and Bn are submonoids of Pn [19, §2], called partial Brauer monoids and Brauer monoids, +respectively. +4.1 +Conjugacy ∼n in Pn, BPn, and Bn +Let b ∈ Pn. As in previous work on Pn, a special role is played by the equivalence relation ker(b)∨coker∧(b). +We say that b is connected if ker(b) ∨ coker∧(b) is the universal relation on {1, . . . , n}. Let s be a block of +b. We say that s is transversal if s ∩ n ̸= ∅ and s ∩ n′ ̸= ∅. If b does not have any transversal blocks, it is +called transversal free; if it has exactly one transversal block, it is called 1-transversal. +Let A ⊆ n be not empty. For b ∈ Pn, we denote by bA the partition of A ∪ A′ (that is, an element of +PA) with [x]bA = [x]b ∩ (A ∪ A′) and [x′]bA = [x′]b ∩ (A ∪ A′), for all x ∈ A. We call bA the subpartition of b +induced by A. In this context, for a block s of b, we use the notation sA = s ∩ (A ∪ A′), and we agree that +any such use is meant to imply that s is a block of b. +A subpartition bA is called trivial if |A| = 1. The definitions of bA being connected, transversal free, and +1-transversal are obtained by adjusting their definitions for b to the index set A in the obvious way. Similarly +we extend the definitions of ker, coker, ker∧, and coker∧ to bA. +For the following results, it will be useful to represent an intermediate step in the calculation of a +partition product. Let n∗ = {1∗, . . . , n∗}. For partitions a, b ∈ Pn, we denote by (a, b)∗ the partition of the +set n ∪ n∗ ∪ n′ that corresponds to the situation before the final deletion of the middle row, where n, n∗, n′ +represent the top, middle, and bottom row, respectively. When a, b are represented by specific graphs, we +represent (a, b)∗ as the graph obtained by identifying corresponding vertices in the lower row of a with those +in the upper row of b, followed by the merging of all double edges. +Recall that we are identifying partitions with their corresponding equivalence relations. For example we +might write (x, y) ∈ b instead of y ∈ [x]b. +Lemma 4.1. Let b ∈ Pn such that bA is connected and transversal-free, it contains blocks sA ⊆ A and +tA ⊆ A′, and for every block rA /∈ {sA, tA}, rA = r. Fix y ∈ A and define c ∈ Pn as follows: +• [y]c = (s \ A) ∪ {y} and [y′]c = (t \ A′) ∪ {y′}; +28 + +• [x]c = {x} and [x′]c = {x′}, for all x ∈ A \ {y}; +• [x]c = [x]b if [x]b does not intersect A ∪ A′, and [x′]c = [x′]b if [x′]b does not intersect A ∪ A′. +Then b ∼n c. +Proof. Define g ∈ Pn by [x]g = [x]b for x ∈ A \ s, [x]g = sA ∪ {y′} for x ∈ sA, [x′]g = {x′} for x′ ∈ A′ \ {y′}, +and [x]g = [x′]g = {x, x′} for x /∈ A. +Define h ∈ Pn by [x′]h = [x′]b for x ∈ A′ \ t, [x′]h = tA ∪ {y} for x′ ∈ tA, [x]h = {x} for x ∈ A \ {y}, and +[x]h = [x′]h = {x, x′} for x /∈ A. +It is easy to see that (gh)A is obtained from bA by merging the upper block sA with the lower block tA, +while outside of A∪A′, gh acts as the identity. Hence, since bA is connected, A∗ is included in a single block +of (gh, b)∗. Note that y∗ ∈ A∗ and that, by the definition of g, (z, y∗) ∈ (gh, b)∗ for every z ∈ sA. +We claim that ghb = b. For any b-block other than s, it is straightforward to check that it is also a +ghb-block (using the hypothesis that rA = r for every block rA ̸= sA, tA). Regarding the block s, select +any z ∈ sA. We want to prove that [z]ghb = s. Let x ∈ s. If x ∈ sA, then x ∈ [z]ghb since sA ⊆ [z]ghb. +Suppose x ∈ s \ sA. Then, (z, y∗), (y∗, z∗), and (z∗, x∗) are in ((gh), b)∗. Since (x, x′) ∈ gh, we also have +(x∗, x) ∈ (gh, b)∗. Thus, by the definition of the product in Pn, (z, x) ∈ ghb. Finally, let x′ ∈ s. Then, +(z, y∗), (y∗, z∗), and (z∗, x′) are in (gh, b)∗, and so (z, x′) ∈ ghb. We have proved that s ⊆ [z]ghb, and equality +s = [z]ghb follows as all other blocks of b are also blocks of ghb. Hence ghb = b. +A similar argument shows that b = bgh. +We now have g(hbg) = (ghb)g = bg, h(b)g = hbg, and +g(hbg)h = (gh)(bgh) = ghb = b. Thus, hgb and b satisfy (i), (iii), and (iv), and so hbg ∼n b by Proposition 2.2. +A straightforward calculation now shows that hbg = c, and so b ∼n c. +The following result is similar to Lemma 4.1, except that the blocks sA and tA are merged. +Lemma 4.2. Let b ∈ Pn such that bA is connected, it has exactly one transversal block sA, and for every +block rA ̸= sA, rA = r. Fix y ∈ A and define c ∈ Pn as follows: +• [y]c = (s \ (A ∪ A′)) ∪ {y, y′}; +• [x]c = {x} and [x′]c = {x′}, for all x ∈ A \ {y}; +• [x]c = [x]b if [x]b does not intersect A ∪ A′, and [x′]c = [x′]b if [x′]b does not intersect A ∪ A′. +Then b ∼n c. +Proof. Define g ∈ Pn by [x]g = [x]b for x ∈ A \ s, [x]g = (sA ∩ A) ∪ {y′} for x ∈ (sA ∩ A), [x′]g = {x′} for +x ∈ A′ \ {y′}, and [x]g = [x′]g = {x, x′} for x /∈ A. +Define h ∈ Pn by [x′]h = [x′]b for x ∈ A′ \ s, [x′]h = (sA ∩ A′) ∪ {y} for x′ ∈ (sA ∩ A′), [x]h = {x} for +x ∈ A \ {y}, and [x]h = [x′]h = {x, x′} for x /∈ A. +Then, as in the proof of Lemma 4.1, we can show that b = ghb = bgh and c = hbg. Hence b ∼n c. +Definition 4.3. Let b ∈ Pn. We say that b is in n-normal form if the following conditions hold: +1. in every non-trivial, connected, transversal-free subpartition bA of b, there exist distinct blocks sA, tA +with sA ̸= s and tA ̸= t, such that either sA, tA ⊆ A or sA, tA ⊆ A′; +2. in every non-trivial, connected, 1-transversal subpartition bA of b, with transversal sA, there exists a +block tA ̸= sA such that t ̸= tA. +Remark 4.4. Applying Lemmas 4.1 and 4.2 to non-trivial connected sets A will result in a partition with an +increased number of singleton blocks. It follows that this process must stop, and hence every n-conjugacy +class contains an element in normal form. +We will next show that in each n-conjugacy class, any partitions a and b in normal form can be obtained +from each other by a permutation of the underlying set n. +29 + +Lemma 4.5. Let a, p ∈ Pn such that ap = pa = a and p is an idempotent. Suppose that there are k, l ∈ n +with (k, l′) ∈ p. Then (k, k∗) ∈ (p, a)∗ and (l∗, l′) ∈ (a, p)∗. +Proof. Suppose that p is represented by the simple graph with the largest possible number of edges. Since +p = p2, (k, l′) is in pp, and hence it is also in (p, p)∗. Since (k, l′) ∈ p, we have (l′, k∗) ∈ (p, p)∗. Hence +(k, k∗) ∈ (p, p)∗. +Let k − · · · − k∗ be a shortest path from k to k∗ in the graph representing (p, p)∗, as obtained from the +maximal graph representing p. Suppose to the contrary that this path contains a vertex j′ ∈ A′. Then, the +path has a subpath i∗ +1 − j′ +1 − · · · − j′ +t − i∗ +2, where t ≥ 1. But t must be 1 since j′ +1 − i∗ +2 (by the fact that p is +represented by the graph with the largest number of edges) and k − · · · − k∗ is a shortest path from k to k∗. +We then have i∗ +1 − j′ +1 − i∗ +2, which implies (i1, j′ +1), (j′ +1, i2) ∈ p. Hence (i1, i2) ∈ p, and so (i∗ +1, i∗ +2) ∈ (p, p)∗. This +a contradiction since we can replace i∗ +1 − j′ +1 − i∗ +2 with i∗ +1 − i∗ +2 obtaining a shorter path from k to k∗. +Now, let a also be represented by the graph with the maximal number of edges. Then because a = pa, +every edge in the graph for (p, p)∗ with no vertex from A′ is also an edge in the graph for (p, a)∗. Thus, the +path k − · · · − k∗ above is also a path in the graph for (p, a)∗. Hence (k, k∗) ∈ (p, a)∗. +Dually, we obtain (l∗, l′) ∈ (a, p)∗. +Lemma 4.6. Let a, p ∈ Pn such that pa = ap = a and p is an idempotent. Let A be a non-empty subset +of n such that aA is connected, ker(aA) = ker(pA), and coker(aA) = coker(pA). Then: +(1) there is at most one a-block s intersecting A such that s is transversal or s is not a block of p; +(2) there is at most one a-block v intersecting A′ such that v is transversal or v is not a block of p. +Proof. Since aA is connected and coker(pA) = coker(aA), the set A∗ is included in a single block of (p, a)∗. +Suppose to the contrary that (1) is false. Then there are three possible cases. +Case 1. There are distinct transversal a-blocks s and t intersecting A. +We then have g, k′ ∈ s and h, l′ ∈ t, where g, h ∈ A. Thus (g∗, k′), (h∗, l′) ∈ (p, a)∗, and so [k′](p,a)∗ = +[l′](p,a)∗ (as A∗ lies within one block). It follows that (k′, l′) ∈ pa, and so (k′, l′) ∈ a since pa = a. This is a +contradiction since s ̸= t. +Case 2. There are a-blocks s and t intersecting A such that s is transversal, t is not transversal, and t is +not a p-block. +As in Case 1, we have g, k′ ∈ s, where g ∈ A. Select h ∈ t ∩ A. Now, [h]p needs to be a transversal block, +for otherwise [h]p = [h]pa = [h]a = t and t is not a p-block. Hence, by Lemma 4.5, (h, h∗) ∈ (p, a)∗. We now +have (g∗, k′), (h∗, h) ∈ (p, a)∗, which implies (h, k′) ∈ pa, and so (h, k′) ∈ a. This is a contradiction since t is +not transversal. +Case 3. There are distinct non-transversal a-blocks s and t intersecting A that are not p-blocks. +Select g ∈ s∩A and h ∈ t∩A. As in Case 2, we obtain (g, g∗), (h, h∗) ∈ (p, a)∗, leading to the contradiction +(g, h) ∈ a. +We have proved (1). Statement (2) follows by a dual argument. +The following result is crucial for proving our characterization of ∼n in Pn. +Proposition 4.7. Let a ∈ Pn be in normal form, and let p ∈ Pn be such that pa = a = ap. Then the kernel +and cokernel of p consist of singletons. +Proof. Suppose, by way of contradiction, that the conclusion is false, that is, there are distinct k, l ∈ n +such that (k, l) ∈ p or (k′, l′) ∈ p. By replacing p with its idempotent power, we may assume that p is an +idempotent. +Suppose (k, l) ∈ p. Then, since pa = a, we have (k, l) ∈ a. Since a is in normal form, it follows that +(k′, l′) /∈ a. Thus, (k′, l′) /∈ p since ap = a. It follows that ker(a{k,l}) = ker(p{k,l}) and coker(a{k,l}) = +coker(p{k,l}). By a dual argument, these equalities also hold if (k′, l′) ∈ p. +30 + +Let A be a subset of n of maximum size such that aA is connected and it satisfies ker(aA) = ker(pA), +coker(aA) = coker(pA). We have |A| ≥ |{k, l}| = 2, so aA is not trivial. +By Lemma 4.6, aA has at most one transversal block, there exists at most one a-block s intersecting A +such that s is transversal or s is not a block of p, and there exists at most one a-block v intersecting A′ such +that v is transversal or v is not a block of p. +Consider the set H = {h ∈ n \ A : [h]a ∩ A ̸= ∅, [h]a ̸= s} (here and in the following, we ignore conditions +of the form [h]a ̸= s if no exceptional block s exist). We claim that for each h ∈ H, there exists lh ∈ A such +that (h′, l′ +h) ∈ a. +For h ∈ H, let t = [h]a. Then t intersects A. Since t ̸= s, t is also a block of p, and hence ker(aA∪{h}) = +ker(pA∪{h}). Moreover, aA∪{h} is connected, and hence by the maximality of the size of A, we conclude +that coker(aA∪{h}) ̸= coker(pA∪{h}). This implies that there is an lh ∈ A such that (l′ +h, h′) ∈ a, (l′ +h, h′) /∈ p. +(Note that coker(pA∪{h}) ⊆ coker(aA∪{h}) since ap = a.) +Consider the set +B = {x ∈ n ∩ s : [x′]a ∩ A′ ̸= ∅} ∪ +� +{u : u is an a-block with u ∩ A ̸= ∅, u ̸= s}. +(If no exceptional block s exists, interpret the first set as ∅, and ignore the condition u ̸= s). +By the +definition of B, we have A ⊆ B (so aB is not trivial), aB is connected, and every a-block intersecting B also +intersects A. Hence, by Lemma 4.6, s is the only a-block intersecting B such that s is transversal or s is not +a block of p. In particular, aB has at most one transversal block, which, if it exists, equals sB. +Moreover, every a-block intersecting B′ also intersects A′. Indeed, let r be an a-block intersecting B′, +say g′ is in the intersection. If g lies in the first set from the definition of B, then r intersects A′ by the +definition of B. Suppose g ∈ u, where u is an a-block included in the second set of the definition of B. If +g ∈ A, then g′ ∈ r ∩ A′. Otherwise, g ∈ u \ A. Since u ̸= s and u ∩ A ̸= ∅, g ∈ H. Hence (l′ +g, g′) ∈ a, with +l′ +g ∈ A′, and so r intersects A′. +By Lemma 4.6 and the fact that every a-block intersecting B′ also intersects A′, v, if it exists, is the only +a-block intersecting B′ such that v is transversal or v is not a block of p. +Suppose aB has a transversal block, which must be equal to both sB and vB. Then s = v and, since a +is normal, there is an a-block w such that w ̸= s (so w ̸= v), w intersects B ∪ B′, and w ̸= wB. The block +w cannot intersect B (by the definition of B), so it intersects B′. Suppose aB is transversal free. Then we +have either two distinct a-blocks intersecting B and extending beyond B ∪ B′, or two blocks intersecting B′ +and extending beyond B ∪ B′. The former is not possible, because only s can extend beyond B ∪ B′ (by the +definition of B). In the second case, one of these blocks, say w, must differ from v. +In either case, we have an a-block w such that w ̸= v, w intersects B′, and w ̸= wB. Since v is the only +a-block intersecting B′ such that v is transversal or v is not a block of p, w ⊆ n′ and w is a block of p. Since +w ̸= wB, there is m′ ∈ w \ B′. +Consider the set A ∪ {m}. Because w is also a block of p and it intersects A′, we have coker(aA∪{m}) = +coker(pA∪{m}). +Thus, by the maximality of the size of A, ker(aA∪{m}) ̸= ker(pA∪{m}). +However, our +construction of B shows that [m]a does not intersect B, and hence it does not intersect A. Because pa = a, +this also holds for [m]p, which implies ker(aA∪{m}) = ker(pA∪{m}). This is a contradiction, which completes +the proof. +Let Sn be the symmetric group of permutations on n = {1, . . . , n}. +Then Sn acts on Pn by aσ +(a ∈ Pn, σ ∈ Sn), where aσ is obtained by replacing x by xσ and y′ by (yσ)′ in each block of a. +For example, if a = {{1, 3}, {2, 4′}, {1′, 2′}, {3′, 4, 5}, {5′}} ∈ P5 and σ = (1 2 5)(3 4) ∈ S5, then aσ = +{{2, 4}, {5, 3′}, {2′, 5′}, {4′, 3, 1}, {1′}}. +For σ ∈ Sn, define λσ = {{x, (xσ)′} : x ∈ n} ∈ Pn. Then Sn = {λσ ∈ Pn : σ ∈ Sn} is the group of units +of Pn, which is isomorphic to Sn. The mapping σ → λσ is an isomorphism for Sn to Sn. Note that for all +a ∈ Pn and σ ∈ Sn, aσ = λ−1 +σ aλσ. +We can now characterize the natural conjugacy ∼n in Pn. +Theorem 4.8. In the partition monoid Pn, every n-conjugacy class contains an element in normal form. +Moreover, if a, b ∈ Pn are in normal form, then a ∼n b if and only if b = aσ for some permutation σ ∈ Sn. +31 + +Proof. The first statement follows by repeated applications of Lemmas 4.1 and 4.2. To simplify the notation +in the proof of the second statement, we will identify any σ ∈ Sn with λσ ∈ Sn. In particular, when we write +σ−1aσ, where a ∈ Pn, we will mean λ−1 +σ aλσ. Let a, b ∈ Pn be in normal form. It is clear that if b = aσ for +some σ ∈ Sn, then a ∼n b. +For the converse, suppose that a ∼n b and let g, h ∈ Pn be conjugators (elements from the definition +of ∼n) for a and b. +Let g1 = (gh)ig, where i ≥ 0 is an integer such that g1h is an idempotent. +It is +straightforward to check that g1 and h are also conjugators for a and b. Now, let h1 = (hg1)jh, where j ≥ 0 +is an integer such that h1g1 is an idempotent. Again, we can check that g1 and h1 are conjugators for a +and b. By a routine calculation, we can show that g1h1 is also an idempotent. Therefore, we may assume +that gh and hg are idempotents. +By Proposition 4.7, the kernel and cokernel of gh and of hg both consist of singletons. It follows that the +same statement holds for g and h. Hence, for every x ∈ n, [x]g = {x, y′} or [x]g = {x}, and [x′]g = {x′, y} or +[x′]g = {x′}, for some y ∈ n. The same statement is true for h. Since gh is an idempotent, for every x ∈ n, +either [x]gh = {x, x′} or [x]gh = {x} and [x′]gh = {x′}. The same statement is true for hg. +Define σ : n → n by +xσ = +� y +if [x]g = {x, y′} or [x′]h = {x′, y}, +x +if [x]g = {x} and [x′]h = {x′}. +By the properties of g, h, gh, and hg stated above, σ is well defined and σ ∈ Sn. By the definition of σ, we +have g ⊆ σ and h ⊆ σ−1. To conclude the proof, it suffices to show that σbσ−1 = a. +Since g ⊆ σ and h ⊆ σ−1, we have a = gbh ⊆ σbσ−1. For the reverse inclusion, let x ∈ n. We will prove +that [x]σbσ−1 ⊆ [x]a and [x′]σbσ−1 ⊆ [x′]a. +Suppose z ∈ [x]σbσ−1. If z = x, then z ∈ [x]a. Suppose z ̸= x. Then, z ∈ [x]σbσ−1 can only happen when +xσ = y1, (y1, y2) ∈ b, and zσ = y2, for some y1, y2 ∈ n. Note that y1 ̸= y2. We have [y1]hg = {y1, y′ +1} or +[y1]hg = {y1}. The latter is impossible since we would have [y1]hgb = {y1}, but hgb = b and y2 ∈ [y1]b. Thus +[y1]hg = {y1, y′ +1}, so there is l ∈ n such that (y1, l′) ∈ h and (l, y′ +1) ∈ g. Hence lσ = y1, which implies l = x +(since xσ = y1), and so (x, y′ +1) ∈ g. By symmetry, (z, y′ +2) ∈ g. We now have (x, y′ +1) ∈ g, (y1, y2) ∈ b, and +(z, y′ +2) ∈ g, which implies z ∈ [x]gbh, and so z ∈ [x]a. +Suppose z′ ∈ [x]σbσ−1. Then, xσ = y, (y, k′) ∈ b, and kσ−1 = z (that is, zσ = k), for some y, k ∈ n. We +have [y]hg = {y, y′} or [y]hg = {y}. The latter is impossible since we would have [y]hgb = {y}, but hgb = b +and k′ ∈ [y]b. Thus [y]hg = {y, y′}, so there is l ∈ n such that (y, l′) ∈ h and (l, y′) ∈ g. Hence lσ = y, which +implies l = x (since xσ = y), and so (x, y′) ∈ g. Further, we have [k′]hg = {k, k′} or [k′]hg = {k′}. The latter +is impossible since we would have [k′]bhg = {k′}, but bhg = b and y ∈ [k′]b. Thus [k′]hg = {k, k′} = [k]hg, so +there is m ∈ n such that (k, m′) ∈ h and (m, k′) ∈ g. Hence mσ = k, which implies m = z (since zσ = k), +and so (k, z′) ∈ h. We now have (x, y′) ∈ g, (y, k′) ∈ b, and (k, z′) ∈ h, which implies z′ ∈ [x]gbh, and so +z′ ∈ [x]a. +We have proved that [x]σbσ−1 ⊆ [x]a. By a dual argument, we obtain [x′]σbσ−1 ⊆ [x′]a. It follows that +σbσ−1 = a, and so b = σ−1aσ, that is, b = aσ. +We next prove some consequences of our classification. +Recall that ∼n⊆ D. +In Pn, the D-classes +correspond to partitions of the same rank. The following characterizes ∼n on partitions of small rank. +Corollary 4.9. In Pn the partitions of rank 0 form one ∼n-class. +Proof. Clearly, the singleton partition is in ∼n-normal form. We claim that it is the only such partition of +rank 0 +If b is any other rank 0 partition, it contains a non-trivial connected subset. Consider a maximal such +subset A. Then any block B in bA must be a block of b for otherwise b would have to be a transversal by +the maximality of B. However, this is impossible as b has rank 0. The set B now witnesses that b is not in +normal form, as required. +Corollary 4.10. In Pn, the partitions of rank 1 form two 2 distinct ∼n-classes, if n ≥ 2, and of a single +∼n-class, if n = 1. +32 + +Proof. Let n ≥ 2. Consider the set T of paritions bx,y′ that contain a single 2-element transversal {x, y′} +and consists of singletons otherwise. Clearly the elements of T are ∼n-normal. By Theorem 4.8 the elements +of T lie in two different ∼n-classes depending on whether x = y or not. +If b is any other rank 1 transformation, it contains a non-trivial connected subset, and hence a maximal +such subset A. Similar to Corollary 4.9 we see that bA can contain at most one block that is not a block of +b. Moreover, this must be the transversal block of bA, if one is present. It follows that A witnesses that b is +not in normal form, as required. +The result for n = 1 is trivial. +We remark that the classes of the corollary can be characterized by the existence or absence of a 1- +transversal connected subpartition. +Corollary 4.11. As n → ∞, the number of ∼n-classes of Pn consisting of rank 2 partitions is not bounded. +Proof. In Pn, consider all partitions consisting of singletons and a subpartition from the following list and +its infinite generalization: +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +It is straightforward to check that all such partitions are in normal form, and pairwise non-conjugate. The +result follows. +The above results explains why it is likely not possible to give a more explicit description of the ∼n-classes +of Pn. If d ≥ 2, we can construct increasingly complex connected, ∼n-normal, and non-conjugate partitions +with rank d. +For checking practical examples, our results imply which connected subpartitions A of a given size can +appear in an ∼n-normal partition (together with information about which blocks t satisfy tA ̸= t). Without +proof, all such subpartitions of size 2 and 3 are given below, up to vertical and horizontal permutation. For +this list only, a pointed arrow indicates that the corresponding block t satisfies tA ̸= t, while the absence of +such an arrow allows both tA = t and tA ̸= t. +• +• +• +• +� +• +� +• +� +• +• +• +• +• +• +� +• +� +• +� +• +• +• +� +• +� +• +• +� +33 + +• +• +• +• +� +• +• +� +• +• +• +� +• +• +• +• +• +• +• +• +• +We now extend our results to the Brauer monoid Bn and the partial Brauer monoid BPn. When it is +necessary for distinction, we write ∼nP , ∼nB and ∼nP B for the natural conjugacy relation in Pn, Bn and +BPn, respectively. Similarly, we will use expression such as “nP B-normal form”. Clearly, ∼nB⊆∼nP B⊆∼nP . +It is straightforward to check that in Lemmas 4.1 and 4.2, if b ∈ BPn, so are the conjugators g, h. As +conjugation by a unit is identical in BPn and Pn, it follows that two partitions are in BPn are conjugate if +and only if they are conjugate in Pn. We are moreover able to give a simpler description of our normal form +in the case of BPn. +Definition 4.12. Let b ∈ BPn. We say that b is in n-normal form if the following conditions hold: +1. If {x, y} is a block, then x′ and y′ lie in transversal blocks; +2. If {x′, y′} is a block, then x and y lie in transversal blocks. +Theorem 4.13. In the partial Brauer monoid BPn, every n-conjugacy class contains an element in normal +form. Moreover, if a, b ∈ Pn are in normal form, then a ∼n b if and only if b = aσ for some permutation +σ ∈ Sn. +Proof. By the above considerations, it suffices to show that an element b ∈ BPn is in ∼nP B-normal form if +and only if it is in ∼nP -normal form. +Suppose that b is in ∼nP B-normal form. Then any non-trivial connected subset A has size 2, is transversal- +free, and one of the 2 conditions from Definition 4.12 hold on A. It follows that b is in ∼nP -normal form. +Conversely, let b be in ∼nP -normal form. Suppose that {x, y} is a block. By normality, x′ and y′ lie +in distinct non-singleton b-blocks. Suppose one, say x′, does not lie in a transversal block. Then there is a +z ̸= z, y such that {x′, z′} is a block. Consider B = {x, y, z}. We have that B is connected and non-trivial. If +{y, z′} is a b-block, then b would violate the second condition of Definition 4.3, for a contradiction. However, +if {y, z′} is not a block, then bB is transversal free, and it is not possible to satisfy the first condition of +Definition 4.3. By contraction, both x′ and y′ lie in transversal blocks. +If {x′, y′} is a block, then a dual argument shows that x and y lie in transversal blocks. The result +follows. +We now turn to the Brauer monoid Bn. Unlike in the previous case, we need a modified version of +Lemmas 4.1 and 4.2. +Lemma 4.14. Let b ∈ Bn such that bA is connected with |A| = 3, say A = {x, y, z} with blocks {x, y} and +{y′, z′}. +If {x′, z} is not a block, then b ∼n c, where c contains the blocks {x, y}, {x′, y′}, [z]b, ([x′]b ∪ z) \ {x′} as +well as all b-blocks not intersecting A ∪ A′ ∪ [z]b ∪ [x′]b. +If {x′, z} is a block, then b ∼n c, where c contains the blocks {x, y}, {x′, y′}, {z, z′} as well as all b-blocks +not intersecting A ∪ A′. +Proof. Define g ∈ Bn with blocks {x, y}, {z, z′}, {x′, y′} and {w, w′} for all w ̸∈ A; define h ∈ Bn with blocks +{x, y}, {z, x′}, {y′, z′} and {w, w′} for all w ̸∈ A. In either of the above cases, it is straightforward to check +that g, h witness b ∼n c. +34 + +Definition 4.15. Let b ∈ Bn. We say that b is in n-normal form if the following conditions hold: +1. If {x, y} is a block, then either {x′, y′} is a block, or x′ and y′ lie in transversal blocks; +2. If {x′, y′} is a block, then either {x, y} is a block, or x and y lie in transversal blocks. +Theorem 4.16. In the Brauer monoid Bn, every n-conjugacy class contains an element in normal form. +Moreover, if a, b ∈ Pn are in normal form, then a ∼n b if and only if b = aσ for some permutation σ ∈ Sn. +Proof. Let b ∈ Bn. If B is a connected subset of b with |B| ≥ 3, then there is a connected set A ⊆ B that +satisfies the conditions of Lemma 4.14. Any application of the lemma will increase the number of maximal +connected subsets. Hence, after repeated application of the lemma we reach a conjugate c of b that only +contains connected subsets of size at most 2. This is equivalent to c being in normal form. +Assume now that b ∼nB c with b, c in nB-normal form. Then b ∼nP c. Let b∗, c∗ be some nP -normal +forms of b, c that are obtained by repeated application of Lemmas 4.1 and 4.2. +By Theorem 4.8, b∗ = λωc∗λ−1 +ω +for some permutation ω. By replacing c with cω we may assume w.l.o.g. +that b∗ = c∗. Because b, c are in ∼nB-normal form, the only non-trivial applications of Lemmas 4.1 and 4.2 +to b, c involve Lemma 4.1 on a connected set A = {x, y} with blocks {x, y} and {x′, y′}. The same also holds +for the outcome of such an application. It follows that b∗, c∗ are obtained from b, c by replacing all blocks in +such subpartitions with singletons. +Let D ⊆ n be the largest set for which b∗ +D = c∗ +D consist of singleton blocks. Then |D| is even, and there +are two partition Db +i, Dc +j of D into blocks of size two such that bDb +i , cDc +j consist of two non-transversal blocks +each, for all i and j. In addition, on the complement ¯D = n \ D, we have that b ¯ +D = b∗¯ +D = c∗¯ +D = c ¯ +D. The +result now follows. +4.2 +Conjugacy ∼tr in Pn, BPn, and Bn +To characterize trace conjugacy ∼tr (see (1.8)) in Pn, we first need to describe the group elements of +Pn. Let S be any semigroup. The maximal subgroups of S are the H-classes He of S such that e is an +idempotent [15, Ex. 1, p. 61]. An element a ∈ S is a group element of S if a ∈ He for some idempotent +e ∈ S. These element are also called completely regular, as in Section 2.4. +Lemma 4.17. Let a, b ∈ Pn. Then: +(1) a R b ⇐⇒ ker(a) = ker(b) and kert(a) = kert(b); +(2) a L b ⇐⇒ coker(a) = coker(b) and cokert(a) = cokert(b). +Proof. By [22, Prop. 4.2], (1) and (2) are true if kert and cokert are replaced by dom and codom∧, respectively. +If ker(a) = ker(b), then dom(a) = dom(b) +⇐⇒ +kert(a) = kert(b); and if coker(a) = coker(b), then +codom∧(a) = codom∧(b) ⇐⇒ cokert(a) = cokert(b). The result follows. +We also have a D b ⇐⇒ rank(a) = rank(b), and D = J [22, Prop. 4.2]. +For equivalence relations ρ1 and ρ2 on X, the join ρ1 ∨ρ2 of ρ1 and ρ2 is the smallest equivalence relation +containing the union ρ1 ∪ ρ2. To describe the group elements of Pn, we will need the join ker(a) ∨ coker∧(a), +where a ∈ Pn. +First, the idempotents of Pn were described in [19, Thm. 5]. +Lemma 4.18. Let e ∈ Pn. Then, e is an idempotent if and and only if the following two conditions are +satisfied: +(1) for every transversal block A ∪ B′ of e, there exists a block P (necessarily unique) of ker(e) ∨ coker∧(e) +such that A ∪ B′ ⊆ P ∪ P ′; +(2) for every block P of ker(e) ∨ coker∧(e), P ∪ P ′ contains at most one transversal block of e. +35 + +Proposition 4.19. Let a ∈ Pn. Then, a is an element of a group H-class of Pn if and only if for every +block P of ker(a) ∨ coker∧(a) one of the following conditions holds: +(a) neither P nor P ′ intersects a transversal block of a; or +(b) each of P and P ′ intersects exactly one (not necessarily the same) transversal block of a. +Proof. Suppose that a is an element of a group H-class H of Pn. Let e be the identity of H, so a H e. By +Lemma 4.17, ker(a) ∨ coker∧(a) = ker(e) ∨ coker∧(e), kert(a) = kert(e), and cokert(a) = cokert(e). Let P be +a block of ker(a) ∨ coker∧(a). +Suppose that P does not intersect any transversal block of a. Suppose to the contrary that P ′ intersects +some transversal block A∪B′ of a. Then B′ ⊆ P ′ and B′ ∈ cokert(a). Since cokert(a) = cokert(e), it follows +by Lemma 4.18 that there is C ∈ kert(e) such that C ∪ B′ ⊆ P ∪ P ′. Since kert(e) = kert(a) and C ⊆ P, +the block P intersects some transversal block of a, which is a contradiction. We have proved that if P does +not intersect any transversal block of a, then (a) holds. Similarly, (a) holds if P ′ does not intersect any +transversal block of a. +Suppose (a) does not hold. Then P intersects some transversal block A ∪ B′ of a. If it also intersected +another transversal block of a, say C ∪ D′, then we would have A, C ∈ ker(e), A, C ⊆ P, and A ̸= C, which +would contradict Lemma 4.18(2). A similar argument can be applied to P ′, which implies that (b) holds. +Conversely, suppose that for every block P of ker(a)∨coker∧(a), (a) or (b) holds. Let k(a) be the number +of blocks P such that P intersects a transversal block A ∪ B′ of a, and P ′ intersects a different transversal +block C ∪ D′ of a. If k(a) = 0, then a is an idempotent (and so a group element) by Lemma 4.18. Let +k(a) ≥ 1 and consider P, A ∪ B′, and C ∪ D′ as above. Then, A ⊆ P, D′ ⊆ P ′, B′ ⊆ Q′, and C ⊆ R, where +Q and R are blocks of ker(a) ∨ coker∧(a) such that P /∈ {Q, R}. Construct a1 ∈ Pn by replacing in a the +transversal blocks A∪B′ and C ∪D′ by A∪D′ and C ∪B′. Then k(a1) < k(a) (since P and P ′ both intersect +the same transversal block of a1, namely A∪D′), and it is straightforward to check, using Lemma 4.17, that +a H a1. Applying this construction repeatedly, we obtain (after at most k(a) steps) an element e ∈ Pn such +that k(e) = 0 (so e is an idempotent) and a H e. Hence a is a group element. +Let σ ∈ Sm, where Sm is the symmetric group of permutations on [m] = {1, . . . , m}. We allow m to be +zero, in which case [m] = ∅, Sm = {∅}, and σ = ∅. The cycle type of σ is the sequence (k1, . . . , km), where ki +is the number of cycles of length i in the cycle-decomposition of σ. If m = 0, then we define the cycle type +of σ as (0). +Definition 4.20. Let a ∈ Pn be a group element. By Proposition 4.19, for every block P of ker(a)∨coker∧(a), +either P does not intersect any transversal block of a or there is a unique A ∈ kert(a) such that A ⊆ P. Let +{P1, . . . , Pm} be the set of all blocks of ker(a) ∨ coker∧(a) that intersect some transversal block of a. For +each i ∈ [m], let Ai be a unique element of kert(a) such that Ai ⊆ Pi. Note that kert(a) = {A1, . . . , Am}. +By Proposition 4.19 again, each P ′ +i contains a unique B′ +i ∈ cokert(a) and cokert(a) = {B′ +1, . . . , B′ +m}. Note +that m can be 0, which happens when kert(a) = cokert(a) = ∅. +Define τa : [m] → [m] by +iτa = j +⇐⇒ +Ai ∪ B′ +j is a transversal block of a . +By Proposition 4.19, τa ∈ Sm. We define the cycle type of a to be the cycle type of τa. Note that τa depends +on the ordering of {P1, . . . , Pm}, but the cycle type of τa is the same regardless of an ordering. +Let e be the idempotent in the group H-class of a. Then the transitive blocks of e are A1∪B′ +1, . . . , Am∪B′ +m, +and the transitive blocks of a are A1 ∪ B′ +1τa, . . . , Am ∪ B′ +mτa. +Lemma 4.21. Let e, f, g, h ∈ Pn such that e and f are idempotents, gh = e, hg = f, ghg = g, and hgh = h. +Then kert(g) = kert(e) and cokert(g) = cokert(f). +Proof. We have g R e (since gh = e and eg = ghg = g) and g L f (since hg = f and gf = ghg = g). Thus, +by Lemma 4.17, kert(g) = kert(e) and cokert(g) = cokert(f). +36 + +We can now characterize the trace conjugacy ∼tr in Pn. +Theorem 4.22. Let a, b ∈ Pn. Then a ∼tr b if and only if aω+1 and bω+1 have the same cycle type. +Proof. Let e = aω, f = bω, u = aω+1, and v = bω+1. Suppose that a ∼tr b. By (1.8), there exist g, h ∈ Pn +such that +ghg = g, hgh = h, gh = e, hg = f, and hug = v. +We also have gvh = ghugh = eue = u. +By Lemma 4.21 and the fact that u H e and v H f, we have +kert(g) = kert(e) = kert(u), cokert(g) = cokert(f) = cokert(v), kert(h) = kert(f) = kert(v), and cokert(h) = +cokert(e) = cokert(u). Let m = | kert(e)|. Then, by the above equations, | kert(f)| = | kert(u)| = | kert(v)| = +| kert(g)| = | kert(h)| = m. +Let {P1, . . . , Pm} be the set of all blocks of ker(e) ∨ coker∧(e) that intersect some transversal block of e, +and let {Q1, . . . , Qm} be the set of all blocks of ker(f) ∨ coker∧(f) that intersect some transversal block of f +(see Definition 4.20). (We have the same m since | kert(e)| = | kert(f)| = m.) Since e and f are idempotents, +the transversal blocks of e and of f are, respectively, Ai ∪ B′ +i with Ai ⊆ Pi and B′ +i ⊆ P ′ +i , and Ci ∪ D′ +i with +Ci ⊆ Qi and D′ +i ⊆ Q′ +i, where i ∈ [m]. Since u ∈ He and v ∈ Hf, the transversal blocks of u and of v +are, respectively, Ai ∪ B′ +iτu and Ci ∪ D′ +iτv, where i ∈ [m] (see Definition 4.20). Since kert(g) = kert(e) and +cokert(g) = cokert(f), there is σ ∈ Sm such that the transversal blocks of g are Ai ∪ D′ +iσ, where i ∈ [m]. +Finally, since kert(h) = kert(f) and cokert(h) = cokert(e), there is δ ∈ Sm such that the transversal blocks +of h are Ci ∪ B′ +iδ, where i ∈ [m]. +We claim that σ = δ−1. Let i ∈ [m]. Since Ai ∪ D′ +iσ is a block of g and Ciσ ∪ B′ +i(σδ) is a block of h, +we conclude that Ai ∪ B′ +i(σδ) is a block of gh. Further, e = gh and Ai ∪ B′ +i is a block of e, which implies +i(σδ) = i. Hence σ = δ−1. +Our second claim is that στuδ = τv. Let i ∈ [m]. Since Ai ∪ D′ +iσ is a block of g and Ciσ ∪ D′ +i(στv) is +a block of v, we conclude that Ai ∪ D′ +i(στv) is a block of gv. Thus, since Ci(στv) ∪ B′ +i(στvδ) is a block of h, +it follows that Ai ∪ B′ +i(στvδ) is a block of gvh. But, gvh = u and Ai ∪ B′ +iτu is a block of u, which implies +i(στvδ) = iτu. Hence στuδ = τv. +Thus, δ−1τuδ = τv, and so τu and τv are group conjugate in Sm. Hence, τu and τv have the same cycle +type, and so aω+1 (= u) and bω+1 (= v) have the same cycle type (see Definition 4.20). +Conversely, suppose that aω+1 and bω+1 have the same cycle type. Then τu and τv are group conjugate +in Sm, that is, there are σ, δ ∈ Sm such that σ = δ−1 and στuδ = τv. With the notation for the transversal +blocks of e, f, u, and v as in the first part of the proof, let g ∈ Pn be such that ker(g) = ker(e) (= ker(u)), +coker(g) = coker(f) (= coker(v)), and the transversal blocks of g are Ai ∪ Diσ, where i ∈ [m]. Similarly, +let h ∈ Pn be such that ker(h) = ker(f) (= ker(v)), coker(h) = coker(e) (= coker(u)), and the transversal +blocks of h are Ci ∪Biδ, where i ∈ [m]. Simple calculations (similar to the ones in the first part of the proof) +show that ghg = g, hgh = h, gh = e, hg = f, and hug = v. Hence a ∼tr b. +Turning to BPn and Bn, it is clear that ∼trB⊆∼trP B⊆∼trP , and hence for two ∼tr-conjugate partitions +a, b ∈ BPn or Bn, aω+1 and bω+1 have the same cycle type. Conversely, if a, b are two such partitions in BPn +[in Bn], it is straightforward to check that the conjugators g, h constructed in the second part of Theorem +4.22 lie in BPn [in Bn]. Hence we obtain the following characterization. +Theorem 4.23. Let a, b ∈ Pn or a, b ∈ Bn. Then a ∼tr b if and only if aω+1 and bω+1 have the same cycle +type. +4.3 +Conjugacy ∼∗ +p in Pn, BPn, and Bn +In any epigroup, ∼∗ +p ⊆ ∼tr [4, Thm. 4.8]. The reverse inclusion is not true in the class of epigroups [4, +Thm. 4.15]. The goal of this subsection is to show that in Pn, ∼∗ +p = ∼tr. +(See (1.2) and (1.4) for the +definitions of ∼p and ∼∗ +p.) +37 + +Lemma 4.24. Let a ∈ Pn, and s ⊆ n a non-transversal a-block, such that s′ intersects one (or more) +transversal a-blocks. Then a has a ∼p-conjugate c ∈ Pn such that cs is transversal free, and such that c has +more blocks than a. +Proof. Let u ∈ Pn have the blocks s, {z′}, where z ∈ s, and {k, k′}, where k /∈ s. By straightforward +calculations, we check that ua = a. The partition c = au has blocks t \ s′, for every a-block t satisfying +t ̸⊆ s′, and {z′} for z ∈ s. Clearly cs is transversal-free. As we assumed that at least one transversal a-block +intersects s′, c has more blocks than a. +Clearly, a dual result holds if s′ is a non-transversal block such that s intersects a transversal block. +Lemma 4.25. Let a ∈ Pn, s an a-block, A = s ∩ n, such that A′ intersect two different a-blocks t1, t2 (one +of which might be s). Then a ∼p c, where c is obtained from a by merging the blocks t1, t2. +Proof. Let x, y ∈ A, with x′ ∈ t1, y′ ∈ t2. +Let v ∈ Pn have the blocks {x, y, x′, y′} and {z, z′}, where +z /∈ {x, y}. By straightforward calculations, we check that va = a and that av has the desired properties. +Once again, clearly the dual version of the Lemma 4.25 holds as well. +Proposition 4.26. Let a ∈ Pn. Then, there exists a group element c ∈ Pn such that a ∼∗ +p c. +Proof. We recursively apply Lemma 4.24 [or its dual] to a, as long as we find a non-transversal block s [resp. +s′] such that s′ [resp. s] intersects a transversal nlocks. Because the number of blocks increases at each step, +this process must stop with a partition b ∼∗ +p a for which dom(b) = codom∧(b). +We next apply Lemma 4.25 (or its dual) to all cases in which the involved blocks t1, t2 are transversal +(note that this means that s is also transversal). Each such application will preserve the condition dom(·) = +codom∧(·), as only transversal blocks will be merged. As this decreases the number of blocks, this process +will stop with an element c ∼∗ +p b ∼∗ +p a such that +1. dom(c) = codom∧(c); +2. if s is a transversal c-block, A = s ∩ n, then A′ intersects at most one transversal c-block; +3. if s is a transversal c-block, A′ = s ∩ n′, then A intersects at most one transversal c-block. +We will show that these conditions imply that c is a group element. Let P be a block of ker(c) ∨ coker∧(c). +If P does not intersect any transversal block of c, then, by 1., neither does P ′ (and vice versa). +Suppose that s = A∪B′ is a transversal c-block, and let P and Q be the blocks of ker(c)∨coker∧(c) such +that A ⊆ P and B′ ⊆ Q′. We claim that s = P ∪Q′. By 1., any block intersected by A′ must be transversal. +Thus, by 2., there exists a transversal c-block t such that A′ ⊆ C′, where C′ = t ∩ n′. Applying the dual +argument to C′ and using 3., we obtain a transversal c-block w such that C ⊆ D, where D = w ∩ n. Since +A′ ⊆ C′, we have A ⊆ C ⊆ D, so A ⊆ s ∩ w. Thus, s = w, A = C = D, and A′ = C′ = D′. +We will now prove that A = P. Let x ∈ P and select any y ∈ A. Since A ⊆ P, we have (y, x) ∈ +ker(c) ∨ coker∧(c), and so there are y = z0, z1, . . . , zk = x in n such that for every i ∈ {0, . . . , k − 1}, +either (zi, zi+1) ∈ ker(c) or (zi, zi+1) ∈ coker∧(c). +Let i ∈ {0, . . ., k − 1} and suppose that zi ∈ A. +If +(zi, zi+1) ∈ ker(c), then zi+1 ∈ A. Suppose (zi, zi+1) ∈ coker∧(c), that is, (z′ +i, z′ +i+1) ∈ coker(c). Then x′ +i ∈ C′ +(since A′ = C′), and so x′ +i+1 ∈ C′ (since C′ ⊆ t). Thus zi+1 ∈ C, and so zi+1 ∈ A. Since y = z0 ∈ A, it +follows that x = zk ∈ A, and so P = A. +By a dual argument, B′ = Q′, and so s = P ∪ Q′. Hence, c is a group element by Proposition 4.19. +Theorem 4.27. In Pn, ∼∗ +p = ∼tr. That is, for a, b ∈ Pn, a ∼∗ +p b if and only if aω+1 and bω+1 have the +same cycle type. +Proof. Let a, b ∈ Pn. Suppose that a ∼tr b. By Proposition 4.26, there are group elements c and d of Pn such +that a ∼∗ +p c and b ∼∗ +p d. Since ∼∗ +p ⊆ ∼tr, we have c ∼tr a ∼tr b ∼tr d, and so c ∼tr d. By [4, Thm. 4.15], as +relations on the group elements of any semigroup, ∼p = ∼∗ +p = ∼tr. Thus, c ∼p d, and so a ∼∗ +p c ∼p d ∼∗ +p b, +which implies a ∼∗ +p b. We have proved that ∼tr ⊆ ∼∗ +p. Since ∼∗ +p ⊆ ∼tr in any epigroup, ∼∗ +p = ∼tr. +38 + +Let a, b ∈ Pn. We can check if a and b are p∗-conjugate (equivalently, tr-conjugate) in two ways. We +can calculate the successive positive powers of a and b until we obtain idempotents e and f, respectively. +Then we check if ea (= aω+1) and fb (= bω+1) have the same cycle type. Or, using Proposition 4.26 and +Lemmas 4.24 and 4.25, we calculate group elements c, d such that a ∼∗ +p c and b ∼∗ +p d, and we check if c and d +have the same cycle type. +We now turn to BPn and Bn. Let a ∈ BPn. In this case, the partition u constructed in Lemma 4.24 is +an element of BPn as well, and therefore Lemma 4.24 and its dual also hold in BPn. We can now repeat +the proof of Proposition 4.26, noting that the situations in which Lemma 4.25 or its dual are used cannot +arise in BPn: if s is transversal, than A = s ∩ n is a singleton, so A′ cannot intersect different blocks t1, t2. +As in Theorem 4.27, we obtain: +Theorem 4.28. In BPn, ∼∗ +p = ∼tr. That is, for a, b ∈ BPn, a ∼∗ +p b if and only if aω+1 and bω+1 have the +same cycle type. +Lemma 4.29. Suppose that a ∈ Bn, {x, y} ⊆ n is a block of a, such that x′, y′ lie in (necessarily distinct) +transversal blocks. Then a ∼p c, for some c ∈ Bn with lower rank than a. +Proof. Let {v, x′}, {w, y′} be the blocks containing x′, y′, and k the number of upper blocks of a. As a is a +partition in Bn, k is also the number of lower blocks. Consider u ∈ Bn with the following blocks: s and s′ +for each upper block s of a, and {z, z′} for each z ∈ n that does not intersect an upper block of a. +It is straightforward to check that ua = a. Let c = au, so c ∼p a. The k upper blocks of a are also upper +blocks of c. In addition, {v, w} is an upper block of c. So c has more than k upper blocks, and hence also +more than k lower blocks. It follows that it has fewer transversal blocks than a, as required. +Clearly, the dual version of Lemma 4.29 holds as well. +Proposition 4.30. Let a ∈ Bn. Then there exists a group element c ∈ Bn such that a ∼∗ +p c. +Proof. Recall that ∼n ⊆ ∼∗ +p. Let a ∈ Bn. Then a ∼n b (and hence a ∼∗ +p b) for some b in n-normal form. +Suppose that there is a b-block {x, y} as in Lemma 4.29. We can then use Lemma 4.29 to obtain an element +c such that b ∼∗ +p c and c has a lower rank than b. If instead there is a b-block {x′, y′} such that x, y lie +in transversal b-blocks, than we can find such c using the dual version of Lemma 4.29. We next obtain a +partition a1 ∈ Bn in n-normal form satisfying c ∼n a1. Note that c and a1 have the same rank as ∼n ⊆ D +(by Proposition 2.4). +We have constructed an element a1 ∈ Bn in n-normal form such that a ∼∗ +p a1 and a1 has a lower rank +than a. We keep repeating this construction until we obtain a partition d ∈ Bn such that a ∼∗ +p d, d is in +n-normal form, and neither Lemma 4.29 nor its dual can be applied to d. (Note that d may be b if neither +Lemma 4.29 nor its dual can be applied to b.) By Definition 4.15, this means that {x, y} is an upper block +of d if and only if {x′, y′} is a lower block of d. Hence d is a group element. +As in Theorem 4.27, we obtain: +Theorem 4.31. In Bn, ∼∗ +p = ∼tr. That is, for a, b ∈ Bn, a ∼∗ +p b if and only if aω+1 and bω+1 have the +same cycle type. +4.4 +Conjugacies ∼o and ∼c in Pn, BPn, and Bn +The conjugacy ∼o (1.3) is the largest of the conjugacies considered in this paper. In any semigroup, ∼n ⊆ ∼p +⊆ ∼∗ +p ⊆ ∼o and ∼n ⊆ ∼c ⊆ ∼o [38, Prop. 2.3]. In any epigroup, ∼n ⊆ ∼p ⊆ ∼∗ +p ⊆ ∼tr ⊆ ∼o [4, Thm 4.8]. +Moreover, for any semigroup S, ∼o is the universal relation if S has a zero, and ∼o = ∼c if S has no zero. +It is known that ∼o is the identity relation on a semigroup S if and only if S is commutative and +cancellative [4, Thm. 5.6]. There is no characterization of the semigroups (with no zero) in which ∼o is the +universal relation. In the finite partition monoids, which have no zero, ∼o is the universal relation. +Theorem 4.32. In Pn, ∼o = Pn × Pn. +39 + +Proof. Let e = {{x, x′} : x ∈ [n]} be the identity in Pn and let a ∈ Pn be arbitrary. We want to find g ∈ Pn +such that ag = ge. Consider g ∈ Pn such that ker(g) = ker(aω), coker(g) = {{x′} : x′ ∈ [n′]}, and g does not +have any transversal blocks. Then ker(ag) = ker(aaω) = ker(aω+1) = ker(aω) = ker(g), where the last but +one equality follows from the fact that aω+1 H aω. Since coker(g) is trivial and g has no transversal blocks, +coker(ag) is also trivial and ag has no transversal blocks either. Thus ag = g = ge. Similarly, for h ∈ Pn +such that coker(h) = coker(aω), ker(h) = {{x} : x ∈ [n]}, and h does not have any transversal blocks, we +have ha = h = eh. We have proved that for every a ∈ Pn, a ∼o e. Hence ∼o = Pn × Pn since ∼o is an +equivalence relation. +In the case that a ∈ BPn, the elements g and h constructed as above are in BPn as well. Hence we +immediately obtain the following classification. +Theorem 4.33. In BPn, ∼o = BPn × BPn. +We now consider ∼o for a Brauer moniod Bn. As ∼tr⊆∼o, it follows from Theorem 4.23 that there is a +partition Q of the set of available cycle types, such that a ∼o b if and only if the cycle types of aω+1 and +bω+1 lie in the same part of Q. Moreover, as ∼n⊆∼o, Theorem 4.16 shows that a has a ∼o-conjugate c in +n-normal form (see Definition 4.15). We will show below that this element can be chosen as a group element. +The following lemma provides a description of such partitions, which follows directly from Theorem 4.16 +and Definition 4.15. +Lemma 4.34. Suppose that c ∈ Bn is both a group element and in n-normal form. Then there is a partition +n = A∪B, such that A∪A′ contains all transversal b-blocks and B ∪B′ contains all non-transversal b-blocks +(where we allow A = ∅ or B = ∅). +Moreover, there is a parition of B into subsets Bi of size 2, such that Bi and B′ +i are blocks for all i. +We remark that |B| is even, and that we may identify cA with a permutation in SymA. +Lemma 4.35. Let a ∈ Bn be a group element. Then there is a partition b ∈ Bn in n-normal form such that +b is a group element with the same cycle type as a. +Proof. Let k be the number of blocks of ker(a)∨coker∧(a) that are used in the construction of the permutation +corresponding to a (that is, the blocks of ker(a) ∨ coker∧(a) that intersect a transversal block of a). Pick a +k-subset A of n. Using only transversal blocks, we can construct a partition bA on A ∪ A′ that has the same +cycle type as a (and which we might consider to be an element of SymA). +In Bn, a block of ker(a) ∨ coker∧(a) that intersects one transversal of a has odd cardinality, while a block +of ker(a) ∨ coker∧(a) that does not has even cardinality. It follows that |n \ A| is even. +Partitioning B = n \ A into 2-element sets Bi, we can extend ba to a partition b ∈ Bn by adding the +blocks Bi, B′ +i for each i. The result follows. +If the permutation associated with bA contains a cycle of size l, it is clear that we may identify a subset +C of A such that bC represents this cycle. In the following, when we speak of such a representation, we will +always assume that |C| = l (so unlike in the standard use of “cycle”, we do not allow any additional 1-cycles +to be represented in C). +Lemma 4.36. Let a ∈ Bn be a group element in n-normal form, and suppose that C ⊆ n is such that aC +represents a cycle of even length l. Then there is a partition of C into 2-subsets Ci and b ∈ Bn such a ∼o b, +b contains the blocks Ci, C′ +i for all i, and aD = bD for D = n \ C. +Proof. Order the elements of C as c1, . . . , cl, such that the a-blocks intersecting C are {cl, c′ +1} and {ci, c′ +i+1} +for i = 1, . . . , l − 1. +Partition C into blocks Ci = {ci, ci+l/2} for i = 1, . . . l/2, define g ∈ Bn with blocks Ci, C′ +i and {z, z′} for +z /∈ C, and set g = h. It is straightforward to check that g, h witness a ∼o b. +Lemma 4.37. Let a ∈ Bn be a group element in n-normal form, and suppose that C, D ⊆ n, C ̸= D are +such that aC, aD represents cycles of the same length l. Then there is a partition of C ∪ D into 2-subsets Gi +and b ∈ Bn such a ∼o b, b contains the blocks Gi, G′ +i for all i, and aL = bL for L = n \ (C ∪ D). +40 + +Proof. Suppose that C = {c1, c2, . . . , cl}, D = {d1, d2, . . . , dl} are ordered such that {cl, c′ +1}, {dl, d′ +1}, +{ci, c′ +i+1} and {di, d′ +i+1}, i = 1, . . . , l − 1, are the a-blocks intersecting C ∪ D. +Partition C ∪ D into blocks Gi = {ci, di} for i = 1, . . . l, define g ∈ Bn to have blocks Gi, G′ +i and {z, z′} +for z /∈ C ∪ D, and set g = h. It is straightforward to check that g, h witness a ∼o b. +Theorem 4.38. Let a, b ∈ Bn, such that aω+1 and bω+1 have cycle types (k1, . . . , kn) and (l1, . . . , ln), +respectively. Then a ∼o b if and only if ki ≡ li mod 2 for each odd i. +Proof. Suppose that ki ≡ li mod 2 for each odd i. +Because ∼tr⊆∼o, and by Lemma 4.35, there exist +partitions a′ ∼o a, b′ ∼o b, such that a′, b′ are both group elements in ∼n-normal form with the same cycle +type as a, b. +By repeated applications of the constructions from Lemmas 4.36 and 4.37, we obtain partitions a′′ ∼o +a′, b′′ ∼o b′, such that a′, b′ are both group elements in ∼n-normal form, and such the permutations corre- +sponding to a′′, b′′ contain no even cycles and at most one j-cycle for each odd j. Moreover, a′′ [b′′] contains +an odd j-cycle exactly if kj [lj] is odd. As we assumed that ki ≡ li mod 2 for each odd i, we see that a′′ +and b′′ have the same cycle type. It follows that a′′ ∼tr b′′, thus a′′ ∼o b′′, and hence a ∼o b, as required. +Assume now that ki ̸≡ li mod 2 for some odd i. Let a′′ ∼o a, b′′ ∼o b be constructed as in the first +part, and construct a′′′ and b′′′ from a′′, b′′ by replacing all blocks of the form {x, y}, {x′, y′} with blocks +{x, x′}, {y, y′}. As this introduces an even number of 1-cycles, it follows that a′′′ ∼o a, b′′′ ∼o b by the first +part of this proof, and moreover that the condition ki ̸≡ li mod 2 carries over to the cycle types of a′′′ and +b′′′. Moreover, a′′′, b′′′ are unit elements whose corresponding permutations only contains odd cycles with at +most one j-cycle for j ̸= 1. +By abuse of notation, we will rename a′′′, b′′′ as a, b. +Our aim os to show that a ̸∼o b. +By way of +contradiction, assume that g, h ∈ Bn witness a ∼o b. +Let Xa, Xb ⊆ n be the set of values z for which {z, z′} is a block of a or b, respectively (i.e. the values +corresponding to 1-cycles of a, b.) We claim that |Xa| = |Xb|. +Consider z ∈ Xa, and assume that z lies in a transversal block {z, u′} of g. Then {z, u′} is a block of +ag = gb. Hence {u, u′} is a block of b, and u ∈ Xb. A dual argument shows that if z ∈ Xb and the g-block +{z′, u} containing z′ is a transversal, then u ∈ Xa. Hence g induces a bijection between subsets Za ⊆ Xa, +Zb ⊆ Xb, where Za, Z′ +b consists of those elements of Xa, Xb that lie in transversal blocks of g. +It follows that the elements of Xa \ Za, and X′ +b \ Z′ +b lie in non-transversal blocks of g. As g ∈ Bn, it has +the same number of upper and lower non-transversal blocks. Hence to show the claim, it suffices to show +that all non-transversal blocks of g lie in Xa or X′ +b. +Let {x, y} be an upper block of g. Then {xa−1, ya−1} is an upperblock of ag = gb. As b is a unit, this is +only possible if {xa−1, ya−1} is an upper g-block. Repeating this argument, we see that {xa−i, ya−i} is an +upper g-block for all i. +Now suppose that x, y lie in some set C ⊆ n such that C sorresponds to one l-cycle of a with l ̸= 1. It +follows that C is a union of upper blocks of g. However, l is odd, so this is not possible. +Assume instead that x ∈ C, y ∈ D, such that C, D represents a-cycles of different size. Then there is an +i such that, w.l.og. xa−i = x, ya−i ̸= y, contradicting that {x, y} is a g-block. +It follows that {x, y} ⊆ Xa. By a dual argument, if {x′, y′} is a lower block of g, then x, y ∈ Xb. The +claim follows, and so |Xa| = |Xb| = k1 = l1, which also implies that i ̸= 1. +By replacing b with a conjugate of the form ubu−1 for a suitable unit u and g with gu−1, we may assume +w.l.o.g. that Xa = Xb (we once again abuse notation and name this new partitions b and g.) This process +preserves the cycle type of b. +Applying the above considerations to our new value of g, we see that all g-blocks intersecting Xa ∪ X′ +a +are subsets of Xa ∪ X′ +a, and that, moreover, all non-transversal g-blocks lie in Xa ∪ X′ +a. It follows that all +g-blocks intersecting Y = n\ Xa are transversal blocks and intersect n\ X′ +a. Hence the induced subpartition +gY is a unit element of BY , corresponding to a permutation of Y . Trivially, this is also true for aY , bY . +Moreover the cycles types of aY , bY agree with those of a, b, except for the first position. +In BY , we have aY gY = gY bY . Working in the unit group of By, we obtain that g−1 +Y aY gY = bY , which +is an equation of permutations. However, this is not possible, as we assumed that ki ̸≡ li mod 2 for some +41 + +odd i, i ̸= 1. +By contradiction, a ̸∼o b, as required. +Since ∼c = ∼o in any semigroup that does not have a zero, we obtain the following result. The listed +exceptional cases contain a zero and can be confirmed by direct calculation (See (1.5) for the definition of +∼c.) +Theorem 4.39. In Pn, BPn, and Bn, ∼o = ∼c, except for P1, PB1, B2, where ∼c is equality. +That is, on Pn and BPn, ∼c is the universal relation, except for P1, PB, where ∼c is equality. +If a, b ∈ Bn, n ̸= 2, such that aω+1 and bω+1 have cycle types (k1, . . . , kn) and (l1, . . . , ln), then a ∼c b if +and only if ki ≡ li mod 2 for each odd i. On B2, ∼c is equality. +5 +Conjugacy growth in polycyclic monoids +The study of conjugation in polycyclic monoids was initiated in [3] by some of the authors of this article. +Polycyclic monoids are inverse monoids with zero so ∼o is the universal relation and ∼i = ∼n. In [3] the +notions of ∼p (1.2), and ∼c (1.5) were characterized. In this section we intend to present a study on ∼n +(1.7). +The conjugacy growth function of a finitely generated group G counts the number of conjugacy classes +intersecting the ball of radius n in the Cayley graph of G centered at the identity, for all n ≥ 0. It has +been studied for free groups [16,52,53], hyperbolic groups [17,18], solvable groups [9], linear groups in [10], +acylindrically hyperbolic groups [1,36], certain branch groups [27], in the higher Heisenberg groups in [24], +and several other classes of groups [31]. +Given a notion of conjugation for monoids that is an equivalence relation, the conjugacy growth function +of the groups can be extended to finitely presented monoids. In this section we will present the conjugacy +growth functions of the polycyclic monoids, for the conjugations ∼n, ∼c, and ∼∗ +p. +In the last few years, the conjugacy growth series (the generating series associated with the conjugacy +growth functions) have been computed for several classes of groups based on the description of sets consisting +of minimal length representatives from all conjugacy classes [1,11–14]. The paper [23] supports the conjecture +that virtually abelian groups are the only ones with rational conjugacy series. Historically, one of the initial +motivations for counting conjugacy classes of a given length came from counting closed geodesics of bounded +length in compact Riemannian manifolds [46]. +We first need some preliminaries. +5.1 +Characterization of the conjugacy relations in Pn +Let n ≥ 2. Consider a set An = {p1, . . . , pn}, and denote by A−1 +n +a disjoint copy {p−1 +1 , . . . , p−1 +n }. +Let +Σ = An ∪ A−1 +n . The polycyclic monoid Pn is the monoid with zero defined by the monoid presentation +Pn = ⟨Σ0 | p−1 +i pi = 1 and p−1 +i pj = 0, i ̸= j}⟩, where Σ0 = Σ ∪ {0} and 0 is a symbol that is not in Σ that is +interpreted as the zero of the monoid by what we consider implicit the multiplications by 0. +Given x ∈ Σ, we define x−1 to be p−1 +i +if x = pi ∈ An, and to be pi if x = p−1 +i +∈ A−1 +n . We define 1−1 = 1 +and (xw)−1 = w−1x−1, for all x ∈ An and w ∈ A∗ +n. It is well known (e.g., [45, subsection 9.3]) that every +nonzero element of Pn has a unique representation of the form yx−1 with y, x ∈ A∗ +n. Whenever we write +a = yx−1, it will be understood that x, y ∈ A∗ +n. We will identify nonzero elements of Pn with the words +of this form. The explicit multiplication is provided by the following lemma. We say that words x, v ∈ A∗ +n +prefix comparable if one is a prefix of the other. +Lemma 5.1. ([3, Lem. 3.2]) Consider nonzero elements yx−1 and vu−1 of Pn. Then: +(1) yx−1 · vu−1 ̸= 0 iff x and v are prefix comparable; +42 + +(2) if yx−1 · vu−1 ̸= 0, then +yx−1 · vu−1 = +� +yzu−1 +if v = xz , +y(uz)−1 +if x = vz . +(3) y = v in Pn iff y = v in A∗ +n, and x−1 = u−1 in Pn iff x = u in A∗ +n. +A word w ∈ Pn is said to be cyclically reduced if w = 0 or w = yx−1, where x and y have no common +prefix other than 1. Every nonzero element of Pn can be written in the form ryx−1r−1, with r ∈ A∗ +n and +yx−1 a cyclically reduced word. From any a ∈ Pn, we compute a cyclically reduced word �a in the following +way: if a = 0, we let �a be equal to 0; otherwise, a = ryx−1r−1 as above, so we let �a be the (possibly empty) +cyclically reduced word yx−1. +We will now characterize conjugacy ∼n in Pn. +Since Pn is an inverse monoid, we have ∼n=∼i by +Proposition 2.11, that is, for all a, b ∈ Pn, a ∼n b if and only if there exists g ∈ Pn such that g−1ag = b and +gbg−1 = a. +Theorem 5.2. Let a, b ∈ Pn. Then a ∼n b if and only if a = b = 0 or �a = �b. +Proof. Since [0]n = {0}, it remains to establish criteria for nonzero a, b ∈ Pn to be n-conjugate. In the +calculations below, it will be convenient to write a = yx−1 as a = a+a−1 +− . +Let a = a+a−1 +− , b = b+b−1 +− ∈ Pn with a ∼n b. Then there exists g = g+g−1 +− ∈ Pn such that +g−g−1 ++ a+a−1 +− g+g−1 +− = b+b−1 +− +and +g+g−1 +− b+b−1 +− g−g−1 ++ = a+a−1 +− . +(5.7) +Since b+b−1 +− ̸= 0, it follows by Lemma 5.1 that a− and g+ are prefix-comparable, g+ and a+ are also prefix +comparable, and +g−g−1 ++ a+a−1 +− g+g−1 +− = + + + + + + + +g−g−1 ++ a+rg−1 +− +if g+ = a−r, = +� +g−sg−1 +− +if a+r = g+s +g−(g−s)−1 +if g+ = a+rs +g−g−1 ++ a+(g−r)−1 +if a− = g+r, = +� g−(g−rs)−1 +if g+ = a+s +g−s(g−r)−1 +if a+ = g+s, +where r, s ∈ A∗ +n. By these calculations, first equality in (5.7), and Lemma 5.1(4), we obtain: +g−s = b+ and g− = b− +if +a+r = g+s and g+ = a−r, +g− = b+ and g−s = b− +if +g+ = a+rs and g+ = a−r, +g− = b+ and g−rs = b− +if +g+ = a+s and a− = g+r, +g−s = b+ and g−r = b− +if +a+ = g+s and a− = g+r. +Thus we have have four cases to consider, and in each case we can draw conclusions using the second equality +in (5.7) and Lemma 5.1(4). +Case 1. g−s = b+, g− = b−, a+r = g+s, g+ = a−r. +Then a+a−1 +− = g+g−1 +− b+b−1 +− g−g−1 ++ = g+sg−1 ++ , so r = 1, and hence a = g+sg−1 ++ +and b = g−sg−1 +− . +Case 2. g− = b+, g−s = b−, g+ = a+rs, g+ = a−r. +Then a+a−1 +− = g+g−1 +− b+b−1 +− g−g−1 ++ = g+(g+s)−1, so s = r = 1, and hence a = g+g−1 ++ +and b = g−g−1 +− . +Case 3. g− = b+, g−rs = b−, g+ = a+s, a− = g+r. +Then a+a−1 +− = g+g−1 +− b+b−1 +− g−g−1 ++ = g+(g+rs)−1, so s = 1, and hence a = g+(g+r)−1 and b = g−(g−r)−1. +Case 4. g−s = b+, g−r = b−, a+ = g+s, a− = g+r. +Then a+a−1 +− = g+g−1 +− b+b−1 +− g−g−1 ++ = g+s(g+r)−1, and hence a = g+s(g+r)−1 and b = g−s(g−r)−1. +Note that the forms of a and b deduced in Cases 1–3 are special cases of the forms deduced in Case 4. +Therefore, if a ∼n b, then a = g+s(g+r)−1 and b = g−s(g−r)−1 for some g+, g−, r, s ∈ A∗ +n. Conversely, if +a = g+s(g+r)−1 and b = g−s(g−r)−1 for some g+, g−, r, s ∈ A∗ +n, then it is straightforward to verify g−1ag = b +and gbg−1 = a for g = g+g−. We have proved the result. +43 + +Note that for any representative a ∈ Pn we have a ∼n �a. This gives the following corollary. +Corollary 5.3. The set of cyclically reduced words is a set of representatives of minimal length of the +partition Pn/∼n. +For a nonzero representative a = yx−1 ∈ Pn, we denote by ρ(a) the representative word of x−1y in Pn. +We also set ρ(0) = 0. Note that ρ(a) ∈ A∗ +n ∪ (A−1 +n )∗ ∪ {0}, for any representative a ∈ Pn. Also note that +ρ(a) = �a if and only if �a ∈ A∗ +n ∪ (A−1 +n )∗ ∪ {0}. +Let us recall the characterizations of ∼c and ∼p from [3]. +Lemma 5.4. ([3, Thm. 3.9]) Let a, b ∈ Pn. Then a ∼c b if and only if one of the following conditions is +satisfied: +(a) a = b = 0; +(b) �a = �b; or +(c) �a,�b ∈ (A−1 +n )∗ and �a ∼p �b in the free monoid (A−1 +n )∗. +In particular, if an element of Pn is not in (A−1 +n )∗ ∪ {0} then it is ∼c-conjugate to a unique element yx−1 +such that y ̸= 1 and x and y have no common prefix other than 1. +For a given alphabet X, let Lp(X) denote a set of representatives of minimal length of the partition +resulting of the quotient of free monoid on X by the equivalence relation ∼p on X∗. +Corollary 5.5. The set of cyclically reduced words with a prefix in An ∪ {0} together with the set Lp(A−1 +n ), +is a set of representatives of minimal length of the partition Pn/∼c. +Any two different a, b ∈ Pn such that a, b ∈ A∗ +n or a, b ∈ (A−1 +n )∗ are never n-conjugate. This shows that +in Pn, conjugacy ∼n is strictly included in ∼c and ∼p (see [3, Corollary 3.10]). +Lemma 5.6. ([3, Thm. 3.6]) Let a, b ∈ Pn. Then a ∼p b if and only if one of the following conditions is +satisfied: +(a) a = ρ(b) = 0 or ρ(a) = b = 0; +(b) ρ(a) = ρ(b) = 0 and �a = �b; +(c) �a,�b ∈ A∗ +n and �a ∼p �b in the free monoid A∗ +n; or +(d) �a,�b ∈ (A−1 +n )∗ and �a ∼p �b in the free monoid (A−1 +n )∗. +From Lemma 5.6 and other results in [3], we can deduce a characterization of ∼∗ +p in Pn. +Proposition 5.7. Let a, b ∈ Pn. Then a ∼∗ +p b if and only if one of the following conditions is satisfied: +(a) ρ(a) = ρ(b) = 0; +(b) �a,�b ∈ A∗ +n and �a ∼p �b in the free monoid A∗ +n; or +(c) �a,�b ∈ (A−1 +n )∗ and �a ∼p �b in the free monoid (A−1 +n )∗. +Proof. Suppose a ∼∗ +p b. Then, by [3, Thm. 3.7], either a ∼p b or a ∼p 0 ∼p b. In the former case, (a), (b), +or (c) is satisfied by Lemma 5.6. Suppose a ∼p 0 ∼p b. Then ρ(a) = ρ(b) = 0 by [3, Lem. 3.4], and so (a) is +satisfied. +Conversely, suppose that one of (a), (b), (c) holds. If (b) or (c) holds, then a ∼p b by Lemma 5.6, and +so a ∼∗ +p b. Suppose (a) is satisfied. Then, by [3, Lem. 3.4] again, a ∼p 0 ∼p b, and so a ∼∗ +p b. +In particular, if a representative element of Pn is not in A∗ +n ∪ (A−1 +n )∗, then it is ∼∗ +p-conjugate to 0. +Corollary 5.8. The set Lp(An) ∪ Lp(A−1 +n ) ∪ {0, 1}, is a set of representatives of minimal length of the +partition Pn/∼∗ +p. +44 + +5.2 +Conjugacy growth functions in Pn +Let M be a monoid generated by a finite set X. Then every element of M can be represented as a word in +X∗. The length of an element a ∈ M is the minimum length of a word that represent y, written |a|X or just +|a| if the context is clear. Since X is finite, for every integer m ≥ 0, there are only finitely many elements of +M that are of length m. This leads us to the following definition. +Definition 5.9. For a monoid M with finite generating set X, we define the strict growth function of M +(with respect to X) respectively as +σM,X(n) = #{a ∈ M : |a|X = n} +for any n ∈ N0. +Regarding the characterization of representatives of the polycyclic monoid given in the previous subsec- +tion, we obtain the following result: +Proposition 5.10. The polycyclic monoid on n generators Pn, has strict growth function given by +σPn,Σ0(0) = 1, σPn,Σ0(1) = 2n + 1, and σPn,Σ0(m) = (m + 1)nm for m ≥ 2 . +Let ∼j be a conjugacy in M that is an equivalence relation. For a ∈ M, we denote by [a]∼j the ∼j- +conjugacy class of a, and we write M/∼j for the set of ∼j-conjugacy classes in M. For a ∈ M, we define the +length of the conjugacy class [a]∼j by +|[a]∼j|X = min{|b|X : b ∈ [a]∼j}. +Definition 5.11. For a monoid M with finite generating set X, and a conjugacy ∼j in M that is an +equivalence relation, we define the strict conjugacy growth function of M relative to ∼j (with respect to X) +respectively as +∼jσ M,X(n) = #{a ∈ M : |[a]∼j|X = n} +for any n ∈ N0. +We will now compute the conjugacy growth functions of the polycyclic monoids for the conjugacies ∼n, +∼c, and ∼∗ +p. +Theorem 5.12. The polycyclic monoid on n generators Pn, has strict conjugacy growth function relative to +∼n given by +∼n +σ Pn,Σ0(0) = 1, +∼nσ Pn,Σ0(1) = 2n + 1, and +∼n +σ Pn,Σ0(m) = 2nm + (m − 1)nm−1(n − 1), for m ≥ 2. +Proof. We use Corollary 5.3 to deduce the result. The cases for m = 0 and m = 1 are easy. For m ≥ 2, we +can distinguish the case when the cyclically reduced word is in A∗ +n ∪ (A−1 +n )∗, for which we get 2nm ciclically +reduced words of length m, from the cases where the cyclically reduced word of lenght m has the form yx−1, +with x and y non-empty and with no common prefix. +To be able to compute the conjugacy growth functions of ∼c and ∼∗ +p we need to compute the ∼p-conjugacy +growth function of the free monoid on a given alphabet X. +Theorem 5.13. Let X be an alphabet with |X| = n. The ∼p-conjugacy growth function of the free monoid +on X is +∼∗ +pσ X∗,X(m) = +� +d|m +� +e|d +µ +�d +e +� ne +d , +m ≥ 1, +where µ is the M¨obius function. +45 + +Proof. The number of words in X∗ of length m is nm. Given a word a in X of length m, a ∼p-conjugate +word to a will be a cyclic permutation of a, that is, it will be some b ∈ X∗ with a = uv and b = vu, for some +u, v ∈ X∗. So, how many distinct cyclic permutations of a we may have? We know that, a = uv = vu, with +u, v ̸= 1, if and only if a = wk, for some w ̸= 1, and k > 1 [44, Corollary 5.3]. +A word p is called primitive if whenever p = wk, for some w ∈ X∗, then k = 1. The root of a word a, +denoted √a, is the unique primitive word p such that a = pk. Hence, a word a has |√a|X distinct cyclic +permutations. +Denote by f(d) the number of primitive words in X of length d. Then the number am of ∼p-conjugate +elements in X∗ of length m is +am = +� +d|m +f(d) +d +. +Now, the number of words in X∗ of length m can be given by +nm = +� +d|m +f(d). +Therefore, by the M¨obius inversion formula +f(m) = +� +d|m +µ +�m +d +� +nd, +where µ is the M¨obius function. +The result follows. +Theorem 5.14. The polycyclic monoid on n generators Pn, has strict conjugacy growth function relative to +∼c given by +∼cσ Pn,Σ0(0) = 1, +∼cσ Pn,Σ0(1) = 2n+1, and +∼cσ Pn,Σ0(m) = nm +(m−1)nm−1(n−1)+ +∼∗ +pσ A∗n,An(m), +for m ≥ 2. +Proof. We use Corollary 5.5 and the previous theorem to deduce the result. The proof follows the same +reasoning of the proof of Theorem 5.12. +Theorem 5.15. The polycyclic monoid on n generators Pn, has strict conjugacy growth function relative to +∼∗ +p given by +∼∗ +pσ Pn,Σ0(0) = 1, +∼∗ +pσ Pn,Σ0(1) = 2n + 1, and +∼∗ +pσ Pn,Σ0(m) = 2 +∼∗ +pσ A∗ +n,An(m), for m ≥ 2. +Proof. The result follows from Corollary 5.8 and Theorem 5.13. +5.3 +Conjugacy growth series of Pn +In this subsection we describe the different growth series of the polyclyclic monoids. We begin by introducing +the concepts. +Definition 5.16. Let M be a monoid generated by a finite set X. The standard growth series of M is the +following power series with indeterminate z: +ΞM,X(z) = +� +m≥0 +σM,X(m)zm, +where σM,X is the strict growth function of M with respect to X. +Definition 5.17. Let M be a monoid generated by a finite set X, and let ∼j be a conjugacy in M that is an +equivalence relation. The ∼j-conjugacy growth series of M is the following power series with indeterminate +z: +∼j +Ξ M,X(z) = +� +m≥0 +∼jσ M,X(m)zm, +where +∼jσ M,X is the strict growth function of M with respect to X. +46 + +Note that even if one cannot define in growth function for infinitely generated groups, the paper [6] gives +the conjugacy growth series for some infinitely generated groups. +From Theorem 5.13 we deduce the following: +Theorem 5.18. Let X be an alphabet with |X| = n. The ∼p-conjugacy growth series of the free monoid on +X is +∼∗ +p +Ξ X∗,X(z) = +� +r,s≥1 +nr +rs ϕ (s) zrs, +where ϕ is the totient Euler formula. +We can now give an explicit formula for the conjugacy growth series of the polycyclic monoids Pn for the +conjugacies ∼n, ∼c and ∼∗ +p. +Theorem 5.19. The n-conjugacy growth series of Pn is +∼n +Ξ Pn,Σ0(z) = +1 − nz2 +(1 − nz2)2 + z. +Proof. According to Corollary 5.3, we have to count the number of words sr−1, where r and s do not have +a common prefix other than the empty word, plus the element 0. The conjugacy class of 0 contributes z. +We can do the former by counting all words yx−1 ∈ Pn, and then removing those for which x and y have at +least one common beginning letter from An. This gives +z + +1 +(1 − nz)2 − nz2 +1 +(1 − nz)2 , +which completes the proof. +Theorem 5.20. The ∼c-conjugacy growth series of Pn is given by +∼c +Ξ Pn,Σ0(z) = +1 +1 − nz + z + (n2 − n)z2 +(1 − nz)2 + +∼∗ +p +Ξ A∗ +n,An(z). +Proof. By Corollary 5.5, we have to count the number of cyclically reduced words with a prefix in An ∪ {0} +and the words in the set Lp(A−1 +n ). The conjugacy classes of the elements of A∗ +n contribute +1 +1−nz to the +series, and the conjugacy class of 0 contributes z. Further, there are the conjugacy classes of the elements +yx−1 such that both x and y are not empty and have no common prefix other than 1. They contribute +(nz)2 +(1−nz)2 − +nz2 +(1−nz)2 to the series. Finally, we have the conjugacy classes of the elements in (A−1 +n )∗ \ {1}, which +contribute +∼∗ +p +Ξ A∗ +n,An(z). +For completeness, we present the analogous result for the ∼∗ +p-conjugacy. +Theorem 5.21. The ∼∗ +p-conjugacy growth series of Pn is given by +∼∗ +p +Ξ Pn,Σ0(z) = 1 + z + 2 +∼∗ +p +Ξ A∗n,An(z). +Proof. The conjugacy class of the empty word contributes 1 to the series, and the conjugacy class of 0 +contributes z. Further, there are the conjugacy classes of the elements of A∗ +n \ {1} and the conjugacy classes +of the elements in (A−1 +n )∗ \ {1}, which both contribute +∼∗ +p +Ξ A∗n,An(z). +47 + +6 +Questions +We characterized the conjugacy classes (for several different notions of conjugation) in the partition monoid +and two of its friends. +Question 6.1. Characterize the conjugacy relations for the other friends of the partition monoid (Planar, +Jones, Kauffman, Martin, Temperley and Lieb, etc.). +Question 6.2. Characterize the partial inner automorphisms for the partition monoid and its friends. +We know that there exist finitely generated groups for which the word problem is solvable, but the +conjugacy problem is not. Hence there exist semigroups for which the word problem is solvable, while (for +various notions of conjugacy) the conjugacy problem is not. This leads us to the following question. +Question 6.3. Is there a finitely generated semigroup with solvable n-conjugacy problem and with unsolvable +word problem? +We note that because of Remark 2.3, given a monoid with some nonidempotent elements, we cannot +embed it injectively into a larger monoid such that all of its elements become n-conjugate. +Hence the +construction in the proof of [3, Theorem 5.2] will not work for n-conjugacy. +Question 6.4. Can we identify the set of n-normal forms as a species in the sense of [8] in such a way to +count the number of n-conjugacy classes in the partition monoid by the counting the isomorphism type series +of this species? +Acknowledgements +We thank Laura Ciobanu and Susan Hermiller for the idea of studying conjugacy growth in finitely generated +groups, which extends naturally to finitely generated monoids. +JA, MK, AM and VM were supported by the Funda¸c˜ao para a Ciˆencia e a Tecnologia (Portuguese Founda- +tion for Science and Technology) through the projects UIDB/MAT/00297/2020 and UIDP/MAT/00297/2020 +(Centro de Matem´atica e Aplica¸c˜oes) and the FCT Project PTDC/MAT-PUR/31174/2017. +MK was also supported by Simons Foundation Collaboration Grant 359872. +VM was also supported by the FCT Project UID/MAT/00297/2019 (Centro de Matem´atica e Aplica¸c˜oes) +and the FCT Project PTDC/MHC-FIL/2583/2014. +References +[1] Y. Antol´ın and L. 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J. +Algebra Comput. 2 (1992), 209–220. +51 + diff --git a/JdE2T4oBgHgl3EQf_wnL/content/tmp_files/load_file.txt b/JdE2T4oBgHgl3EQf_wnL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..781b5d70f55f6513c5926ecbf0f2033bc4f6d599 --- /dev/null +++ b/JdE2T4oBgHgl3EQf_wnL/content/tmp_files/load_file.txt @@ -0,0 +1,3087 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf,len=3086 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='04252v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='GR] 11 Jan 2023 Conjugacy in Semigroups: the Partition and Brauer Diagram Monoids, Conjugacy Growth, and Partial Inner Automorphisms Jo˜ao Ara´ujo, Wolfram Bentz, Michael Kinyon, Janusz Konieczny, Ant´onio Malheiro, and Valentin Mercier Abstract The conjugacy relation plays an important role in group theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If a and b are elements of a group G, a is conjugate to b if g−1ag = b for some g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Group conjugacy extends to inverse semigroups in a natural way: for a and b in an inverse semigroup S, a is conjugate to b if g−1ag = b and gbg−1 = a for some g ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The fourth author has recently defined a conjugacy for an arbitrary semigroup S that coincides with inverse semigroup conjugacy if S is an inverse semigroup, and is included in all existing semigroup conjugacy relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will call it the natural conjugacy for semigroups, and denote it by ∼n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The first purpose of this paper is to study ∼n in various contexts, chiefly the partition monoid and some of its friends (Brauer and partial Brauer monoids), and also to characterize ∼n in several important classes of semigroups, transformation semigroups and in the polycyclic monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The second purpose of this paper is to show how the notion of natural conjugacy leads to the definition of the inverse semigroup of partial automorphisms of an arbitrary semigroup (in the same way conjugation in groups induces the notion of inner automorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Attached to the majority of mathematical objects there is a notion of morphism and hence notions of automorphism and endomorphism that often encode relevant information about the original object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Our approach allows to attach to the endomorphisms of a mathematical object an inverse semigroup that hopefully will bring the deep results on inverse semigroups to help the study of the original object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Finally we extend the notion of conjugacy growth from groups to semigroups and give closed formulas for the conjugacy growth series of the polycyclic monoid, for ∼n and two other semigroup conjugacies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The paper ends with some open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 20M10, 20M20, 20M15, 05C20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Keywords: Conjugacy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' partial inner automorphisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' transformation semigroups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' partition monoids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' poly- cyclic monoids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' conjugacy growth series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 1 Introduction In a semigroup S, define a relation ∼n, which we will call natural conjugacy, as follows: for all a, b ∈ S, a ∼n b ⇐⇒ ∃g,h∈S1 ( ag = gb, bh = ha, hag = b, and gbh = a ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (∼n) The main goals of this paper are the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Describe the natural conjugacy classes in the partition monoid and some of its friends;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' these monoids (Partition, Brauer, Jones, Kauffman, Martin, Temperley and Lieb, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=') belong to the general family of diagram monoids and (with the associated algebras and categories) arise in many areas of mathe- matics such as invariant theory, classical groups, representation theory, logic, knot theory or statistical mechanics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' [7,30,32,33,40,41,59];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' for an excellent overview on the literature and interconnections of these areas please see the introduction of [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Given the importance of these objects, about one third of the paper is dedicated to the description of the conjugacy classes in the partition monoid, the 1 Brauer monoid and the partial Brauer monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We describe the classes for ∼n and for several other notions of conjugacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As conjugation in groups induces in a natural way the group of inner automorphisms (a → g−1ag), the notion ∼n induces on every semigroup the inverse semigroup of partial automorphisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' when the semigroup is a group, then this object is the group of inner automorphisms with a zero adjoined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Computing this object for a given semigroup will be challenging in general;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' here we computed it for the full transformation monoid, the symmetric inverse semigroup and for a completely simple semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Extend to monoids the group theory notion of conjugacy growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As a proof of concept, investigate the conjugacy growth in the polycyclic monoids (a natural family of finitely generated infinite monoids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Prove for the natural conjugacy results similar to the ones proved in [4] for other notions of conjugacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In addition to these general goals, this paper explores many other paths as we now explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a and b be conjugate elements of a group G, that is, g−1ag = b for some g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' There are equivalent formulations that avoid inverses, for example, ag = gb for some g ∈ G or a = uv and b = vu for some u, v ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The latter formulations have been used to define relations ∼l (left conjugate) [51,60,61] and ∼p (primary conjugate) [43] on an arbitrary semigroup S: a ∼l b ⇐⇒ ∃g∈S1 ag = gb, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1) a ∼p b ⇐⇒ ∃u,v∈S1 a = uv and b = vu, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2) where S1 is S with an identity adjoined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In a general semigroup S, the relation ∼l is reflexive and transitive, but not symmetric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' while ∼p is reflexive and symmetric, but not transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' However, these relations can serve as a conjugacy in the class of free semigroups: if S is a free semigroup, then ∼l and ∼p are equivalence relations, and they coincide [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The relation ∼l has been restricted to ∼o [51], and ∼p has been extended to ∼∗ p [42,43], in such a way that the modified relations are equivalences on an arbitrary semigroup S: a ∼o b ⇐⇒ ∃g,h∈S1 ag = gb and bh = ha, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3) ∼∗ p = the transitive closure of ∼p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4) The relation ∼o reduces to S × S for any semigroup S with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' This deficiency has been remedied in [5], where the following relation has been defined on an arbitrary semigroup S: a ∼c b ⇐⇒ ∃g∈P(a)∃h∈P(b) ag = gb and bh = ha, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='5) where for a ̸= 0, P(a) = {g ∈ S1 : ∀m∈S1 (ma ̸= 0 ⇒ (ma)g ̸= 0)}, and P(0) = {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (See [5, Section 2] for a motivation of this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=') The relation ∼c is an equivalence, it does not reduce to S × S if S has a zero, and it is equal to ∼o if S does not have a zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The relations ∼o, ∼∗ p, and ∼c are not satisfactory as conjugacies when applied to inverse semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be an inverse semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then the following relation ∼i on S is a natural extension of the group conjugacy [2]: a ∼i b ⇐⇒ ∃g∈S1 g−1ag = b and gbg−1 = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6) However, none of the relations ∼o, ∼∗ p, or ∼c reduces to ∼i when S is an inverse semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In 2018, the fourth author [38] defined a conjugacy ∼n on any semigroup S by (∼n) above, that is, a ∼n b ⇐⇒ ∃g,h∈S1 ( ag = gb, bh = ha, hag = b, and gbh = a ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='7) The relation ∼n is an equivalence relation on any semigroup S, it does not reduce to S × S if S has a zero, and it coincides with ∼i if S is an inverse semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In fact, it is the smallest of all conjugacies defined up to this point for general semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For these reasons, we will call ∼n the natural conjugacy for semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 2 Note that each of the relations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='7) reduces to group conjugacy when S is a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' However, assuming we require conjugacy to be an equivalence relation on general semigroups, only ∼∗ p, ∼o, ∼c, and ∼n can provide possible definitions of conjugacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' There are equivalence relations, however, that can serve as conjugacies for special classes of semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For example, as we have already mentioned, each of ∼l and ∼p can serve as a conjugacy in the class of free semigroups (in which they coincide).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Another such relation, called trace conjugacy, originally defined for finite monoids, defines a notion of conjugacy in the class of epigroups [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' A semigroup S is called an epigroup if for every a ∈ S, there exists a positive integer n such that an belongs to a subgroup of S, that is, the H-class H = Han of an is a group (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We denote by aω the identity in the group H [54, §2], and we set aω+1 = aωa (which is also an element of H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Every finite semigroup, or more generally, every periodic semigroup S is an epigroup, and in this case, aω itself is a power of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We define the relation ∼tr on any epigroup S as follows [4]: a ∼tr b ⇐⇒ ∃g,h∈S1 ghg = g, hgh = h, gh = aω, hg = bω, and haω+1g = bω+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='8) The relation ∼tr, called trace conjugacy, is an equivalence relation on any epigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Its definition was inspired by the representation theory of finite monoids (see [55] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In any semigroup, we have ∼n ⊆ ∼∗ p ⊆ ∼o and ∼n ⊆ ∼c ⊆ ∼o, and, with respect to inclusion, ∼∗ p and ∼c are not comparable [38, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For detailed comparison and analysis in various classes of semigroups, of the conjugacies ∼∗ p, ∼o, ∼c, and ∼tr, see [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As noted above, the aim of this paper is to study conjugacy ∼n in various classes of semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1, we provide various alternative definitions of ∼n, which we will use throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It was stated in [4] that “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' in general, Green’s relations and the conjugacies under consideration are not comparable with respect to inclusion.” However, in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2, we will show a very nice feature of ∼n, namely that in any semigroup, ∼n is included in Green’s relation D, and that ∼n and D coincide when restricted to idempotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3– 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4, we study ∼n in inverse and stable semigroups, and in epigroups and completely regular semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='5, we characterize ∼n in several well-known semigroups of transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The definition of ∼n was not available during the work that led to [4], so this section can be viewed as an extension of [4] that includes the investigation of properties of ∼n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In particular, it seems clear that ∼n has very nice features, when compared with the notions treated in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The next three sections contain the most important results of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In §3, we show how the notion of the natural conjugacy ∼n leads to the definition of partial inner automorphisms of an arbitrary semigroup (in analogy with the inner automorphisms of an arbitrary group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Therefore, we are able to assign to each semigroup (linear, topological, or any other kind) a natural inverse semigroup that in many cases will encode important information about the original semigroup and will hopefully be tractable using techniques of inverse semigroup theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In particular, we describe this inverse semigroup for the full transformation monoid and for a Rees matrix semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Section §4 characterizes ∼n in several finite partition monoids, namely the partition monoid itself, the Brauer monoid and the partial Brauer monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We also characterize the other notions of conjugation (∼tr, ∼∗ p, ∼o, and ∼c) in these monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Finally, in §5, we characterize ∼n in the finite polycyclic monoids, and give closed formulas for the conjugacy growth series of the polycyclic monoid for ∼n, ∼∗ p, and ∼o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 2 General results on ∼n The goal of this section is to study ∼n in a manner analogous to what was carried out for the other notions in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1 Alternative definitions of ∼n For a semigroup S, a, b ∈ S and g, h ∈ S1, consider the following equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 3 (i) ag = gb (ii) bh = ha (iii) hag = b (iv) gbh = a (v) hg · b = b (vi) gh · a = a (vii) b · hg = b (viii) a · gh = a Our definition of ∼n is based on the set {(i),(ii),(iii),(iv)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We now give some alternative characterizations which will be useful later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In particular, we could have defined ∼n less symmetrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be a semigroup, and let a, b ∈ S and g, h ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then: (a) (i) =⇒ ( (iii) ⇐⇒ (v) );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (b) (i) =⇒ ( (iv) ⇐⇒ (viii) );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (c) (ii) =⇒ ( (iv) ⇐⇒ (vi) );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (d) (ii) =⇒ ( (iii) ⇐⇒ (vii) );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (e) {(iii),(vi)} =⇒ {(i),(v)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (f) {(iv),(v)} =⇒ {(ii),(vi)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (g) {(iv),(vii)} =⇒ {(i),(viii)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (h) {(iii),(viii)} =⇒ {(ii),(vii)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If (i) holds, then hg · b = hag and a · gh = gbh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The first of these implies (a), the second implies (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If (ii) holds, then gh · a = gbh and b · hg = hag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The first of these implies (c), the second implies (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For (e), ag = ghag = gb and then (v) follows from (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For (f), bh = hgbh = ha and then (vi) follows from (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For (g), gb = gbhg = ag and then (viii) follows from (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For (h), ha = hagh = bh and then (vii) follows from (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be a semigroup, and let a, b ∈ S and g, h ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Each of the following sets of equations implies all of (i)–(viii), and thus a ∼n b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (1) {(i),(iii),(iv)} (2) {(i),(iii),(viii)} (3) {(i),(v),(viii)} (4) {(ii),(iii),(iv)} (5) {(ii),(iii),(vi)} (6) {(ii),(iv),(vii)} (7) {(iii),(vi),(viii)} (8) {(iv),(v),(vii)} Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Each case follows from tracking implications in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We prove case (1) and leave the rest to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus assume (i),(iii),(iv) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then (v) and (viii) hold by parts (a) and (b) of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then (ii) holds by part (f), and so (vi) and (vii) hold by parts (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It turns out that any subset of {(i),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' ,(viii)} which is sufficient to prove all eight equations must contain one of the subsets listed in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We omit the unenlightening list of counterexamples necessary to establish this claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For a semigroup S, if a, b ∈ S satisfy a ∼n b, then there exist g, h ∈ S1 satisfying all of the conditions (i)–(viii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For brevity, we will say that g, h are conjugators for a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We shall also use (i)–(viii) freely in calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As already noted, we refer to ∼n as natural conjugacy or just n-conjugacy, for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For a ∈ S we write [a]n = {b ∈ S : b ∼n a} for the conjugacy class of a relative to ∼n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that in any semigroup with a zero, [0]n = {0}, and in any monoid M, [1]n = {gh ∈ M : hg = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2 Conjugacy ∼n and Green’s relations If S is a semigroup and a, b ∈ S, we say that a L b if S1a = S1b, a R b if aS1 = bS1, and a J b if S1aS1 = S1bS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We define H as the intersection of L and R, and D as the join of L and R, that is, the smallest equivalence relation on S containing both L and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' These five equivalence relations are known as Green’s relations [35, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The relations L and R commute [35, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3], and consequently D = L ◦ R = R ◦ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If S is finite, then D = J [35, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Green’s relations are one of the most important tools in studying semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Because D = R ◦ L, we may express D equationally as follows: a D b ⇐⇒ ∃g1,g2,h1,h2∈S1( ag1 = g2b, ag1h1 = a, h2g2b = b ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Comparing this observation with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2, we immediately have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In a semigroup, ∼n ⊆ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' From Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4 and [38, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3], we have ∼n ⊆ D ∩ ∼p ∩ ∼c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (Although the cited reference states ∼n ⊆ ∼∗ p, it actually proves the stronger result ∼n ⊆ ∼p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=') This inclusion is strict in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Consider the monoid S defined by the Cayley table 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 2 0 2 6 6 3 2 6 2 3 0 3 6 6 3 2 6 2 4 0 4 6 6 4 5 6 5 5 0 5 6 6 4 5 6 5 6 0 6 6 6 6 6 6 6 7 0 7 2 3 4 5 6 7 We have 2 = 3 · 7 and 3 = 7 · 3, so 2 ∼p 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Next, 2 · 4 = 3 and 3 · 5 = 2, and so 2 R 3 (and thus certainly 2 D 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Finally, for all x, y ∈ S\\{0}, xy ̸= 0, and thus x ∼c y in S if and only if x ∼o y in S\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In the latter semigroup, ∼o is the universal relation because 6 is a zero, and so 2 ∼c 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' However, 2 ≁n 3 because, as can be checked, there are no suitable conjugators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Next we consider how n-conjugacy interacts with idempotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' First we note that if an n-conjugacy class contains an idempotent, then it consists only of idempotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be a semigroup, let e, a ∈ S, and assume e is an idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If e ∼n a, then a is also an idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let g, h ∈ S1 be conjugators for a and e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then aa = aagh = ageh = geeh = geh = agh = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Restricted to idempotents, n-conjugacy and the D-relation turn out to coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' A pair g, h of elements of a semigroup S are said to be mutually inverse if ghg = g and hgh = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be a semigroup and let e, f ∈ S be idempotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then e ∼n f if and only if e D f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' When this is the case, there exist mutually inverse conjugators g, h of e, f in the same D-class as e, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' One direction is covered by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4, so assume e D f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We just follow the proof of [35, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4], noting that the construction therein gives mutually inverse conjugators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Indeed, by assumption, there exist g, h1, h2 ∈ S1 such that eg = g = gf, gh1 = e and h2g = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (Here we are using the fact that an idempotent e is a left identity element for the R-class Re and a right identity element for the L-class Le [35, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=') Set h = fh1e and check that gh = gfh1e = gh1e = ee = e and hg = fh1eg = fh1g = h2gh1g = h2eg = h2g = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since eg = gf, egh = e and hgf = f, it follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2 that e ∼n f with g, h as conjugators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Finally ghg = eg = g and hgh = fh = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 5 Recall that a band is a semigroup in which every element is an idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In any band, ∼n = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We conclude this section with a brief discussion of the two extreme cases: where n-conjugacy is the universal relation, that is, ∼n= S × S, and where ∼n is the equality relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In neither case will we arrive at a complete characterization, but each case still entails interesting necessary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' A semigroup is bisimple if D is the universal relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' A rectangular band is an idempotent semigroup satisfying xyx = x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' every rectangular band is isomorphic to one of the form I × J for sets I, J with multiplication (i, j) · (k, ℓ) = (i, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If S is a semigroup in which ∼n is universal, then S is bisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If, in addition, S has an idempotent, then S is a rectangular band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The first assertion follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4 and the second follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' At the other extreme, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be a semigroup in which ∼n is the equality relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then each D-class has at most one idempotent, and each regular D-class is an H-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The first assertion follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For the second, assume e is an idempotent and c D e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then c is regular and hence there exists an idempotent f such that c L f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' But then f D e and so by assumption e = f, that is, c L e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By a similar argument, c R e and so c H e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As noted in the introduction, in ( [4], §3), it was shown that Green’s relations and the four notions of conjugation considered are not particularly well related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The results of this subsection show that ∼n tells a completely different story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (See also Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3 Conjugacy ∼n in inverse and stable semigroups As we pointed out in §1, of the known conjugacy relations for general semigroups, ∼n is the only one that coincides with the conjugacy ∼i (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6) in inverse semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' This was first proved in [38, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6] using the Wagner-Preston representation of inverse semigroups as semigroups of partial injective transformations [35, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Here we present a purely equational proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In inverse semigroups, ∼n = ∼i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be an inverse semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The inclusion ∼i ⊆ ∼n follows from [2, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3], but we give a brief proof here to keep the discussion self-contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose a ∼i b for some a, b ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then g−1ag = b and gbg−1 = a for some g ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have a · gg−1 = gbg−1gg−1 = gbg−1 = a and gg−1 · a = gg−1gbg−1 = gbg−1 = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now condition (7) of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2 holds with h = g−1 and so a ∼n b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now suppose a ∼n b for some a, b ∈ S, and let g, h ∈ S1 be conjugators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then g−1 · ag ���� = g−1g · b (by (i)) = g−1g · bb−1 � �� � ·b = b ���� b−1 · g−1g · b (since idempotents commute) = hg · bb−1 · g−1g· � �� � b (by (v)) = h · gg−1g � �� � · bb−1b � �� � (since idempotents commute) = hg · b = b (by (v)) The equality gbg−1 = a is proved similarly, and so a ∼i b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 6 The natural partial order (or Mitsch order) ≤ in a semigroup S is defined as follows: a ≤ b ⇐⇒ ∃s,t∈S1 sa = a = sb and at = a = bt ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' see [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We now consider how natural conjugacy and the natural partial order interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' A semigroup S is left stable if, for all a, b ∈ S, S1a ⊆ S1ab implies S1a = S1ab, that is, a L ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' This can be equivalently formulated as a ∈ S1ab implies ab ∈ S1a for all a, b ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Right stability is defined dually, and a semigroup is said to be stable if it is both left and right stable [15, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' I, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Every periodic semigroup, and in particular every finite semigroup, is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be a stable semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then ∼n ∩ ≤ is the identity relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Assume a ∼n b and a ≤ b for some a, b ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let g, h ∈ S1 be conjugators for a, b and let s, t ∈ S1 witness a ≤ b, that is, sa = a = sb and at = a = bt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have a = sb = shag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By stability, there exists u ∈ S1 such that ag = ua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus ua = uat = agt = gbt = ga, hence ag = ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now a = bt = hgbt = hga = hag = b, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4 Conjugacy ∼n in epigroups and completely regular semigroups An element a of a semigroup S is an epigroup element (or a group-bound element) if there exists a positive integer n such that an is contained in a subgroup of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The smallest n for which this is satisfied is the index of a, and for all k ≥ n, ak is contained in the group H-class of an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The set of all epigroup elements of S is denoted by Epi(S) and the subset consisting of elements of index no more than n is denoted by Epin(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have Epim(S) ⊆ Epin(S) for m ≤ n and Epi(S) = � n≥1 Epin(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The elements of Epi1(S) are called completely regular (or group elements);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' thus Epi1(S) is the union of all group H-classes of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For a ∈ Epin(S), let e denote the identity element of the group H-class H of an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then ae = ea is in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The pseudo-inverse a′ of a is a′ = (ae)−1, the inverse of ae in the group H [54, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have the following characterization: a ∈ Epi(S) if and only if there exists a positive integer n and a (unique) a′ ∈ S such that the following hold [54, §2]: a′aa′ = a′ , aa′ = a′a , an+1a′ = an, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='9) where the smallest n such that an+1a′ = an is the index of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If a is an epigroup element, then so is a′ with a′′ = aa′a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The element a′′ is always completely regular and a′′′ = a′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We set aω = aa′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We also have aω = a′′a′ = a′a′′, (a′)ω = (a′′)ω = aω, and more generally aω = (aa′)m = (a′)mam = am(a′)m, for all m > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For finite semigroups, aω is usually called the idempotent power of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' A semigroup S is said to be an epigroup if Epi(S) = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If Epi1(S) = S (that is, if S is a union of groups), then S is called a completely regular semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For n > 0, the class En consists of all epigroups S such that S = Epin(S);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' thus E1 is the class of completely regular semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' ([4, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1]) Let S be a semigroup and suppose that uv, vu ∈ Epi(S) for some u, v ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then (uv)′u = u(vu)′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='10) As a relation on the set Epi1(S) of completely regular elements of a semigroup S (that is, as the restriction to Epi1(S) × Epi1(S)), ∼p is transitive (that is, ∼p = ∼∗ p) and coincides with ∼tr [4, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We extend this result to ∼n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be a semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then on Epi1(S), ∼n = ∼p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The inclusion ∼n ⊆ ∼p holds in all semigroups [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For the converse, suppose a ∼p b, where a, b ∈ Epi1(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then a = uv and b = vu for some u, v ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Set g = u and h = v(uv)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then ag = uvu = gb, bh = vuv(uv)−1 = v(uv)−1uv = ha and hag = v(uv)−1uvu = vu(vu)−1vu = vu = b, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus a ∼n b by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 7 Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In a completely regular semigroup, ∼n = ∼p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' An epigroup in which ∼n = ∼p need not be completely regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For example, a null semi- group S (S has a zero and ab = 0 for all a, b ∈ S) of order greater than 1 is not completely regular, but ∼p, and hence ∼n, are both identity relations in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be a regular epigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then S is completely simple if and only if ∼n = ∼o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' From [4, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='22], we know that a regular epigroup is completely simple if and only if ∼p = ∼o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' This is stated in the cited reference with the additional assumption that the epigroup does not have a zero, and we now take the opportunity to point out that this assumption was never used in the proof of [4, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose that S is completely simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then S is completely regular [35, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2], and so ∼n = ∼p, by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='15, and ∼p = ∼o, by [4, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='22], so ∼n = ∼o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Conversely, suppose that ∼n = ∼o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then ∼p = ∼o since ∼n ⊆ ∼p ⊆ ∼o in any semigroup, and so S is completely simple by [4, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be a semigroup in which ∼n = ∼p and let c be a regular epigroup element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then c is completely regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let c∗ denote an inverse of c, that is, cc∗c = c and c∗cc∗ = c∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let c′ denote the epigroup pseudoinverse of c, so cn+1c′ = cn for some n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will prove that cnc′ = cn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It will then follow by induction that c ∈ Epi1(S), that is, c is completely regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since c∗c · c ∼p c · c∗c = c and ∼n = ∼p, it follows that c∗c2 ∼n c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus there exist conjugators g, h ∈ S1 for c∗c2, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3, g, h are also conjugators for (c∗c2)k, ck for any positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that (c∗c2)k = c∗ck+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus gck = c∗ck+1g, which we will use multiple times in the calculation that follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have gcnc′ = c∗cn+1gc′ = c∗c · cngc′ = c∗c · c′cn+1gc′ = c∗c′ · cn+2gc′ = c∗c′ · c c∗cn+2g � �� � c′ = c∗c′cg cn+1c′ � �� � = c∗c′cgcn = c∗c′ cc∗cn+1 � �� � g = c∗ c′cn+1 � �� � g = c∗cng = gcn−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus cnc′ = hgcnc′ = hgcn−1 = cn−1, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Combining Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='18 with Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='15, we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be a regular epigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then S is completely regular if and only if ∼n = ∼p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Form the previous result and [4, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='21] we get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be a completely simple semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then ∼n = ∼p = ∼∗ p = ∼tr = ∼o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For an element a in a completely regular semigroup S, it is customary to denote the unique idempotent aω in the H-class of a by a0, that is, a0 = aa−1 = a−1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We know by Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='7 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4 that group H-classes He and Hf, where e and f are idempotents, are isomorphic via mutually inverse conjugators of e, f in the D-class of e and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The next result shows that we may select those conjugators to be the same as those for a, b for any a ∈ He and b ∈ Hf such that a ∼n b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a, b be completely regular elements of a semigroup S such that a ∼n b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then there exist mutually inverse conjugators in the D-class of a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let e = a0, f = b0, and let g, h ∈ S1 be conjugators of a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4, φg,h is an isomorphism of Ha onto Hb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In particular, e ∼n f with the same conjugators g, h, so eg = gf, fh = he, heg = f, and gfh = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Set ¯g = eg and ¯h = fh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then a¯g = aeg = ag = gb = gfb = ¯gb, a¯g¯h = aegfh = aee = e, and ¯h¯gb = fhegb = ffb = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus ¯g, ¯h are conjugators of a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Finally, ¯g¯h¯g = egfheg = egff = egf = eeg = eg = ¯g and ¯h¯g¯h = fhegfh = fffh = fh = ¯h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 8 We also have a characterization of ∼n in a completely regular semigroup S in terms of a single conjugator g ∈ S1 instead of a pair g, h ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' First we need a bit of notation and a lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that for positive integers m, (am)−1 = (a−1)m, and so we may denote this by a−m unambiguously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be a completely regular semigroup and suppose a, b ∈ S, g ∈ S1 satisfy ag = gb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then for all integers m, amg = gbm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We first verify the case m = 0: a0g = a−1 ag ���� = a−1gb = a−1 gb ���� b0 = a0gb0 = a0 gb ���� b−1 = a0agb−1 = ag ���� b−1 = gbb−1 = gb0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Next we check m = −1: a−1g = a−1a0g = a−1gb0 = a−1 gb ���� b−1 = a−1agb−1 = a0gb−1 = gb0b−1 = gb−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The remaining cases follow by an easy induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be a completely regular semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then, for all a, b ∈ S, a ∼n b ⇐⇒ ∃g ∈ S1 ( ag = gb, g0a = a, bg0 = b ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Fix a, b ∈ S, assume a ∼n b and let g, h ∈ S1 be conjugators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then g0a = g0gha = gha = a and bg0 = bhgg0 = bhg = b , using (vi) and (vii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For the converse, assume that there exists g ∈ S1 such that ag = gb, g0a = a and bg0 = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Set h = bg−1a−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='22 (with m = −1) in the following: hg = bg−1 a−1g � �� � = bg−1gb−1 = bg0 ���� b−1 = bb−1 = b0 and gh = gb ���� g−1a−1 = agg−1a−1 = ag0 a−1a � �� � a−1 = a g0a ���� a−1a−1 = aaa−1a−1 = a0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus hg · b = b and a · gh = a, and so condition (3) of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Therefore a ∼n b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have already seen that n-conjugacy is equivalent to i-conjugacy in inverse semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now we discuss the analog of i-conjugacy for completely regular semigroups, this time using the commuting inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For a completely regular semigroup S, we define ∼i by: a ∼i b ⇐⇒ ∃g ∈ S1( g−1ag = b and gbg−1 = a ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The relation ∼i cannot be regarded as a conjugacy in the class of completely regular semigroups because it is not, in general, transitive in this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The following multiplication table defines a smallest example of a completely regular semi- group in which ∼i is not transitive: 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 0 1 0 3 3 5 5 4 2 1 2 4 4 6 6 5 1 0 1 5 5 3 3 6 1 2 1 6 6 4 4 9 The commuting inverse is just the identity map: x−1 = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Set a = 0, b = 1, c = 2, g = 5, and h = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have g−1ag = 5 · 0 · 5 = 1 = b and gbg−1 = 5 · 1 · 5 = 0 = a, and so a ∼i b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Also h−1bh = 6 · 1 · 6 = 2 = c and hch−1 = 6 · 2 · 6 = 1 = b, and so b ∼i c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose, however, that x−1ax = c and xcx−1 = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then, we must have x = 2 or x = 4, but 2c2 = 2 · 2 · 2 = 2 ̸= 0 = a and 4c4 = 4 · 2 · 4 = 2 ̸= 0 = a, so a ̸∼i c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' However, one can check that in the variety of completely regular semigroups defined by the identity xx(yxx)−1 = x(yx)−1 (which includes Clifford semigroups), the relation ∼i is transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In this class, ∼i is strictly included in ∼n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We conclude this subsection by characterizing n-conjugacy in 0-Rees matrix semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let Γ be a group, I and Λ two nonempty sets, and P a Λ×I matrix with entries in Γ∪{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let M0(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' I, Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' P) be the 0-Rees matrix semigroup induced by Γ, I, Λ and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let (A, a, α), (B, b, β) ∈ M0(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' I, Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' P) \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then (A, a, α) ∼n (B, b, β) iff pβB ̸= 0 ̸= pαA & ∃g∈Γ pβBb = g−1apαAg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We start by proving the direct implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By definition, (A, a, α) ∼n (B, b, β) implies that there exist (G, g, γ), (H, h, η) ∈ M0(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' I, Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' P) such that (A, a, α)(G, g, γ) = (G, g, γ)(B, b, β) (B, b, β) = (H, h, η)(A, a, α)(G, g, γ) (A, a, α) = (G, g, γ)(B, b, β)(H, h, η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' From the first equality we get G = A and γ = β, from the second we get H = B, and from the third we get η = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Therefore, (A, apαAg, β) = (A, a, α)(A, g, β) = (A, g, β)(B, b, β) = (A, gpβBb, β) (B, b, β) = (B, h, α)(A, a, α)(A, g, β) = (B, hpαAapαAg, β) (A, a, α) = (A, g, β)(B, b, β)(B, h, α) = (A, gpβBbpβBh, α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The second line of equalities implies that pαA ̸= 0 (otherwise (B, b, β) would equal 0 in M0(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' I, Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' P), contrary to our assumptions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Similarly, the third line implies that pβB ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The first line implies that apαAg = gpβBb, that is, g−1apαAg = pβBb as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Conversely, let (A, a, α), (B, b, β) ∈ M0(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' I, Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' P) such that pβB ̸= 0 ̸= pαA and there exists g ∈ Γ such that pβBb = g−1apαAg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Consider the elements (A, g, β), (B, p−1 βBg−1p−1 αA, α) ∈ M0(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' I, Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then (A, a, α)(A, g, β) = (A, apαAg, β) apαAg=gpβBb = (A, gpβBb, β) = (A, g, β)(B, b, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' On the other hand, (B, p−1 βBg−1p−1 αA, α)(A, a, α)(A, g, β) = (B, p−1 βBg−1p−1 αApαAapαAg, β) = (B, p−1 βBg−1apαAg, β) = (B, b, β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Similarly, (A, g, β)(B, b, β)(B, p−1 βBg−1p−1 αA, α) = (A, gpβBbpβBp−1 βBg−1p−1 αA, α) = (A, gpβBbg−1p−1 αA, α) = (A, a, α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='5 Conjugacy ∼n in semigroups of transformations Let X be a non-empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In [38], n-conjugacy was characterized in the semigroup P(X) of partial transfor- mations on X, the semigroup T (X) of full transformations on X, the symmetric inverse semigroup I(X) of partial injective transformations on X, and the semigroup J (X) of full injective transformation on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In this 10 section, we describe ∼n for other basic transformation semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As in [38], we will use the representation of transformations by directed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' A directed graph (or a digraph) is a pair Γ = (A, E) where A is a set (not necessarily finite and possibly empty) and E is a binary relation on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Any element x ∈ A is called a vertex of Γ, and any pair (x, y) ∈ E is called an edge of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' A vertex x of Γ is called initial if there is no vertex y such that (y, x) ∈ E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' x is called terminal if there is no vertex y such that (x, y) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let Γ = (A, E) and Υ = (B, F) be digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' A function φ : A → B is called a homomorphism from Γ to Υ if for all x, y ∈ A, (x, y) ∈ E implies (xφ, yφ) ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' A bijection φ : A → B is called an isomorphism from Γ to Υ if for all x, y ∈ A, (x, y) ∈ E if and only if (xφ, yφ) ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will say that Γ and Υ are isomorphic, written Γ ∼= Υ, if there exists an isomorphism from Γ to Υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let α ∈ P(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We denote by dom(α) and im(α) the domain and image of α, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We define the span of α, written span(α), to be dom(α) ∪ im(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Any α ∈ P(X) can be represented by the digraph Γ(α) = (A, E), where A = span(α) and for all x, y ∈ A, (x, y) ∈ E if and only if x ∈ dom(α) and xα = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (We apply transformations on the right and compose from left to right: x(αβ) = (xα)β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=') Any digraph Γ = (A, E) such that Γ = Γ(α) for some α ∈ P(X), where A ⊆ X, is called a functional digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For the structure of functional graphs, see [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The following definitions and theorem are fundamental to studying n-conjugacy in semigroups of trans- formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let Γ = (A, E) be a digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' An initial vertex x of Γ will be called bottom initial if for all vertices y, z of Γ, if (x, y) ∈ E and (z, y) ∈ E, then z is initial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let α ∈ P(X), x be a bottom initial vertex of Γ(α) = (A, E), and y be a unique vertex in Γ(α) such that (x, y) ∈ E (y = xα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will call the set yα−1 = {z ∈ A : (z, y) ∈ E} the initial bundle in Γ(α) containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that every vertex in an initial bundle in Γ(α) is bottom initial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For example, the functional digraph presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1 on the left has four initial bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' ([38, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1]) Let Γ = (A, E) and Υ = (B, F) be digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' A homomorphism φ : A → B is called a restricted homomorphism (or an r-homomorphism) from Γ to Υ if: (1) for every terminal vertex x of Γ, xφ is a terminal vertex of Υ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (2) for every bottom initial vertex x of Γ, xφ is an initial vertex of Υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' ([38, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4]) Let S be a subsemigroup of P(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will say that S is closed under restrictions to spans if for all α, β ∈ S such that span(α) ⊆ dom(β), β|span(α) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that every semigroup of full transformations on X is closed under restrictions to spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' ([38, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='5]) Let S be a subsemigroup of P(X) that is closed under restrictions to spans, and let α, β ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then α ∼n β in S if and only if there are φ, ψ ∈ S1 such that φ is an r-homomorphism from Γ(α) to Γ(β), ψ is an r-homomorphism from Γ(β) to Γ(α), y(φψ) = y for every non-initial vertex y of Γ(α), and v(ψφ) = v for every non-initial vertex v of Γ(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Conjugacy ∼n in P(X) and T (X) was characterized in [38] in terms of a trim of a functional digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' ([38, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3]) For α ∈ P(X), we define a trim of Γ(α) as a digraph obtained from Γ(α) by removing all initial vertices except that we retain exactly one vertex from each initial bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Any two trims of Γ(α) are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We denote by Γt(α) any trim of Γ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In the semigroups P(X) and T (X), α ∼n β if and only if Γt(α) ∼= Γt(β) [38, Thms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='8 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The concept of a trim of Γ(α), where α ∈ P(X), can be replaced by a simpler concept of the prune of Γ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let α ∈ P(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The digraph Γp(α) obtained from Γ(α) by removing all initial vertices of Γ(α) will be called the prune of Γ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 11 • • • • .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' � � �⑧ ⑧ ⑧⑧ ⑧⑧ � �❄❄❄❄❄❄ �✞✞✞✞✞ � �✼✼✼✼✼ �✞✞✞✞✞ �✗✗✗✗ �✬✬✬✬ �✼✼✼✼✼ �⑧ ⑧ ⑧⑧ ⑧⑧ �❄❄❄❄❄❄ � � � � �✿✿✿✿ �❄❄❄ �❄❄❄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' � � �⑧ ⑧⑧ ⑧⑧ ⑧ �❄❄❄❄❄❄ �✞✞✞✞✞ �✼✼✼✼✼ �⑧ ⑧⑧ ⑧⑧ ⑧ � � � � �❄❄❄ �❄❄❄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' � � �⑧⑧ ⑧⑧ ⑧ ⑧ �❄❄❄❄❄❄ �⑧⑧ ⑧⑧ ⑧ ⑧ � � � �❄❄❄ Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1: A functional digraph (left), its trim (middle), and its prune (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The prune of Γ(α), where α ∈ P(X), is a subgraph of a trim of Γ(α) since in the latter some initial vertices of Γ(α) may be preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that the prune of Γ(α) is unique (not just unique up to isomorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1 presents an example of a functional digraph, its trim, and its prune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For a function f : A → B and A1 ⊆ A, denote by f|A1 the restriction of f to A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For all α, β ∈ P(X), Γt(α) ∼= Γt(β) if and only if Γp(α) ∼= Γp(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let α, β ∈ P(X) with Γt(α) = (At, Et), Γp(α) = (Ap, Ep), Γt(β) = (Bt, Ft), and Γp(β) = (Bp, Fp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose Γt(α) ∼= Γt(β) and let σ : At → Bt be an isomorphism from Γt(α) to Γt(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The set Ap consists of the non-initial vertices of Γt(α), and the subgraph of Γt(α) induced by Ap is equal to Γp(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The corresponding statement is true for β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since σ maps the set of non-initial vertices of Γt(α) onto the set of non-initial vertices of Γt(β), it follows that σ|Ap is an isomorphism from Γp(α) to Γp(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Conversely, suppose Γp(α) ∼= Γp(β) and let δ : Ap → Bp be an isomorphism from Γp(α) to Γp(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , yk, where k ≥ 0, be the initial vertices of Γp(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , vk, where vi = yiδ for each i, are the initial vertices of Γp(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By the definitions of a trim and the prune of a functional graph, for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , k}, there is a unique initial vertex xi of Γt(α) such that (xi, yi) ∈ E, and x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , xk are the only initial vertices of Γ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Similarly, for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , k}, there is a unique initial vertex ui of Γt(β) such that (ui, vi) ∈ E, and u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , uk are the only initial vertices of Γ(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence σ : At → Bt that extends δ in such a way that xiσ = ui, for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , k}, is an isomorphism from Γt(α) to Γt(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The following theorem follows immediately from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='32 and the characterizations of ∼n in P(X) and T (X) (stated above) obtained in [38] in terms of trims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In the semigroups P(X) and T (X), α ∼n β if and only if Γp(α) ∼= Γp(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We are now ready to characterize ∼n in some transformation semigroups not considered in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will begin with the semigroups of transformations whose image is restricted by a prescribed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Such semigroups have been studied extensively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' see, for example, [48,50,56–58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let X be an arbitrary set and ∅ ̸= Y ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then T (X, Y ) = {α ∈ T (X) : im(α) ⊆ Y } is a subsemigroup of T (X), consisting of transformations whose image is restricted by Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will now describe n-conjugacy in T (X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be a subsemigroup of P(X) and let α, β ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose φ, ψ ∈ S1 are r-homomorphisms as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let Ap and Bp be the sets of vertices of Γp(α) and Γp(β), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then φ|Ap is an isomorphism from Γp(α) to Γp(β) and (φ|Ap)−1 = ψ|Bb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By [38, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6], for every non-initial vertex y of Γ(α), yφ is not initial in Γ(β), and an analogous statement is true for ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus, φ|Ap is a homomorphism from Γp(α) to Γp(β), and ψ|Bp is a homomorphism from Γp(β) to Γp(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Moreover, φ|Ap and ψ|Bb are inverses of each other, which implies that they are isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 12 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let X and Y be sets such that ∅ ̸= Y ⊆ X, and let α, β ∈ T (X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then α ∼n β in T (X, Y ) if and only if Γp(α) ∼= Γp(β), and if Z is an initial bundle in Γ(α) or in Γ(β), then Z ∩ Y ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let Γ(α) = (X, E), Γ(β) = (X, F), Γp(α) = (A, Ep), and Γp(β) = (B, Fp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose α ∼n β in T (X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let φ, ψ ∈ T (X, Y ) be r-homomorphisms as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='29, where S = T (X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='34, Γp(α) ∼= Γp(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let Z be an initial bundle in Γ(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then Z = vβ−1 for some initial vertex v in Γp(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let y = vψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then y is an initial vertex in Γp(α) (since, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='34, ψ|B is an isomorphism form Γp(β) to Γp(α)), and yα−1 is an initial bundle in Γ(α) (by [38, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let x ∈ yα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since φ is a homomorphism and (x, y) ∈ E, we have (xφ, v) = (xφ, v(ψφ)) = (xφ, yφ) ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus xφ ∈ Z, and so Z ∩Y ̸= ∅ since xφ ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By symmetry, we have Z ∩ Y ̸= ∅ for every initial bundle Z in Γ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Conversely, suppose that Γp(α) ∼= Γp(β), and if Z is an initial bundle in Γ(α) or in Γ(β), then Z ∩Y ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let δ: A → B be an isomorphism from Γp(α) to Γp(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let v ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If v is not initial in Γp(β), then fix v∗ ∈ B such that (v∗, v) ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If v is initial in Γp(β), then fix v∗ ∈ Y such that (v∗, v) ∈ F (possible since Z = {u ∈ X : (u, v) ∈ F} is an initial bundle in Γ(α), and so Z ∩ Y ̸= ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Define φ : X → X by xφ = � xδ if x ∈ A, (yδ)∗ if x is initial in Γ(α) and (x, y) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It is straightforward to check that φ ∈ T (X, Y ) and φ is an r-homomorphism from Γ(α) to Γ(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Symmet- rically, we can define ψ ∈ T (X, Y ) such that ψ is an r-homomorphism from Γ(β) to Γ(α) with vψ = vδ−1 for every v ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then α ∼n β in T (X, Y ) by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Next, we consider the semigroup of full order-preserving transformations on a chain with n elements, where n ≥ 1, say Xn = {1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' < n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Viewing Xn as a set, we denote by Tn the semigroup T (Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let On be the subset of Tn consisting of full order-preserving transformations, that is, On = {α ∈ Tn : ∀x,y∈Xn(x ≤ y ⇒ xα ≤ yα)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The semigroup On has been studied in numerous papers since the 1960s (see [29, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will now describe n-conjugacy in On.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let α, β ∈ P(Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose Γ′(α) = (A′, E′) and Γ′(β) = (B′, F ′) are subgraphs of Γ(α) and Γ(β), respectively, where A′ = {x1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' < xk} and B′ = {y1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' < yk} (k ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We denote by Γ′ β(α) the digraph obtained from Γ′(α) by replacing every vertex xi with yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let α, β ∈ On, with Γ(α) = (X, E), Γ(β) = (X, F), Γp(α) = (A, Ep), and Γp(β) = (B, Fp), where A = {x1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' < xk} and B = {y1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' < ym} (k, m ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then α ∼n β in On if and only if k = m and Γp β(α) = Γp(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose α ∼n β in On.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let φ, ψ ∈ On be r-homomorphisms as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='34, φp = φ|A is an isomorphism from Γp(α) to Γp(β), ψp = ψ|B is an isomorphism from Γp(β) to Γp(α), and ψp = φ−1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' This gives k = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Further, Γp β(α) = (B, E0), where (yi, yj) ∈ E0 if and only if (xi, xj) ∈ Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It remains to show that E0 = Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since φp preserves order, we have x1φp < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' < xkφp, which implies xiφp = yi for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The equality E0 = Fp follows since for all i, j, (xi, xj) ∈ Ep if and only if (yi, yj) = (xiφp, xjφp) ∈ Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence Γp β(α) = Γp(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Conversely, suppose that k = m and Γp β(α) = Γp(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Fix y∗ i ∈ X such that (y∗ i , yi) ∈ F (such a y∗ i exists since yi is not initial in Γ(β)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let Ai = {xj : (xj, xi) ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let x be an initial vertex in Γ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then xα = xi (so (x, xi) ∈ E) for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that x is bottom initial in Γ(α) if and only if Ai = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose Ai ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Write Ai = {xj1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' < xjw}, where w ≥ 1, and define mx ∈ {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , jw} as follows: mx = j1 if x < xj1, mx = jw if xw < x, and mx = js if xjs < x < xjs+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now, define φ : X → X by xφ = \uf8f1 \uf8f2 \uf8f3 yi if x = xi, y∗ i if x is bottom initial in Γ(α) (so Ai = ∅) and (x, xi) ∈ E, ymx if x is initial, but not bottom initial, in Γ(α) (so Ai ̸= ∅) and (x, xi) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 13 Note that xiφ = yi for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' First, we will prove that φ is an r-homomorphism from Γ(α) to Γ(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since Γp β(α) = Γp(β), (xi, xj) ∈ E if and only if (yi, yj) ∈ F, for all i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Moreover, for every i, (y∗ i , yi) ∈ F and if x is initial, but not bottom initial, in Γ(α) with xα = xi, then (ymx, yi) ∈ F (since (xmx, xi) ∈ E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that φ is a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since Γ(α) does not have any terminal vertices, (1) of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='27 is vacuously satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let x be a bottom initial vertex of Γ(α) and let xi = xα (so (x, xi) ∈ E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose to the contrary that xφ is not initial in Γ(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then xφ = yj, for some j, and (yj, yi) = (xφ, xiφ) ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus (xj, xi) ∈ E, which is a contradiction since (x, xi) ∈ E and x is bottom initial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence xφ is initial in Γ(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Therefore, φ is an r-homomorphism from Γ(α) to Γ(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Next, we will prove that φ ∈ On.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let x, z ∈ X with x < z, and let xi = xα and xj = zα (so (x, xi) ∈ E and (z, xj ∈ E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since α ∈ On, we have xi ≤ xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We want to prove that xφ ≤ zφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Consider three possible cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' x and z are not initial in Γ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then x = xs and z = xt, for some s and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus xs < xt, and so xφ = xsφ = ys < yt = xtφ = zφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' x or z is initial in Γ(α), and i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then xi < xj, and so yi < yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since φ is a homomorphism from Γ(α) to Γ(β), we have (xφ, yi) = (xφ, xiφ) ∈ F and (zφ, yj) = (zφ, xjφ) ∈ F, that is, (xφ)β = yi and (zφ)β = yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since β ∈ On, zφ ≤ xφ would imply yj ≤ yi, which would contradict yi < yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence xφ < zφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' x or z is initial in Γ(α), and i = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If Ai = ∅, then both x and z are bottom initial in Γ(α), and so xφ = y∗ i = zφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let Ai = {xj1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' < xjw} ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose x is initial in Γ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then xφ = ymx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose z is not initial in Γ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then z = xjq for some q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since x < z = xjq, we have xmx ≤ xjq (by the definition of mx), and so xφ = ymx ≤ yjq = xjqφ = zφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose z is initial in Γ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then zφ = ymz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since x < z, xmx ≤ xmz, and so xφ = ymx ≤ ymz = zφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If z is initial in Γ(α), then we obtain xφ ≤ zφ by a similar argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence, in all cases, xφ ≤ zφ, that is, φ ∈ On.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By symmetry, there exists an r-homomorphism ψ from Γ(β) to Γ(α) such that yiψ = xi for all i, and ψ ∈ On.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then for every i, xi(φψ) = xi and yi(ψφ) = yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence φ and ψ are as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='29, and so α ∼n β in On.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Consider α, β, δ ∈ O6 whose digraphs are given in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The prunes of the digraphs are presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3, with the orderings of vertices: 4 < 5 < 6 in Γp(α), 3 < 4 < 5 in Γp(β), and 2 < 4 < 5 in Γp(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Replacing the vertices in Γp(α) according to these orderings, we obtain Γp β(α) and Γp δ(α) as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We can see that Γp β(α) = Γp(β), but Γp δ(α) ̸= Γp(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='37, α and β are n-conjugate in O6, but α and δ are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' � �✄✄✄✄✄ � � �❀❀❀❀❀ � 5 3 2 1 6 4 �• � �⑧⑧ ⑧⑧ ⑧⑧ � � �� 4 3 2 1 5 6 � �⑧⑧ ⑧⑧ ⑧⑧ � � �� 5 4 3 6 2 1 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2: Γ(α) (left), Γ(β) (middle), and Γ(δ) (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In the semigroups I(X) and J (X) of injective transformations on X (partial and full, respectively), α ∼n β if and only if Γ(α) ∼= Γ(β) [38, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2 and Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The latter result is also true for the semigroup Ω(X) of surjective transformations on X, which was studied in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We actually have a stronger result for Ω(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let Sym(X) be the symmetric group of permutations on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be any subsemigroup of P(X) such that Sym(X) ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For α, β ∈ S, we say that α is conjugate to β by permutation if β = σ−1ασ for some σ ∈ Sym(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that the conjugacy-by-permutation is included in ∼n in any such semigroup S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 14 � � � 5 6 4 � � � 4 3 5 � � � 5 4 2 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3: Γp(α) (left), Γp(β) (middle), and Γp(δ) (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' � � � 4 5 3 � � � 4 2 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4: Γp β(α) (left) and Γp δ(α) (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For all α, β ∈ Ω(X), the following conditions are equivalent: (a) α and β are n-conjugate in Ω(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (b) the digraphs Γ(α) and Γ(β) are isomorphic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (c) α and β are conjugate by permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let α, β ∈ Ω(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose that α ∼n β in Ω(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='29 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='34, Γp(α) ∼= Γp(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since the digraph of any surjective transformation does not have any initial vertices, Γp(α) = Γ(α) and Γp(β) = Γ(β), and so Γ(α) ∼= Γ(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence (a) implies (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose that Γ(α) ∼= Γ(β), and let σ be an isomorphism from Γ(α) = (X, E) to Γ(β) = (X, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then clearly σ ∈ Sym(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let u ∈ X and v = uβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then (u, v) ∈ F, and so (uσ−1, vσ−1) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus (uσ−1)α = vσ−1 = (uβ)σ−1, which implies u(σ−1ασ) = u(βσ−1σ) = uβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence β = σ−1ασ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have proved that (b) implies (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Finally, (c) implies (a) since the conjugacy-by-permutation is included in ∼n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The same result is true for the semigroup J (X) of full injective transformations on X [38, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3], and for the finite symmetric inverse semigroup I(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' However, for an infinite set X, the conjugacy-by- permutation in I(X) is strictly included in n-conjugacy in I(X) [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Recall that for an integer n ≥ 1, Xn = {1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' < n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Viewing Xn as a set, we denote by In the symmetric inverse semigroup I(Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let OIn be the subset of In consisting of partial injective order- preserving transformations, that is, OIn = {α ∈ In : ∀x,y∈Xn(x < y ⇒ xα < yα)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then OIn is an inverse semigroup [25,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will now describe n-conjugacy in OIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let Γ be a digraph and let v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , vk, k ≥ 1, be pairwise distinct vertices of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose that v0 → v1 → · · · → vk−1 → v0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1) v0 → v1 → · · · → vk−1 → vk (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2) are sub-digraphs of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We call (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2), respectively, a cycle of length k (or a k-cycle), writ- ten (v0 v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' vk−1), and a chain of length k (or a k-chain), written [v0 v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' vk], in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We can view (v0 v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' vk−1) and [v0 v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' vk] as partial injective transformations on the set of vertices of Γ, both with domain {v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , vk−1}, and the values calculated according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 15 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let α ∈ P(X), where X is any set, and let x ∈ span(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The subgraph of Γ(α) induced by the set {y ∈ span(α) : αk(y) = αm(x) for some integers k, m ≥ 0} is called the component of Γ(α) containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The components of Γ(α) correspond to the connected compo- nents of the underlying undirected graph of Γ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If α ∈ In, then each component of Γ(α) is either a cycle or a chain, that is, Γ(α) is a disjoint union of cycles and chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will use the language “a cycle [chain] in α” to mean “a component in Γ(α) that is a cycle [chain].” If α ∈ OIn, then each cycle in α has length 1, and if [v0 v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' vm] is a chain in α, then either v0 < v1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' < vm or v0 > v1 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' > vm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Recall that for α ∈ P(X), span(α) = dom(α) ∪ im(α) and that span(α) is the set of vertices of Γ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For the meaning of Γβ(α), which appears in the following theorem, see Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let α, β ∈ OIn with span(α) = {x1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' < xk} and span(β) = {y1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' < ym}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then α ∼n β in OIn if and only if k = m and Γβ(α) = Γ(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose α ∼n β in OIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since OIn is closed under restrictions to spans, there is φ ∈ OIn such that φ is an isomorphism from Γ(α) to Γ(β) (by [38, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus k = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let Γ(α) = (A, E) and Γ(β) = (B, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have Γβ(α) = (B, E0), where (yi, yj) ∈ E0 if and only if (xi, xj) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It remains to show that E0 = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since φ preserves order, we have x1φ < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' < xkφ, which implies xiφ = yi for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The equality E0 = F follows since for all i, j, (xi, xj) ∈ E if and only if (yi, yj) = (xiφ, xjφ) ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence Γβ(α) = Γ(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Conversely, suppose that k = m and Γβ(α) = Γ(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Define φ : A → B by xiφ = yi for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then φ ∈ OIn and for all i, j, (xi, xj) ∈ E ⇔ (yi, yj) ∈ E0 ⇔ (yi, yj) ∈ F ⇔ (xiφ, xjφ) ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus, φ is an isomorphism from Γ(α) to Γ(β), and so α ∼n β in OIn by [38, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let α ∈ OIn with span(α) = {x1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' < xk}, k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='41, we can construct the n-conjugacy class [α]n as follows: (a) begin with [α]n = ∅ and Yk = the set of all subchains {y1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' < yk} of Xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (b) select a subchain {y1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' < yk} from Yk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (c) replace each xi in Γ(α) with yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (d) add β to [α]n, where β is the transformation represented by the digraph obtained in (c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (e) remove the subchain {y1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' < yk} selected in (b) from Yk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (f) if Yk ̸= ∅, return to (b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' otherwise STOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By the above algorithm and the fact that [0]n = {0} in any semigroup with zero, we have if α ∈ OIn with | span(α)| = k, then |[α]n| = �n k � for every k ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let ∅ ̸= α ∈ OIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If Γ(α) has s + t components, where σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , σs are 1-cycles and τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , τt are chains, then we will write α = σ1 ⊔ · · · ⊔ σs ⊔ τ1 ⊔ · · · ⊔ τt, where each σi and τj is viewed as an element of OIn, and “⊔” (called the join) is the union of functions viewed as sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Consider α = (1) ⊔ (4) ⊔ [3 5 7] ⊔ [10 9 8] ∈ OI11, and note that we have span(α) = {1 < 3 < 4 < 5 < 7 < 8 < 9 < 10} and | span(α)| = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Select any subchain of X11 with 8 elements, say {2 < 3 < 5 < 6 < 7 < 8 < 10 < 11}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now, replace each x in α, written as above, with the corresponding (according to the orderings) y from that subchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then, β = (2) ⊔ (5) ⊔ [3 6 7] ⊔ [11 10 8] is n-conjugate to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 16 3 Conjugacy ∼n and partial inner automorphisms If G is a group, then any g ∈ G defines an inner automorphism of G by a �→ g−1ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The notion of natural conjugacy ∼n leads us to a definition of a partial inner automorphism of an arbitrary semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be a semigroup, fix g, h ∈ S1, and define Dg,h = {a ∈ S | gh · a = a · gh = a} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that for all a, b ∈ S, a ∼n b with conjugators g and h if and only if a ∈ Dg,h and b = hag (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let ⪯ be a preorder on a set A (that is, ⪯ is a binary relation on A that is reflexive and transitive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We say that a subset B of A is downward directed in ⪯ if for all a ∈ A and b ∈ B, a ⪯ b implies a ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be a semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then the relation ⪯H on S defined by a ⪯H b if sb = a = bt for some s, t ∈ S1 is a preorder on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that if a ⪯H b and b ⪯H a, then a H b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be a semigroup and let g, h ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then: (1) Dg,h is a subsemigroup of S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (2) Dg,h is downward directed in the H-preorder ⪯H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (3) Dg,h is downward directed in the natural partial order ≤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (4) if a ∈ Dg,h, then Ha ⊆ Dg,h, where Ha denotes the H-class of a in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (1) is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For (2), assume a ∈ Dg,h and c ⪯H a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then there exist s, t ∈ S1 such that sa = c = at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have c · gh = s a · gh � �� � = sa = c and gh · c = gh · a � �� � t = at = c, and so c ∈ Dg,h, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now (3) follows from (2) since the natural partial order ≤ refines the H-preorder ⪯H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Finally, (4) also follows from (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now we define a mapping by φg,h : Dg,h → S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' a �→ hag .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that for all a, b ∈ S, a ∼n b with conjugators g and h if and only if aφg,h = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' φg,h is a partial automorphism of S, specifically, it is an isomorphism of Dg,h onto Dh,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For a ∈ Dg,h, set b = aφg,h = hag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2, a ∼n b with g, h as conjugators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus we also have hg·b = b·hg = b, that is, b ∈ Dh,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In addition, gbh = a, that is, bφh,g = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since aφg,hφh,g = ghagh = a and bφh,gφg,h = b, we have φg,h is a bijection from Dg,h to Dh,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Finally we show that φg,h is a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a1, a2 ∈ Dg,h be given and set bi = haig for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since ai ∼n bi, we have (a1a2)φg,h = ha1 a2g ���� = ha1g ���� b2 = b1b2, which establishes the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be a semigroup and suppose a, b ∈ S satisfy a ∼n b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then ak ∼n bk for all positive integers k, and if g, h ∈ S1 are conjugators for a, b, then g, h are also conjugators for ak, bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The bijection φg,h : Dg,h → Dh,g restricts to bijections between H-classes, that is, for a ∈ Dg,h and b = aφg,h, the restriction of φg,h to Ha is a bijection onto Hb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Further, if Ha is a group H-class then φg,h is a group isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Fix c ∈ Ha and let d = cφg,h = hcg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' There exist s1, s2, t1, t2 ∈ S1 such that s1a = c, s2c = a, at1 = c, ct2 = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Set ¯si = hsig and ¯ti = htig for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then ¯s1b = hs1 gb ���� = h s1a ���� g = hcg = d , ¯s2d = hs2 ghc ���� g = hs2cg = hag = b , b¯t1 = bh ���� t1g = h at1 ���� g = hcg = d and d¯t2 = h cgh ���� t2g = h ct2 ���� g = hag = b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 17 This proves d H b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus (Ha)φg,h ⊆ Hb and by symmetry, (Hb)φh,g ⊆ Ha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Finally Hb = (Hb)φh,gφg,h ⊆ (Ha)φg,h ⊆ Hb, so that φg,h is a bijection of Ha onto Hb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The remaining assertion follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It is a basic result in semigroup theory that any two group H-classes in the same D-class of a semigroup are isomorphic [35, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have actually reproved this;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' it follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='7 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Our proofs are certainly more involved but better highlight the role of n-conjugacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' H ◦ ∼n = ∼n ◦ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Say c H a ∼n b and let g, h ∈ S1 be conjugators for a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Set d = (c)φg,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4, we have b H d ∼n c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The other inclusion is similarly proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now we consider the composition of partial automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For gi, hi ∈ S1, i = 1, 2, we have φg1,h1φg2,h2 ⊆ φg1g2,h2h1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The domain of φg1,h1φg2,h2 is C = {a ∈ Dg1,h1 | h1ag1 ∈ Dg2,h2} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If a ∈ C, then g1g2h2h1 · a = g1 g2h2h1ag1 � �� � h1 = g1h1ag1h1 = a and a · g1g2h2h1 = g1 h1ag1g2h2 � �� � h1 = g1h1ag1h1 = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus a ∈ Dg1g2,h2h1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Clearly aφg1,h1φg2,h2 = aφg1g2,h2h1 for a ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In general, the inclusion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1) is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For instance, in the group Z2 written additively, the map φ0,1 is the empty map and thus so is φ0,1φ0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' However, φ0+0,1+1 = φ0,0 is the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let Inn(S) denote the inverse monoid of partial automorphisms generated by the φg,h’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will call Inn(S) the partial inner automorphism monoid of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' This is a natural generalization to semigroups of the inner automorphism group of a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Indeed, suppose S is a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For g, h ∈ S, if Dg,h ̸= ∅, then gh · a = a for some a, so gh = 1, that is, h = g−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' But then Dg,g−1 = S and φg,g−1 is the usual inner automorphism of conjugacy by g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus if S is a nontrivial group, our Inn(S) is a zero group, the union of the usual inner automorphism group of S and the empty mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The case where S is an inverse semigroup is studied in detail in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It turns out that for any g, h ∈ S1, Dg,h ⊆ Dg,g−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In that case, we may just work with the partial inner automorphisms φg,g−1 and for those, the inclusion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1) is an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We then get a homomorphism Φ : S → Inn(S);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' g �→ φg,g−1, whose kernel is precisely the central congruence of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In particular, if S is the symmetric inverse semigroup of partial injective transformations on a set X, then the homomorphism Φ is an isomorphism, and so S ∼= Inn(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It is well known that nonisomorphic groups can have isomorphic automorphism groups (eg, Q8 and S4 both have automorphism groups isomorphic to S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The same happens with partial inner automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The cyclic groups of order 2 and 3, both have the 2-chain as the semigroup of partial inner automorphisms (and the 2-chain is isomorphic to its semigroup of partial inner automorphisms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 18 Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' An elementary observation in group theory is that if two elements a, b are conjugate, then every element of the centralizer Ca of a is conjugate to some element of the centralizer Cb of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' This is not true for ∼n, even in highly structured semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Consider the semigroup defined by this table: e r1 r2 s1 s2 s3 f c e e r1 r2 s1 s2 s3 e s1 r1 r1 r2 e s3 s1 s2 r1 s3 r2 r2 e r1 s2 s3 s1 r2 s2 s1 s1 s2 s3 e r1 r2 s1 e s2 s2 s3 s1 r2 e r1 s2 r2 s3 s3 s1 s2 r1 r2 e s3 r1 f e r1 r2 s1 s2 s3 f c c s1 s2 s3 e r1 r2 c f This is a Clifford semigroup, that is, an inverse semigroup in which the idempotents (in this case, e and f) commute with all elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We see that this semigroup is a union (in fact, semilattice) of the subgroups A = {e, r1, r2, s1, s2, s3} and B = {e, c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since s2 3 = e, the identity element of A, we have that A ⊆ Ds3,s3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now (s1)φs3,s3 = s3s1s3 = s2, and thus s1 ∼n s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We see from the table that Cs1 = {e, f, s1, c} and Cs2 = {e, f, s2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If gh · c = c = c · gh, then from the table, gh = f, and so g = h = f or g = h = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We compute cφf,f = c and cφc,c = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Therefore the n-conjugacy class of c is [c]n = {c}, and so c is not n-conjugate to any element of Cs2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We can use the machinery above to show that in epigroups, we can impose additional restrictions on conjugators without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Recall that elements g, h of a semigroup S are mutually inverse if ghg = g and hgh = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be an epigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then for all g, h ∈ S1, there exist mutually inverse ¯g, ¯h ∈ S1 such that φg,h ⊆ φ¯g,¯h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let g, h ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Setting ¯g = (gh)ωg and ¯h = h(gh)′, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2) we obtain: ¯g¯h = (gh)ωgh(gh)′ = (gh)ω, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3) ¯h¯g = h(gh)′(gh)ωg = h(gh)′g (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='10) = hg(hg)′ = (hg)ω, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4) ¯g¯h¯g = (gh)ω(gh)ωg = (gh)ωg = ¯g, ¯h¯g¯h = h(gh)′(gh)ω = h(gh)′ = ¯h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Therefore ¯g, ¯h are mutually inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now assume aφg,h = b, that is, a ∼n b with g, h as conjugators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will now show that (gh)ωa = a = a(gh)ω and (hg)ωb = b = b(hg)ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='5) Indeed, choose n such that (gh)n(gh)ω = (gh)n+1(gh)′ = (gh)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then a(gh)ω = a(gh)n·(gh)ω = a(gh)n = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The other three equations in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='5) are proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now we use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='5) in the following calculations: a¯g = a(gh)ωg = ag = gb = g(hg)ωb = (gh)ωgb = ¯gb , ¯h¯g · b = (hg)ωb = b , and a · ¯g¯h = a(gh)ω = a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2, ¯g, ¯h are conjugators for a, b, and thus aφ¯g,¯h = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 19 Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In general, the conclusion of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='12 is a strict inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For example, consider the semigroup defined by the multiplication table 1 2 3 4 1 1 1 4 4 2 2 2 3 3 3 3 3 2 2 4 4 4 1 1 Set g = 1 and h = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then ¯g = 1 and ¯h = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For a = 1, b = 2, we have a¯g = 1 = ¯gb, a¯g¯h = 1 = a, ¯h¯gb = 2 = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus a ∼n b with ¯g, ¯h as conjugators, so aφ¯g,¯h = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' However, agh = 3 ̸= a and so a ̸∈ Dg,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If a ∼n b in an epigroup S, then there exist mutually inverse conjugators for a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1 The partial inner automorphism monoid of T(X) Computing the partial inner automorphisms of a given semigroup is a challenge in itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We already observed that the symmetric inverse semigroup is isomorphic to its inverse semigroup of partial inner automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In this subsection, we describe the partial inner automorphism monoid S = Inn(T (X)), for the full transfor- mation monoid of a set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It turns out that the structure of S is essentially isomorphic to the combination of two components, one of which is the symmetric inverse semigroup on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The other component consists of bijections between partitions of X with the same number of parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In the same way that the partial composition operation of the symmetric inverse semigroup is based on the intersection of an image and a domain, the operation of the second component is based on the join ∨ of two partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In the above description, we write “essentially” for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The two components are not entirely independent, but are required to be compatible which each other in a natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In addition, further small adjustments are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The number of elements of Inn(T (X)) that are affected by these adjustments are small relative to the size of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Throughout this subsection, we will blur the distinction between partitions and their corresponding equivalence relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let g, h ∈ T (X) and Dg,h be as defined above, that is, Dg,h = {x ∈ T (X) : ghx = xgh = x} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then there exists a partition P of X, and a partial section I of P, such that Dg,h consists of all transfor- mations t with im t ⊆ I and P ⊆ ker t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Moreover, I, P can be chosen so that every singleton part S of P satisfies S ⊆ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' I is uniquely determined by Dg,h, and if Dg,h contains more than one transformation, then P is uniquely determined by Dg,h as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Conversely, suppose that P is a partition of X and I is a partial section of P such that all singleton parts of P intersect I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then there exist g, h ∈ T (X) such that Dg,h consists of all transformations t ∈ T (X) with im t ⊆ I and P ⊆ ker t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In the above cases, if |I| ≥ 2, then I, P uniquely determine Dg,h, while if |I| ≤ 1, then I uniquely determines Dg,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Assume first that g, h ∈ T (X), and let D = Dg,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Clearly D only depends on the product p = gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let I ⊆ X be the set of points fixed by p, and let P be the collection of connected components of the function graph of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In each part of P, there is at most a single point x with xp = x, and so I is a partial section of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If for some x ∈ X, {x} is a singleton part of P, then xp = x, and so {x} ⊆ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let t ∈ Dg,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Because tp = t, p acts as the identity on the image of t and so t maps into I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Because pt = t, if xp = y, then yt = x(pt) = xt, and so (x, y) ∈ ker t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that the connected component of x in the function graph of p is contained in the kernel of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence P ⊆ ker t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 20 Conversely, if t ∈ T (X) maps into I and P ⊆ ker t, it is straightforward to check that pt = tp = t, and so t ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that D consists of all t with im t ⊆ I and P ⊆ ker t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now, let I and P be any set and partition that characterize D in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then I is the union of all images of transformations in D, and hence is uniquely determined by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If |D| ≥ 2, then |I| ≥ 2 and |P| ≥ 2, the latter because I is a partial section of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose that P ∈ {P1, P2}, where P1, P2 are two distinct partitions of X, each with at least two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='og.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' P1 is a refinement of a 2-partition P ′ of X that does not contain P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Because |I| ≥ 2, there exists a t ∈ T (X) with im t ⊆ I and ker t = P ′ ⊇ P1, but P2 ̸⊆ P ′ = ker t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that P is uniquely determined by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now suppose that P is a partition of X and I is a partial section of P such that all singleton parts of P are contained in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let g ∈ T (X) be the identity, and define h ∈ T (X) as follows: if x ∈ X is in a part B of P intersecting I, then let xh = y were y is the unique element of B ∩ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If B is a part of P not intersecting I then |B| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Pick b1 ̸= b2 ∈ B, and let b1h = b2, xh = b1 for x ∈ B \\ {b1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Applying the construction in the first part of the proof to Dg,h, it is straightforward to verify that we recover the sets I and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence Dg,h contains all transformations t with im t ⊆ I and P ⊆ ker t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The final uniqueness result now also follows from the first part for |I| ≥ 2, and is trivial for |I| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For any X-partition P and I ⊆ X, we will use the notation DP,I to refer to the set of t ∈ T (X) with im t ⊆ I, P ⊆ ker t, where we also include such I, P in which I is not a partial section of P, or for which not all singleton parts of P intersect I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let Dg,h = DP,I and Dh,g = DP ′,I′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then g|I : I → I′ , h|I′ : I′ → I are inverse bijections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The result is clear if I = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Otherwise, pick i ∈ I, and define t ∈ T (X) by [j]P t = j for j ∈ I, xt = i otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Clearly, t ∈ Dg,h and im t = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Because ght = t, im(ht) = I, and because htg ∈ DP ′,I′, we see that g|I maps into I′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Dually, hI′ maps into I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Because t ∈ Dg,h, tgh = t, and so gh acts as the identity on the image I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Applying the argument to a correspondingly constructed element t′ ∈ Dh,g, we get that hg is the identity on I′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let Dg,h = DP,I, Dh,g = DP ′,I′ with |I| ≥ 2 (and therefore |I′| ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then ˆg : P → P ′, given by [p]P ˆg = [pg]P ′, and ˆh : P ′ → P, given by [p′]P ′ˆh = [p′h]P , are well-defined inverse bijections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Moreover, for all B ∈ P, B′ ∈ P ′, we get B ∩ I = ∅ ⇔ Bˆg ∩ I′ = ∅ and B′ ∩ I′ = ∅ ⇔ B′ˆh ∩ I = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Pick distinct i, j ∈ I, and [p] ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Define t ∈ T (X) by [p]P t = j, xt = i otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Clearly, t ∈ Dg,h = DP,I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Because j = pt = p(ght) we see that p(gh) ∈ [p]P , and therefore [p]P (gh) ⊆ [p]P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose that p1, p2 ∈ [p]P are such that [p1g]P ′ ̸= [p2g]P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let t′ ∈ Dh,g be a transformation that maps [p1g]P ′, [p2g]P ′ to distinct elements i′ 1, i′ 2 ∈ I′ (such t′ clearly exists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then gt′h ∈ Dg,h = DP,I, and therefore i′ 1h = p1gt′h = p2gt′h = i′ 2h, which contradicts the injectivity of h|I′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that ˆg is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' A dual argument shows the corresponding claim for ˆh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We already have seen that p(gh) ∈ [p]P , and so [p]P ˆgˆh = [p]P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As [p]P was arbitrary, we see that ˆgˆh acts as the identity on ¯P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' An analogous argument shows that ˆhˆg is the identity on P ′, and hence ˆg and ˆh are inverse bijections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The last claim follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We now can derive a classification theorem for the generating elements φg,h of the partial inner automor- phism monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The partial inner automorphisms of T (X) having the form φg,h, and acting on more than one transformation are in bijective correspondence with the tuples (P, P ′, I, I′, α, β), where P and P ′ are partitions of X, with |P| = |P ′|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 21 I and I′ are partial sections, of P and P ′, respectively, with |I| = |I′| ≥ 2, and intersecting all singleton sets of P, P ′, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' α : I → I′ is a bijection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' β : P → P ′ is a bijection extending the partial bijection between P and P ′ induced by α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' such that The domain of φg,h consists of all transformations t ∈ T (X) with im t ⊆ I, P ⊆ ker t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The image of φg,h consists of all transformations t ∈ T (X) with im t ⊆ I′, P ′ ⊆ ker t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Given t in the domain of φg,h, and x ∈ X, we have (x)(tφg,h) = iα, where i ∈ I is the unique element in (([x]P ′)β−1)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The partial inner automorphisms of T (X) having the form φg,h and acting on at most one transformation consist of all functions mapping one constant transformation on X to another, and (for |X| ̸= 1), the empty mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We first consider the case of the partial inner automorphisms φg,h whose domain contains more than one transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='15, P, I, P ′, I′ exist, have the stated properties and are uniquely determined by Dg,h and Dh,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Set α = g|I, and β = ˆg, where ˆg is defined as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='16 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='17, α and β are bijections, and by its definition, β extends the partial function on P induced by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let t ∈ dom φg,h = DP,I, and x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='17, β−1 = ˆh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Therefore [x]P ′β−1 ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As t ∈ DP,I, (([x]P ′)β−1)t contains a single element i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We now have that x(ht) ∈ ([x]P ′ˆh)t = {i}, and so x(htg) = (x(ht))g = ig = ig|I = iα, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now for any i ∈ I, let ci ∈ DP,I be the constant function with image i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows from the above that ciφg,h = ciα, and hence α is uniquely determined by φg,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Finally suppose that β, β′ : P → P ′ are two bijections, that, together with some φg,h, α, P, I, P ′, I′ satisfy the conditions of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Pick two distinct elements i, j ∈ I, and for each B ∈ P, let tB be the transformation with Bt = {i}, xt = j for x /∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let x ∈ Bβ, then x(tBφg,h) = iα, as ([x]P ′β−1)tB = {i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Because α is injective, it follows that ([x]P ′β′−1)tB = {i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' From the definition of tB this implies ([x]P ′β′−1) = ([x]P ′β−1), and so β−1 and β′−1 agree on Bβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As B was arbitrary, we get β = β′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The final claim about φg,h with |Dg,h| ≤ 1 easily follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will now turn our attention to general elements of Inn(T (X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let P, P ′ be partitions of X, and γ : P → P ′ a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If ¯P = {Bi} is a partition that refines to P, we define ¯γ on ¯P by (∪Bi)¯γ = ∪((Bi)γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It is clear that ¯γ is well-defined, and that its image is a partition that refines to P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let φ ∈ Inn(T (X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then there exist partitions P, P ′ of X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' I, I′ ⊆ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' bijections α : I → I′, β : P → P ′ satisfying [i]P β = [iα]P ′ for all i ∈ I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' such that The domain of φ consists of all transformations t ∈ T (X) with im t ⊆ I, P ⊆ ker t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The image of φ consists of all transformations t ∈ T (X) with im t ⊆ I′, P ′ ⊆ ker t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Given t in the domain of φ, and x ∈ X, we have (x)(tφ) = iα, where i ∈ I is the unique element in (([x]P ′)β−1)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 22 Moreover, if φ1, φ2 ∈ Inn(T (X)) have corresponding parameters (P1, I1, P ′ 1, I′ 1, α1, β1) and (P2, I2, P ′ 2, I′ 2, α2, β2) then φ1φ2 corresponds to ((P ′ 1 ∨ P2)¯β−1 1 , (I′ 1 ∩ I2)α−1 1 , (P ′ 1 ∨ P2)¯β2, (I′ 1 ∩ I2)α2, α1α2, ¯β1 ¯β2) , where α1α2 refers to the partial composition α1|(I′ 1∩I2)α−1 1 α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will show the assertions by structural induction over the involved elements φ, φ1, φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The beginning of the induction corresponds to those φ of the form φg,h, and follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='18 (in the cases with |Dg,h| ≤ 1, we can chose P = P ′ = {X}, β = id{{X}}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose the theorem holds for φ1, φ2 ∈ Inn(T (X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then L := im φ1 ∩ dom φ2 consists of all transforma- tions t with im t ⊆ I′ 1 ∩ I2 and P ′ 1 ∨ P2 ⊆ ker t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It is now straightforward to check that Lφ−1 1 = D(P ′ 1∨P2) ¯β−1 1 ,(I′ 1∩I2)α−1 1 and Lφ2 = D(P ′ 1∨P2) ¯β2,(I′ 1∩I2)α2 and hence these parameters define the domain and image of φ1φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let i ∈ (I′ 1 ∩ I2)α−1 1 ⊆ I, then [i](P ′ 1∨P2) ¯β−1 1 ¯β1 ⊇ [i]P1β1 = [iα1]P ′ , and so [i](P ′ 1∨P2) ¯β−1 1 ¯β1 = [iα1]P ′ 1∨P2 ⊃ [iα1]P2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Because iα1 ∈ I′ 1 ∩ I2 ⊆ I2, we get that [i](P ′ 1∨P2) ¯β−1 1 ¯β1 ¯β2 ⊃ [iα1]P2β2 = [iα1α2]P ′ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence we get [i](P ′ 1∨P2) ¯β−1 1 ¯β1 ¯β2 = [iα1α2](P ′ 1∨P2) ¯β2 , as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let t ∈ Lφ−1 1 , and x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Pick an element y ∈ [x](P ′ 1∨P2) ¯β2 ¯β−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Because ¯β−1 2 is injective, we have [x](P ′ 1∨P2) ¯β2 ¯β−1 2 = [y]P ′ 1∨P 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that ([x](P ′ 1∨P2) ¯β2(¯β1 ¯β2)−1)t = ([x](P ′ 1∨P2) ¯β2 ¯β−1 2 ¯β−1 1 )t = ([y]P ′ 1∨P2 ¯β−1 1 )t = ([y]P ′ 1β−1 1 )t , where the last equality holds because the kernel of t contains (P ′ 1 ∨ P2)¯β−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By induction, this set contains a unique element i such that y(tφ1) = iα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Also by induction, x((tφ1)φ2) = jα2, where j is the unique element in ([x](P ′ 1∨P2) ¯β2 ¯β−1 2 )(tφ1) = ([y]P ′ 1∨P 2)(tφ1) = {y(tφ1)} = {iα1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence x((tφ1)φ2) = (iα1)α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Because i ∈ ([x](P ′ 1∨P2) ¯β2(¯β1 ¯β2)−1), the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We can now obtain results about the structure of Inn(T (X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For a set X, let A(X), B(X) be the set of all bijections between subsets of X, and bijections on partitions of X, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We say that α ∈ A(X), α : I → I′ and β ∈ B(X), β : P → P ′ are compatible, written α ≈ β, if [i]P β = [iα]P ′ for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let V (X) = {(α, β) : α ∈ A(X), β ∈ B(X), α ≈ β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' On V (X) we define a binary operation (α1, β1)(α2, β2) = (α1α2, ¯β1 ¯β2) , 23 where ¯βi is as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='20, and where we fix the domain of α1α2 [of ¯β1 ¯β2] as the largest subset of X [finest partition on X] for which these expressions are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It is easy to check that domains and images of α1α2 and ¯β1 ¯β2 are given by the expressions from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It will follow from our results below that V (X) with this operation is an inverse monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Because for every partial bijection α on X, there is a compatible β, the projection of V (X) to its the first component is essentially the symmetric inverse monoid on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' On V (X), define a binary relation θ = ∆V (X) ∪ {((α, β1), (α, β2)) : α ∈ A(X), | dom α| ≤ 1, β1, β2 ∈ B(X)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Clearly, θ is an equivalence relation, and because {(α, β) : | dom α| ≤ 1} is an ideal of V (X), θ is compatible with the operation on V (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We set W(X) = V (X)/θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For [(α, β)]θ ∈ W(X) we will also use the short notation [α, β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let X be any set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For φ ∈ Inn(T (X)), let αφ, βφ be the bijectionss associated with φ by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then ϕ : Inn(T (X)) → W(X), given by ϕ(φ) = [(αφ, βφ)]θ is an embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In particular Inn(T (X)) is isomorphic to the substructure of W(X) generated by all elements of W(X) that can be represented as [(α, β)]θ such that dom α is a partial section of dom β, and all singleton parts of dom β intersect dom α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Our construction guarantees that ϕ is a homomorphism, provided it is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence let φ ∈ Inn(T (X)), and α, β be the bijections associated with φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Because dom α and im α are the maximal images of all transformations in dom φ and im φ, respectively, they are uniquely determined by φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For each i ∈ dom α, let ci be the constant function with image i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then ci ∈ dom φ, and ciφ = ciα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that α is uniquely determined by φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If | dom α| ≤ 1, then one θ-class contains (α, β) for all choices of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' So assume otherwise, say i, j ∈ dom α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let B ∈ dom β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Because dom φ contains the transformation tB that maps B to i and X \\ B to j, it follows that the parts of dom β are determined by all minimal kernel classes of transformations in dom φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence dom β is unique, and similarly, we see that im β is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Finally, because tBφ maps exactly Bβ to iα, we see that β itself is uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that ϕ is well-defined, and hence a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Moreover, for every t ∈ dom φ, and x ∈ X, we have (x)(tφ) = iα, where i ∈ I is the unique element in (([x]P ′)β−1)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Therefore tφ is uniquely determined by α, β, and hence ϕ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The final assertion follows from the description of the generators φg,h of Inn(T (X)) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='18, noting that in the case of | dom α| ≤ 1, we may always choose β = id{{X}}, in which case the representation [α, β] is as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For a complete classification, it remains to determine the image of the embedding ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will have to distinguish between finite and infinite X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In the following, by the term “generator”, we will mean an element of the form φg,hϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let X be infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then Inn(T (X)) is isomorphic to W(X), and the embedding ϕ from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='21 is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='21, it suffices to show that W(X) is indeed generated by all generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let I ⊆ X, and P be a partition X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Clearly, idI ≈ idP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We first show that [(idI, idP )]θ is in the image of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Chose a bijection σ : X → X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let P1 be the singleton partition on X, P ′ 1 = {({x}×X)σ−1 : x ∈ X}, and define α1 : X → (∆Y )σ−1, β1 : P1 → P ′ 1 by xα1 = (x, x)σ−1, {x}β1 = ({x} × X)σ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It is straightforward to check that [α1, β1] is a generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Next let α2 and β2 be the identities on {(x, x)σ−1 : x ∈ I} and P ′ 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Because P ′ 1 does not contain any singleton blocks, [α2, β2] is once again a generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let β3 be the identity on the partition P3 consisting of all sets of the form {(x, y), (y, x)}σ−1 for x, y ∈ X with [x]P = [y]P , and singletons otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Moreover, let I3 be the union of all singleton sets in P3 and α3 = idI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Once again, (α3, β3) is a generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 24 Finally, let α4 = α−1 1 , β4 = β−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We claim [(idI, idP )]θ = Π4 i=1[(αi, βi)]θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let x ∈ I, then xα1α2α3α4 = ((x, x)σ−1)α2α3α4 = ((x, x)σ−1)α3α4 = ((x, x)σ−1)α4 = x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If x /∈ I, then α2 is undefined at xα1 = ((x, x)σ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence α1α2α3α4 = idI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let B ∈ P, and C ⊆ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then C ¯β1 ¯β2 ¯β3 ¯β4 = ((C × X)σ−1)¯β2 ¯β3 ¯β4 = ((C × X)σ−1)¯β3 ¯β4 = ((B × X)σ−1)¯β4 = B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' From this it follows that the domain of ¯β1 ¯β2 ¯β3 ¯β4 is indeed P (as opposed to a refinement), and that ¯β1 ¯β2 ¯β3 ¯β4 acts as the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence [(idI, idP )]θ is in the image of ϕ, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For the general case, let [α, β]θ ∈ W(X) be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Construct [α′, β′] as follows: If Bi ∈ dom β intersects dom α, choose a partition PBi of Bi that contains exactly one element of dom α in each part, and let dom β′ be the union of the PBi, together with all B ∈ dom β not intersecting dom α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that dom β′ is a refinement of dom β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let im β′ be the refinement obtained from im β in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If B′ i ∈ dom β′ contains a (unique) element i ∈ dom α, then let B′ iβ′ = [iα]im β′, otherwise, set B′ iβ′ = B′ iβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If Bi ∈ dom β does not intersect dom α, choose an element bi ∈ Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let dom α′ be obtained from dom α by adjoining all the elements bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Similarly enlarge im α to im α′ by choosing one element from each Bi ∈ im β that does not intersect im α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now let xα′ be the unique element in im α′ ∩ [x]dom β′β′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then [α′, β′] is a generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since [iddom α, iddom β] ∈ im ϕ, this also holds for [iddom α, iddom β][α′, β′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' A straightforward check shows that this product is [α, β], and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let X be finite, and [α, β]θ ∈ W(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If | dom α| ≥ 2, then [α, β]θ ∈ im ϕ if and only if one of the following holds: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' dom α = X and dom β is the partition of X into singletons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' there exists B ∈ dom β with |B| ≥ 2, B ̸⊆ dom α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If | dom α| ≤ 1, then [α, β]θ ∈ im ϕ, unless |X| = 1 and dom α = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose first that | dom α| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If [α, β] satisfies condition 1, then it is a generator, and hence in the image of ϕ (in fact its preimage will be a unit of T (X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' So assume that there exists a set B ∈ dom β with |B| ≥ 2, B ̸⊆ dom α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let I = dom α, P = dom β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As in the infinite case, we first show that [(idI, idP )]θ is in the image of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Enumerate X as x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , xm, such that the parts of P correspond to consecutive index ranges in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , m}, with xm ∈ B \\ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will use three different types of generators to obtain [idI, idP ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For J ⊆ I \\ {xm}, let QJ be the partition with part J ∪ {xm}, and singletons otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If J = {xj}, we will just write Qxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We set kj = [idI\\{xm}, idQxj ], and lJ = [idI\\J, idQJ ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Moreover, let βj : Qj → Qj+1 be defined by {xj, xm}βj = {xj}, {xj+1}βj = {xj+1, xm}, and the identity otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Set sj = [idI\\{xm}, βj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It is easy to check that all kj, lJ, and sj are generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , Cr = B be the parts of P, in the order of their index ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For each Ci = {xdi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , xei}, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , r−1, let Ji = Ci \\I, and set pi = kdikdi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' keilJisei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For Cr = B = {xdr, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , xm}, let Jr = B\\I and set pr = kdrkdr+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' km−1lJr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We leave it up to the reader to confirm that [idI, idP ] = p1 · · · pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We now can show that im ϕ contains any [α, β] with dom α = I, dom β = P exactly as in the infinite case in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For the converse, suppose that a = [α, β]θ ∈ im ϕ, say a = g1 · · · gn for some generators gi = [αi, βi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If dom α = X, then by finiteness, dom αi = X for all i, and hence (as the gi are generators), dom βi is the partition into singletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' From this, we get that dom α = X and dom β is the partition of X into singletons, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let dom α ̸= X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We may assume that the number of generators n is the smallest possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If dom α1 = X, then it is easy to see that g1g2 is a generator as well (note that this requires finiteness, which forces g1ϕ−1 to be a unit of T (X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 25 Hence by minimality, dom α1 ̸= X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As g1 is a generator, it follows that dom β1 contains a set B′, |B′| ≥ 2 with B′ ̸⊆ dom α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' But then dom β contains a set B with B′ ⊆ B and dom α ∩ B′ ⊆ dom α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that B satisfies the criteria in condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If | dom α| = 1 then [α, β]θ = [α, id{X}]θ, which is a generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If | dom α| = 0 and |X| ̸= 1, then [α, β], which is the empty mapping, is the generator [∅, id{X}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Conversely, if |X| = 1, then Inn(T (X)) only contains the trivial full automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2 The partial inner automorphism monoid of a completely simple semigroup Every completely simple semigroup is isomorphic to a Rees matrix semigroup and hence we assume at the outset of this subsection that our semigroups have this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let Γ be a group, I and Λ two nonempty sets, and P a Λ × I matrix with entries in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let M(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' I, Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' P) be the Rees matrix semigroup induced by Γ, I, Λ and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let (G, g, γ), (H, h, η) ∈ M(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' I, Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then D(G,g,γ),(H,h,η) ̸= ∅ ⇐⇒ h = (pη,G g pγ,H)−1 and D(G,g,γ),(H,(pηG g pγ,H)−1,η) = {G} × Γ × {η}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Regarding the equivalence, we start by proving the direct implication and the second equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let (A, a, α) ∈ M(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' I, Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' P) such that (G, g, γ)(H, h, η)(A, a, α) = (A, a, α) = (A, a, α)(G, g, γ)(H, h, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then A = G and α = η so that D(G,g,γ),(H,h,η) ⊆ {G} × Γ × {η} and hence the two sets are equal (by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1(4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' This proves the last equality in the statement of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now, from (G, g, γ)(H, h, η)(G, a, η) = (G, a, η), we get g pγ,H h pη,G a = a, that is, h = (pη,G g pγ,H)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The direct implication is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For the converse implication, let h = (pη,G g pγ,H)−1 and (G, a, η) ∈ M(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' I, Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then (G, g, γ)(H, p−1 γ,Hg−1p−1 η,G, η)(G, a, η) = (G, a, η) and similarly (G, a, η)(G, g, γ)(H, p−1 γ,Hg−1p−1 η,G, η) = (G, a, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It is proved that D(G,g,γ),(H,h,η) ̸= ∅ and the lemma follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now we can state the main result of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let Γ be a group, I and Λ two nonempty sets, and P a Λ × I matrix with entries in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let M(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' I, Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' P) be the Rees matrix semigroup induced by Γ, I, Λ and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then the semigroup Inn(M(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' I, Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' P)) is generated by the following maps and corresponding inverses: φ(G,g,γ),(H,(pη,G g pγ,H)−1,η) : {G} × Γ × {η} → {H} × Γ × {γ} (G, a, η) �→ (H, (gpγ,H)−1 a (pη,G g), γ), for g ∈ Γ, G, H ∈ I and γ, η ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 26 4 Conjugacies ∼n, ∼tr, ∼∗ p, ∼o, and ∼c in finite partition monoids The partition monoid PX on a set X has the set of all partitions of X ∪ X′ as its underlying set, where X′ is a disjoint copy of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' These monoids originally arose in the study of partition algebras (see, for example, [32,47]) and subsequently attracted the attention of mathematicians working in semigroup theory (see, for example, [20,22,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' One reason for the attention is that PX contains some important semigroups as subsemigroups, such as T (X) and I(X) (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='5), as well as the symmetric group Sym(X) on X [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In this section, we will be interested in the finite partition monoid Pn on a set with n elements, and in the submonoids BPn and Bn of Pn, which are called partial Brauer monoids and Brauer monoids, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Our goal is to characterize the conjugacies ∼n, ∼tr, ∼p, ∼o, and ∼c in these monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (See §1 for the definitions of all these conjugacy relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=') From now on, we will identify an equivalence relation R on a set Y with the partition of Y induced by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It will always be clear from the context how we view R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Using the notation from [20], we let n = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , n} and n′ = {1′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , n′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Symbols x, y, z, , k, l, m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' will always refer to elements in n, and x′, y′, z′, k′, l′, m′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' to the corresponding elements in n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If A ⊆ n, then A′ = {x′ : x ∈ A} ⊆ n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As customary, we represent an element a ∈ Pn (a partition of n ∪ n′) as a simple graph with vertices 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , n in a row, vertices 1′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , n′ directly below, and edges drawn in such a way that the connected components of the graph correspond to the blocks of the partition a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Such a graph is not unique, so we identify two graphs that have the same connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For example, the graph 1 2 3 4 5 ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❧❧❧❧❧❧❧ represents the element a ∈ P5 whose blocks are: {1, 3}, {2, 4′}, {1′, 2′}, {3′, 4, 5}, {5′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For x ∈ n, [x]a will denote the block of a containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Similarly, we write [x′]a for the block containing x′ ∈ n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We multiply elements of Pn as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If a is as above and b is represented by the graph ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' then to obtain the product ab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' we first draw a over b: ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❧❧❧❧❧❧❧ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' then we glue two middle rows: ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❧❧❧❧❧❧❧ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' and finally we remove the middle row,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' keeping in the same block the elements of X ∪ X′ such that there is a path between these elements in the graph with the middle row: ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❢❢❢❢❢❢❢❢❢❢❢❢❢❢ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 27 (See [22, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=') Let a ∈ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Throughout this section, we will need the following definitions: ker a = {[x]a ∩ [n] : x ∈ [n]}, coker a = {[x′]a ∩ [n′] : x′ ∈ [n′]}, dom(a) = {x ∈ X : x belongs to a transversal block of a}, codom∧(a) = {x ∈ X : x′ belongs to a transversal block of a}, coker∧(a) = {A ⊆ [n] : A′ ∈ coker(a)}, rank(a) = the number of transversal blocks of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (We follow [19, §2] and [22, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2], with some changes in names and notation to make our arguments clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=') We will also need the restriction of ker(a) and coker∧(a) to dom(a) and codom∧(a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For a ∈ Pn, we define kert(a) = {A ∈ ker(a) : A ⊆ dom(a)} and cokert(a) = {B ∈ coker∧(a) : B ⊆ codom∧(a)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6) Note that for every A ∈ kert(a), there exists a unique B ∈ cokert(a) such that A ∪ B′ is a transversal block of a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' and that rank(a) = | kert(a)| = | cokert(a)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We now define the following subsets of Pn: BPn = {a ∈ Pn : each block of a has size at most 2}, Bn = {a ∈ Pn : each block of a has size 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The subsets BPn and Bn are submonoids of Pn [19, §2], called partial Brauer monoids and Brauer monoids, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1 Conjugacy ∼n in Pn, BPn, and Bn Let b ∈ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As in previous work on Pn, a special role is played by the equivalence relation ker(b)∨coker∧(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We say that b is connected if ker(b) ∨ coker∧(b) is the universal relation on {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let s be a block of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We say that s is transversal if s ∩ n ̸= ∅ and s ∩ n′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If b does not have any transversal blocks, it is called transversal free;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' if it has exactly one transversal block, it is called 1-transversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let A ⊆ n be not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For b ∈ Pn, we denote by bA the partition of A ∪ A′ (that is, an element of PA) with [x]bA = [x]b ∩ (A ∪ A′) and [x′]bA = [x′]b ∩ (A ∪ A′), for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We call bA the subpartition of b induced by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In this context, for a block s of b, we use the notation sA = s ∩ (A ∪ A′), and we agree that any such use is meant to imply that s is a block of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' A subpartition bA is called trivial if |A| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The definitions of bA being connected, transversal free, and 1-transversal are obtained by adjusting their definitions for b to the index set A in the obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Similarly we extend the definitions of ker, coker, ker∧, and coker∧ to bA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For the following results, it will be useful to represent an intermediate step in the calculation of a partition product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let n∗ = {1∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , n∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For partitions a, b ∈ Pn, we denote by (a, b)∗ the partition of the set n ∪ n∗ ∪ n′ that corresponds to the situation before the final deletion of the middle row, where n, n∗, n′ represent the top, middle, and bottom row, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' When a, b are represented by specific graphs, we represent (a, b)∗ as the graph obtained by identifying corresponding vertices in the lower row of a with those in the upper row of b, followed by the merging of all double edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Recall that we are identifying partitions with their corresponding equivalence relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For example we might write (x, y) ∈ b instead of y ∈ [x]b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let b ∈ Pn such that bA is connected and transversal-free, it contains blocks sA ⊆ A and tA ⊆ A′, and for every block rA /∈ {sA, tA}, rA = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Fix y ∈ A and define c ∈ Pn as follows: [y]c = (s \\ A) ∪ {y} and [y′]c = (t \\ A′) ∪ {y′};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 28 [x]c = {x} and [x′]c = {x′}, for all x ∈ A \\ {y};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' [x]c = [x]b if [x]b does not intersect A ∪ A′, and [x′]c = [x′]b if [x′]b does not intersect A ∪ A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then b ∼n c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Define g ∈ Pn by [x]g = [x]b for x ∈ A \\ s, [x]g = sA ∪ {y′} for x ∈ sA, [x′]g = {x′} for x′ ∈ A′ \\ {y′}, and [x]g = [x′]g = {x, x′} for x /∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Define h ∈ Pn by [x′]h = [x′]b for x ∈ A′ \\ t, [x′]h = tA ∪ {y} for x′ ∈ tA, [x]h = {x} for x ∈ A \\ {y}, and [x]h = [x′]h = {x, x′} for x /∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It is easy to see that (gh)A is obtained from bA by merging the upper block sA with the lower block tA, while outside of A∪A′, gh acts as the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence, since bA is connected, A∗ is included in a single block of (gh, b)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that y∗ ∈ A∗ and that, by the definition of g, (z, y∗) ∈ (gh, b)∗ for every z ∈ sA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We claim that ghb = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For any b-block other than s, it is straightforward to check that it is also a ghb-block (using the hypothesis that rA = r for every block rA ̸= sA, tA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Regarding the block s, select any z ∈ sA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We want to prove that [z]ghb = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let x ∈ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If x ∈ sA, then x ∈ [z]ghb since sA ⊆ [z]ghb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose x ∈ s \\ sA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then, (z, y∗), (y∗, z∗), and (z∗, x∗) are in ((gh), b)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since (x, x′) ∈ gh, we also have (x∗, x) ∈ (gh, b)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus, by the definition of the product in Pn, (z, x) ∈ ghb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Finally, let x′ ∈ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then, (z, y∗), (y∗, z∗), and (z∗, x′) are in (gh, b)∗, and so (z, x′) ∈ ghb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have proved that s ⊆ [z]ghb, and equality s = [z]ghb follows as all other blocks of b are also blocks of ghb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence ghb = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' A similar argument shows that b = bgh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We now have g(hbg) = (ghb)g = bg, h(b)g = hbg, and g(hbg)h = (gh)(bgh) = ghb = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus, hgb and b satisfy (i), (iii), and (iv), and so hbg ∼n b by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' A straightforward calculation now shows that hbg = c, and so b ∼n c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The following result is similar to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1, except that the blocks sA and tA are merged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let b ∈ Pn such that bA is connected, it has exactly one transversal block sA, and for every block rA ̸= sA, rA = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Fix y ∈ A and define c ∈ Pn as follows: [y]c = (s \\ (A ∪ A′)) ∪ {y, y′};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' [x]c = {x} and [x′]c = {x′}, for all x ∈ A \\ {y};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' [x]c = [x]b if [x]b does not intersect A ∪ A′, and [x′]c = [x′]b if [x′]b does not intersect A ∪ A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then b ∼n c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Define g ∈ Pn by [x]g = [x]b for x ∈ A \\ s, [x]g = (sA ∩ A) ∪ {y′} for x ∈ (sA ∩ A), [x′]g = {x′} for x ∈ A′ \\ {y′}, and [x]g = [x′]g = {x, x′} for x /∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Define h ∈ Pn by [x′]h = [x′]b for x ∈ A′ \\ s, [x′]h = (sA ∩ A′) ∪ {y} for x′ ∈ (sA ∩ A′), [x]h = {x} for x ∈ A \\ {y}, and [x]h = [x′]h = {x, x′} for x /∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then, as in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1, we can show that b = ghb = bgh and c = hbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence b ∼n c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let b ∈ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We say that b is in n-normal form if the following conditions hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' in every non-trivial, connected, transversal-free subpartition bA of b, there exist distinct blocks sA, tA with sA ̸= s and tA ̸= t, such that either sA, tA ⊆ A or sA, tA ⊆ A′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' in every non-trivial, connected, 1-transversal subpartition bA of b, with transversal sA, there exists a block tA ̸= sA such that t ̸= tA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Applying Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2 to non-trivial connected sets A will result in a partition with an increased number of singleton blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that this process must stop, and hence every n-conjugacy class contains an element in normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will next show that in each n-conjugacy class, any partitions a and b in normal form can be obtained from each other by a permutation of the underlying set n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 29 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a, p ∈ Pn such that ap = pa = a and p is an idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose that there are k, l ∈ n with (k, l′) ∈ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then (k, k∗) ∈ (p, a)∗ and (l∗, l′) ∈ (a, p)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose that p is represented by the simple graph with the largest possible number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since p = p2, (k, l′) is in pp, and hence it is also in (p, p)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since (k, l′) ∈ p, we have (l′, k∗) ∈ (p, p)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence (k, k∗) ∈ (p, p)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let k − · · · − k∗ be a shortest path from k to k∗ in the graph representing (p, p)∗, as obtained from the maximal graph representing p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose to the contrary that this path contains a vertex j′ ∈ A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then, the path has a subpath i∗ 1 − j′ 1 − · · · − j′ t − i∗ 2, where t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' But t must be 1 since j′ 1 − i∗ 2 (by the fact that p is represented by the graph with the largest number of edges) and k − · · · − k∗ is a shortest path from k to k∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We then have i∗ 1 − j′ 1 − i∗ 2, which implies (i1, j′ 1), (j′ 1, i2) ∈ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence (i1, i2) ∈ p, and so (i∗ 1, i∗ 2) ∈ (p, p)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' This a contradiction since we can replace i∗ 1 − j′ 1 − i∗ 2 with i∗ 1 − i∗ 2 obtaining a shorter path from k to k∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now, let a also be represented by the graph with the maximal number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then because a = pa, every edge in the graph for (p, p)∗ with no vertex from A′ is also an edge in the graph for (p, a)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus, the path k − · · · − k∗ above is also a path in the graph for (p, a)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence (k, k∗) ∈ (p, a)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Dually, we obtain (l∗, l′) ∈ (a, p)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a, p ∈ Pn such that pa = ap = a and p is an idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let A be a non-empty subset of n such that aA is connected, ker(aA) = ker(pA), and coker(aA) = coker(pA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then: (1) there is at most one a-block s intersecting A such that s is transversal or s is not a block of p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (2) there is at most one a-block v intersecting A′ such that v is transversal or v is not a block of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since aA is connected and coker(pA) = coker(aA), the set A∗ is included in a single block of (p, a)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose to the contrary that (1) is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then there are three possible cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' There are distinct transversal a-blocks s and t intersecting A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We then have g, k′ ∈ s and h, l′ ∈ t, where g, h ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus (g∗, k′), (h∗, l′) ∈ (p, a)∗, and so [k′](p,a)∗ = [l′](p,a)∗ (as A∗ lies within one block).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that (k′, l′) ∈ pa, and so (k′, l′) ∈ a since pa = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' This is a contradiction since s ̸= t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' There are a-blocks s and t intersecting A such that s is transversal, t is not transversal, and t is not a p-block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As in Case 1, we have g, k′ ∈ s, where g ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Select h ∈ t ∩ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now, [h]p needs to be a transversal block, for otherwise [h]p = [h]pa = [h]a = t and t is not a p-block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='5, (h, h∗) ∈ (p, a)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We now have (g∗, k′), (h∗, h) ∈ (p, a)∗, which implies (h, k′) ∈ pa, and so (h, k′) ∈ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' This is a contradiction since t is not transversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' There are distinct non-transversal a-blocks s and t intersecting A that are not p-blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Select g ∈ s∩A and h ∈ t∩A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As in Case 2, we obtain (g, g∗), (h, h∗) ∈ (p, a)∗, leading to the contradiction (g, h) ∈ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have proved (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Statement (2) follows by a dual argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The following result is crucial for proving our characterization of ∼n in Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a ∈ Pn be in normal form, and let p ∈ Pn be such that pa = a = ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then the kernel and cokernel of p consist of singletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose, by way of contradiction, that the conclusion is false, that is, there are distinct k, l ∈ n such that (k, l) ∈ p or (k′, l′) ∈ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By replacing p with its idempotent power, we may assume that p is an idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose (k, l) ∈ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then, since pa = a, we have (k, l) ∈ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since a is in normal form, it follows that (k′, l′) /∈ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus, (k′, l′) /∈ p since ap = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that ker(a{k,l}) = ker(p{k,l}) and coker(a{k,l}) = coker(p{k,l}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By a dual argument, these equalities also hold if (k′, l′) ∈ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 30 Let A be a subset of n of maximum size such that aA is connected and it satisfies ker(aA) = ker(pA), coker(aA) = coker(pA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have |A| ≥ |{k, l}| = 2, so aA is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6, aA has at most one transversal block, there exists at most one a-block s intersecting A such that s is transversal or s is not a block of p, and there exists at most one a-block v intersecting A′ such that v is transversal or v is not a block of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Consider the set H = {h ∈ n \\ A : [h]a ∩ A ̸= ∅, [h]a ̸= s} (here and in the following, we ignore conditions of the form [h]a ̸= s if no exceptional block s exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We claim that for each h ∈ H, there exists lh ∈ A such that (h′, l′ h) ∈ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For h ∈ H, let t = [h]a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then t intersects A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since t ̸= s, t is also a block of p, and hence ker(aA∪{h}) = ker(pA∪{h}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Moreover, aA∪{h} is connected, and hence by the maximality of the size of A, we conclude that coker(aA∪{h}) ̸= coker(pA∪{h}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' This implies that there is an lh ∈ A such that (l′ h, h′) ∈ a, (l′ h, h′) /∈ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (Note that coker(pA∪{h}) ⊆ coker(aA∪{h}) since ap = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=') Consider the set B = {x ∈ n ∩ s : [x′]a ∩ A′ ̸= ∅} ∪ � {u : u is an a-block with u ∩ A ̸= ∅, u ̸= s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (If no exceptional block s exists, interpret the first set as ∅, and ignore the condition u ̸= s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By the definition of B, we have A ⊆ B (so aB is not trivial), aB is connected, and every a-block intersecting B also intersects A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6, s is the only a-block intersecting B such that s is transversal or s is not a block of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In particular, aB has at most one transversal block, which, if it exists, equals sB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Moreover, every a-block intersecting B′ also intersects A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Indeed, let r be an a-block intersecting B′, say g′ is in the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If g lies in the first set from the definition of B, then r intersects A′ by the definition of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose g ∈ u, where u is an a-block included in the second set of the definition of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If g ∈ A, then g′ ∈ r ∩ A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Otherwise, g ∈ u \\ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since u ̸= s and u ∩ A ̸= ∅, g ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence (l′ g, g′) ∈ a, with l′ g ∈ A′, and so r intersects A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6 and the fact that every a-block intersecting B′ also intersects A′, v, if it exists, is the only a-block intersecting B′ such that v is transversal or v is not a block of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose aB has a transversal block, which must be equal to both sB and vB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then s = v and, since a is normal, there is an a-block w such that w ̸= s (so w ̸= v), w intersects B ∪ B′, and w ̸= wB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The block w cannot intersect B (by the definition of B), so it intersects B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose aB is transversal free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then we have either two distinct a-blocks intersecting B and extending beyond B ∪ B′, or two blocks intersecting B′ and extending beyond B ∪ B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The former is not possible, because only s can extend beyond B ∪ B′ (by the definition of B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In the second case, one of these blocks, say w, must differ from v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In either case, we have an a-block w such that w ̸= v, w intersects B′, and w ̸= wB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since v is the only a-block intersecting B′ such that v is transversal or v is not a block of p, w ⊆ n′ and w is a block of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since w ̸= wB, there is m′ ∈ w \\ B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Consider the set A ∪ {m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Because w is also a block of p and it intersects A′, we have coker(aA∪{m}) = coker(pA∪{m}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus, by the maximality of the size of A, ker(aA∪{m}) ̸= ker(pA∪{m}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' However, our construction of B shows that [m]a does not intersect B, and hence it does not intersect A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Because pa = a, this also holds for [m]p, which implies ker(aA∪{m}) = ker(pA∪{m}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' This is a contradiction, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let Sn be the symmetric group of permutations on n = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then Sn acts on Pn by aσ (a ∈ Pn, σ ∈ Sn), where aσ is obtained by replacing x by xσ and y′ by (yσ)′ in each block of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For example, if a = {{1, 3}, {2, 4′}, {1′, 2′}, {3′, 4, 5}, {5′}} ∈ P5 and σ = (1 2 5)(3 4) ∈ S5, then aσ = {{2, 4}, {5, 3′}, {2′, 5′}, {4′, 3, 1}, {1′}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For σ ∈ Sn, define λσ = {{x, (xσ)′} : x ∈ n} ∈ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then Sn = {λσ ∈ Pn : σ ∈ Sn} is the group of units of Pn, which is isomorphic to Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The mapping σ → λσ is an isomorphism for Sn to Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that for all a ∈ Pn and σ ∈ Sn, aσ = λ−1 σ aλσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We can now characterize the natural conjugacy ∼n in Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In the partition monoid Pn, every n-conjugacy class contains an element in normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Moreover, if a, b ∈ Pn are in normal form, then a ∼n b if and only if b = aσ for some permutation σ ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 31 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The first statement follows by repeated applications of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' To simplify the notation in the proof of the second statement, we will identify any σ ∈ Sn with λσ ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In particular, when we write σ−1aσ, where a ∈ Pn, we will mean λ−1 σ aλσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a, b ∈ Pn be in normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It is clear that if b = aσ for some σ ∈ Sn, then a ∼n b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For the converse, suppose that a ∼n b and let g, h ∈ Pn be conjugators (elements from the definition of ∼n) for a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let g1 = (gh)ig, where i ≥ 0 is an integer such that g1h is an idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It is straightforward to check that g1 and h are also conjugators for a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now, let h1 = (hg1)jh, where j ≥ 0 is an integer such that h1g1 is an idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Again, we can check that g1 and h1 are conjugators for a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By a routine calculation, we can show that g1h1 is also an idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Therefore, we may assume that gh and hg are idempotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='7, the kernel and cokernel of gh and of hg both consist of singletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that the same statement holds for g and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence, for every x ∈ n, [x]g = {x, y′} or [x]g = {x}, and [x′]g = {x′, y} or [x′]g = {x′}, for some y ∈ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The same statement is true for h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since gh is an idempotent, for every x ∈ n, either [x]gh = {x, x′} or [x]gh = {x} and [x′]gh = {x′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The same statement is true for hg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Define σ : n → n by xσ = � y if [x]g = {x, y′} or [x′]h = {x′, y}, x if [x]g = {x} and [x′]h = {x′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By the properties of g, h, gh, and hg stated above, σ is well defined and σ ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By the definition of σ, we have g ⊆ σ and h ⊆ σ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' To conclude the proof, it suffices to show that σbσ−1 = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since g ⊆ σ and h ⊆ σ−1, we have a = gbh ⊆ σbσ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For the reverse inclusion, let x ∈ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will prove that [x]σbσ−1 ⊆ [x]a and [x′]σbσ−1 ⊆ [x′]a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose z ∈ [x]σbσ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If z = x, then z ∈ [x]a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose z ̸= x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then, z ∈ [x]σbσ−1 can only happen when xσ = y1, (y1, y2) ∈ b, and zσ = y2, for some y1, y2 ∈ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that y1 ̸= y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have [y1]hg = {y1, y′ 1} or [y1]hg = {y1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The latter is impossible since we would have [y1]hgb = {y1}, but hgb = b and y2 ∈ [y1]b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus [y1]hg = {y1, y′ 1}, so there is l ∈ n such that (y1, l′) ∈ h and (l, y′ 1) ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence lσ = y1, which implies l = x (since xσ = y1), and so (x, y′ 1) ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By symmetry, (z, y′ 2) ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We now have (x, y′ 1) ∈ g, (y1, y2) ∈ b, and (z, y′ 2) ∈ g, which implies z ∈ [x]gbh, and so z ∈ [x]a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose z′ ∈ [x]σbσ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then, xσ = y, (y, k′) ∈ b, and kσ−1 = z (that is, zσ = k), for some y, k ∈ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have [y]hg = {y, y′} or [y]hg = {y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The latter is impossible since we would have [y]hgb = {y}, but hgb = b and k′ ∈ [y]b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus [y]hg = {y, y′}, so there is l ∈ n such that (y, l′) ∈ h and (l, y′) ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence lσ = y, which implies l = x (since xσ = y), and so (x, y′) ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Further, we have [k′]hg = {k, k′} or [k′]hg = {k′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The latter is impossible since we would have [k′]bhg = {k′}, but bhg = b and y ∈ [k′]b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus [k′]hg = {k, k′} = [k]hg, so there is m ∈ n such that (k, m′) ∈ h and (m, k′) ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence mσ = k, which implies m = z (since zσ = k), and so (k, z′) ∈ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We now have (x, y′) ∈ g, (y, k′) ∈ b, and (k, z′) ∈ h, which implies z′ ∈ [x]gbh, and so z′ ∈ [x]a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have proved that [x]σbσ−1 ⊆ [x]a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By a dual argument, we obtain [x′]σbσ−1 ⊆ [x′]a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that σbσ−1 = a, and so b = σ−1aσ, that is, b = aσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We next prove some consequences of our classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Recall that ∼n⊆ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In Pn, the D-classes correspond to partitions of the same rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The following characterizes ∼n on partitions of small rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In Pn the partitions of rank 0 form one ∼n-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Clearly, the singleton partition is in ∼n-normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We claim that it is the only such partition of rank 0 If b is any other rank 0 partition, it contains a non-trivial connected subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Consider a maximal such subset A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then any block B in bA must be a block of b for otherwise b would have to be a transversal by the maximality of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' However, this is impossible as b has rank 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The set B now witnesses that b is not in normal form, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In Pn, the partitions of rank 1 form two 2 distinct ∼n-classes, if n ≥ 2, and of a single ∼n-class, if n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 32 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Consider the set T of paritions bx,y′ that contain a single 2-element transversal {x, y′} and consists of singletons otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Clearly the elements of T are ∼n-normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='8 the elements of T lie in two different ∼n-classes depending on whether x = y or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If b is any other rank 1 transformation, it contains a non-trivial connected subset, and hence a maximal such subset A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Similar to Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='9 we see that bA can contain at most one block that is not a block of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Moreover, this must be the transversal block of bA, if one is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that A witnesses that b is not in normal form, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The result for n = 1 is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We remark that the classes of the corollary can be characterized by the existence or absence of a 1- transversal connected subpartition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As n → ∞, the number of ∼n-classes of Pn consisting of rank 2 partitions is not bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In Pn, consider all partitions consisting of singletons and a subpartition from the following list and its infinite generalization: It is straightforward to check that all such partitions are in normal form, and pairwise non-conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The above results explains why it is likely not possible to give a more explicit description of the ∼n-classes of Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If d ≥ 2, we can construct increasingly complex connected, ∼n-normal, and non-conjugate partitions with rank d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For checking practical examples, our results imply which connected subpartitions A of a given size can appear in an ∼n-normal partition (together with information about which blocks t satisfy tA ̸= t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Without proof, all such subpartitions of size 2 and 3 are given below, up to vertical and horizontal permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For this list only, a pointed arrow indicates that the corresponding block t satisfies tA ̸= t, while the absence of such an arrow allows both tA = t and tA ̸= t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' � � � � � � � � � 33 � � � We now extend our results to the Brauer monoid Bn and the partial Brauer monoid BPn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' When it is necessary for distinction, we write ∼nP , ∼nB and ∼nP B for the natural conjugacy relation in Pn, Bn and BPn, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Similarly, we will use expression such as “nP B-normal form”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Clearly, ∼nB⊆∼nP B⊆∼nP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It is straightforward to check that in Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2, if b ∈ BPn, so are the conjugators g, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As conjugation by a unit is identical in BPn and Pn, it follows that two partitions are in BPn are conjugate if and only if they are conjugate in Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We are moreover able to give a simpler description of our normal form in the case of BPn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let b ∈ BPn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We say that b is in n-normal form if the following conditions hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If {x, y} is a block, then x′ and y′ lie in transversal blocks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If {x′, y′} is a block, then x and y lie in transversal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In the partial Brauer monoid BPn, every n-conjugacy class contains an element in normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Moreover, if a, b ∈ Pn are in normal form, then a ∼n b if and only if b = aσ for some permutation σ ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By the above considerations, it suffices to show that an element b ∈ BPn is in ∼nP B-normal form if and only if it is in ∼nP -normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose that b is in ∼nP B-normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then any non-trivial connected subset A has size 2, is transversal- free, and one of the 2 conditions from Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='12 hold on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that b is in ∼nP -normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Conversely, let b be in ∼nP -normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose that {x, y} is a block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By normality, x′ and y′ lie in distinct non-singleton b-blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose one, say x′, does not lie in a transversal block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then there is a z ̸= z, y such that {x′, z′} is a block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Consider B = {x, y, z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have that B is connected and non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If {y, z′} is a b-block, then b would violate the second condition of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3, for a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' However, if {y, z′} is not a block, then bB is transversal free, and it is not possible to satisfy the first condition of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By contraction, both x′ and y′ lie in transversal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If {x′, y′} is a block, then a dual argument shows that x and y lie in transversal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We now turn to the Brauer monoid Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Unlike in the previous case, we need a modified version of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let b ∈ Bn such that bA is connected with |A| = 3, say A = {x, y, z} with blocks {x, y} and {y′, z′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If {x′, z} is not a block, then b ∼n c, where c contains the blocks {x, y}, {x′, y′}, [z]b, ([x′]b ∪ z) \\ {x′} as well as all b-blocks not intersecting A ∪ A′ ∪ [z]b ∪ [x′]b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If {x′, z} is a block, then b ∼n c, where c contains the blocks {x, y}, {x′, y′}, {z, z′} as well as all b-blocks not intersecting A ∪ A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Define g ∈ Bn with blocks {x, y}, {z, z′}, {x′, y′} and {w, w′} for all w ̸∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' define h ∈ Bn with blocks {x, y}, {z, x′}, {y′, z′} and {w, w′} for all w ̸∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In either of the above cases, it is straightforward to check that g, h witness b ∼n c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 34 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let b ∈ Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We say that b is in n-normal form if the following conditions hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If {x, y} is a block, then either {x′, y′} is a block, or x′ and y′ lie in transversal blocks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If {x′, y′} is a block, then either {x, y} is a block, or x and y lie in transversal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In the Brauer monoid Bn, every n-conjugacy class contains an element in normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Moreover, if a, b ∈ Pn are in normal form, then a ∼n b if and only if b = aσ for some permutation σ ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let b ∈ Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If B is a connected subset of b with |B| ≥ 3, then there is a connected set A ⊆ B that satisfies the conditions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Any application of the lemma will increase the number of maximal connected subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence, after repeated application of the lemma we reach a conjugate c of b that only contains connected subsets of size at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' This is equivalent to c being in normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Assume now that b ∼nB c with b, c in nB-normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then b ∼nP c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let b∗, c∗ be some nP -normal forms of b, c that are obtained by repeated application of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='8, b∗ = λωc∗λ−1 ω for some permutation ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By replacing c with cω we may assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' that b∗ = c∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Because b, c are in ∼nB-normal form, the only non-trivial applications of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2 to b, c involve Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1 on a connected set A = {x, y} with blocks {x, y} and {x′, y′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The same also holds for the outcome of such an application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that b∗, c∗ are obtained from b, c by replacing all blocks in such subpartitions with singletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let D ⊆ n be the largest set for which b∗ D = c∗ D consist of singleton blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then |D| is even, and there are two partition Db i, Dc j of D into blocks of size two such that bDb i , cDc j consist of two non-transversal blocks each, for all i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In addition, on the complement ¯D = n \\ D, we have that b ¯ D = b∗¯ D = c∗¯ D = c ¯ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The result now follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2 Conjugacy ∼tr in Pn, BPn, and Bn To characterize trace conjugacy ∼tr (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='8)) in Pn, we first need to describe the group elements of Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let S be any semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The maximal subgroups of S are the H-classes He of S such that e is an idempotent [15, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' An element a ∈ S is a group element of S if a ∈ He for some idempotent e ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' These element are also called completely regular, as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a, b ∈ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then: (1) a R b ⇐⇒ ker(a) = ker(b) and kert(a) = kert(b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (2) a L b ⇐⇒ coker(a) = coker(b) and cokert(a) = cokert(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By [22, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2], (1) and (2) are true if kert and cokert are replaced by dom and codom∧, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If ker(a) = ker(b), then dom(a) = dom(b) ⇐⇒ kert(a) = kert(b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' and if coker(a) = coker(b), then codom∧(a) = codom∧(b) ⇐⇒ cokert(a) = cokert(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We also have a D b ⇐⇒ rank(a) = rank(b), and D = J [22, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For equivalence relations ρ1 and ρ2 on X, the join ρ1 ∨ρ2 of ρ1 and ρ2 is the smallest equivalence relation containing the union ρ1 ∪ ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' To describe the group elements of Pn, we will need the join ker(a) ∨ coker∧(a), where a ∈ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' First, the idempotents of Pn were described in [19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let e ∈ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then, e is an idempotent if and and only if the following two conditions are satisfied: (1) for every transversal block A ∪ B′ of e, there exists a block P (necessarily unique) of ker(e) ∨ coker∧(e) such that A ∪ B′ ⊆ P ∪ P ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (2) for every block P of ker(e) ∨ coker∧(e), P ∪ P ′ contains at most one transversal block of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 35 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a ∈ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then, a is an element of a group H-class of Pn if and only if for every block P of ker(a) ∨ coker∧(a) one of the following conditions holds: (a) neither P nor P ′ intersects a transversal block of a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' or (b) each of P and P ′ intersects exactly one (not necessarily the same) transversal block of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose that a is an element of a group H-class H of Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let e be the identity of H, so a H e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='17, ker(a) ∨ coker∧(a) = ker(e) ∨ coker∧(e), kert(a) = kert(e), and cokert(a) = cokert(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let P be a block of ker(a) ∨ coker∧(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose that P does not intersect any transversal block of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose to the contrary that P ′ intersects some transversal block A∪B′ of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then B′ ⊆ P ′ and B′ ∈ cokert(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since cokert(a) = cokert(e), it follows by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='18 that there is C ∈ kert(e) such that C ∪ B′ ⊆ P ∪ P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since kert(e) = kert(a) and C ⊆ P, the block P intersects some transversal block of a, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have proved that if P does not intersect any transversal block of a, then (a) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Similarly, (a) holds if P ′ does not intersect any transversal block of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose (a) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then P intersects some transversal block A ∪ B′ of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If it also intersected another transversal block of a, say C ∪ D′, then we would have A, C ∈ ker(e), A, C ⊆ P, and A ̸= C, which would contradict Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='18(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' A similar argument can be applied to P ′, which implies that (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Conversely, suppose that for every block P of ker(a)∨coker∧(a), (a) or (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let k(a) be the number of blocks P such that P intersects a transversal block A ∪ B′ of a, and P ′ intersects a different transversal block C ∪ D′ of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If k(a) = 0, then a is an idempotent (and so a group element) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let k(a) ≥ 1 and consider P, A ∪ B′, and C ∪ D′ as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then, A ⊆ P, D′ ⊆ P ′, B′ ⊆ Q′, and C ⊆ R, where Q and R are blocks of ker(a) ∨ coker∧(a) such that P /∈ {Q, R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Construct a1 ∈ Pn by replacing in a the transversal blocks A∪B′ and C ∪D′ by A∪D′ and C ∪B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then k(a1) < k(a) (since P and P ′ both intersect the same transversal block of a1, namely A∪D′), and it is straightforward to check, using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='17, that a H a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Applying this construction repeatedly, we obtain (after at most k(a) steps) an element e ∈ Pn such that k(e) = 0 (so e is an idempotent) and a H e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence a is a group element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let σ ∈ Sm, where Sm is the symmetric group of permutations on [m] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We allow m to be zero, in which case [m] = ∅, Sm = {∅}, and σ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The cycle type of σ is the sequence (k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , km), where ki is the number of cycles of length i in the cycle-decomposition of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If m = 0, then we define the cycle type of σ as (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a ∈ Pn be a group element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='19, for every block P of ker(a)∨coker∧(a), either P does not intersect any transversal block of a or there is a unique A ∈ kert(a) such that A ⊆ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let {P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , Pm} be the set of all blocks of ker(a) ∨ coker∧(a) that intersect some transversal block of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For each i ∈ [m], let Ai be a unique element of kert(a) such that Ai ⊆ Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that kert(a) = {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , Am}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='19 again, each P ′ i contains a unique B′ i ∈ cokert(a) and cokert(a) = {B′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , B′ m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that m can be 0, which happens when kert(a) = cokert(a) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Define τa : [m] → [m] by iτa = j ⇐⇒ Ai ∪ B′ j is a transversal block of a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='19, τa ∈ Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We define the cycle type of a to be the cycle type of τa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that τa depends on the ordering of {P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , Pm}, but the cycle type of τa is the same regardless of an ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let e be the idempotent in the group H-class of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then the transitive blocks of e are A1∪B′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , Am∪B′ m, and the transitive blocks of a are A1 ∪ B′ 1τa, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , Am ∪ B′ mτa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let e, f, g, h ∈ Pn such that e and f are idempotents, gh = e, hg = f, ghg = g, and hgh = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then kert(g) = kert(e) and cokert(g) = cokert(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have g R e (since gh = e and eg = ghg = g) and g L f (since hg = f and gf = ghg = g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='17, kert(g) = kert(e) and cokert(g) = cokert(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 36 We can now characterize the trace conjugacy ∼tr in Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a, b ∈ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then a ∼tr b if and only if aω+1 and bω+1 have the same cycle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let e = aω, f = bω, u = aω+1, and v = bω+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose that a ∼tr b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='8), there exist g, h ∈ Pn such that ghg = g, hgh = h, gh = e, hg = f, and hug = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We also have gvh = ghugh = eue = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='21 and the fact that u H e and v H f, we have kert(g) = kert(e) = kert(u), cokert(g) = cokert(f) = cokert(v), kert(h) = kert(f) = kert(v), and cokert(h) = cokert(e) = cokert(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let m = | kert(e)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then, by the above equations, | kert(f)| = | kert(u)| = | kert(v)| = | kert(g)| = | kert(h)| = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let {P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , Pm} be the set of all blocks of ker(e) ∨ coker∧(e) that intersect some transversal block of e, and let {Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , Qm} be the set of all blocks of ker(f) ∨ coker∧(f) that intersect some transversal block of f (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (We have the same m since | kert(e)| = | kert(f)| = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=') Since e and f are idempotents, the transversal blocks of e and of f are, respectively, Ai ∪ B′ i with Ai ⊆ Pi and B′ i ⊆ P ′ i , and Ci ∪ D′ i with Ci ⊆ Qi and D′ i ⊆ Q′ i, where i ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since u ∈ He and v ∈ Hf, the transversal blocks of u and of v are, respectively, Ai ∪ B′ iτu and Ci ∪ D′ iτv, where i ∈ [m] (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since kert(g) = kert(e) and cokert(g) = cokert(f), there is σ ∈ Sm such that the transversal blocks of g are Ai ∪ D′ iσ, where i ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Finally, since kert(h) = kert(f) and cokert(h) = cokert(e), there is δ ∈ Sm such that the transversal blocks of h are Ci ∪ B′ iδ, where i ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We claim that σ = δ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let i ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since Ai ∪ D′ iσ is a block of g and Ciσ ∪ B′ i(σδ) is a block of h, we conclude that Ai ∪ B′ i(σδ) is a block of gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Further, e = gh and Ai ∪ B′ i is a block of e, which implies i(σδ) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence σ = δ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Our second claim is that στuδ = τv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let i ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since Ai ∪ D′ iσ is a block of g and Ciσ ∪ D′ i(στv) is a block of v, we conclude that Ai ∪ D′ i(στv) is a block of gv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus, since Ci(στv) ∪ B′ i(στvδ) is a block of h, it follows that Ai ∪ B′ i(στvδ) is a block of gvh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' But, gvh = u and Ai ∪ B′ iτu is a block of u, which implies i(στvδ) = iτu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence στuδ = τv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus, δ−1τuδ = τv, and so τu and τv are group conjugate in Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence, τu and τv have the same cycle type, and so aω+1 (= u) and bω+1 (= v) have the same cycle type (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Conversely, suppose that aω+1 and bω+1 have the same cycle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then τu and τv are group conjugate in Sm, that is, there are σ, δ ∈ Sm such that σ = δ−1 and στuδ = τv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' With the notation for the transversal blocks of e, f, u, and v as in the first part of the proof, let g ∈ Pn be such that ker(g) = ker(e) (= ker(u)), coker(g) = coker(f) (= coker(v)), and the transversal blocks of g are Ai ∪ Diσ, where i ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Similarly, let h ∈ Pn be such that ker(h) = ker(f) (= ker(v)), coker(h) = coker(e) (= coker(u)), and the transversal blocks of h are Ci ∪Biδ, where i ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Simple calculations (similar to the ones in the first part of the proof) show that ghg = g, hgh = h, gh = e, hg = f, and hug = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence a ∼tr b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Turning to BPn and Bn, it is clear that ∼trB⊆∼trP B⊆∼trP , and hence for two ∼tr-conjugate partitions a, b ∈ BPn or Bn, aω+1 and bω+1 have the same cycle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Conversely, if a, b are two such partitions in BPn [in Bn], it is straightforward to check that the conjugators g, h constructed in the second part of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='22 lie in BPn [in Bn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence we obtain the following characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a, b ∈ Pn or a, b ∈ Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then a ∼tr b if and only if aω+1 and bω+1 have the same cycle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3 Conjugacy ∼∗ p in Pn, BPn, and Bn In any epigroup, ∼∗ p ⊆ ∼tr [4, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The reverse inclusion is not true in the class of epigroups [4, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The goal of this subsection is to show that in Pn, ∼∗ p = ∼tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (See (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4) for the definitions of ∼p and ∼∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=') 37 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a ∈ Pn, and s ⊆ n a non-transversal a-block, such that s′ intersects one (or more) transversal a-blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then a has a ∼p-conjugate c ∈ Pn such that cs is transversal free, and such that c has more blocks than a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let u ∈ Pn have the blocks s, {z′}, where z ∈ s, and {k, k′}, where k /∈ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By straightforward calculations, we check that ua = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The partition c = au has blocks t \\ s′, for every a-block t satisfying t ̸⊆ s′, and {z′} for z ∈ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Clearly cs is transversal-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As we assumed that at least one transversal a-block intersects s′, c has more blocks than a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Clearly, a dual result holds if s′ is a non-transversal block such that s intersects a transversal block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a ∈ Pn, s an a-block, A = s ∩ n, such that A′ intersect two different a-blocks t1, t2 (one of which might be s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then a ∼p c, where c is obtained from a by merging the blocks t1, t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let x, y ∈ A, with x′ ∈ t1, y′ ∈ t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let v ∈ Pn have the blocks {x, y, x′, y′} and {z, z′}, where z /∈ {x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By straightforward calculations, we check that va = a and that av has the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Once again, clearly the dual version of the Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='25 holds as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a ∈ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then, there exists a group element c ∈ Pn such that a ∼∗ p c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We recursively apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='24 [or its dual] to a, as long as we find a non-transversal block s [resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' s′] such that s′ [resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' s] intersects a transversal nlocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Because the number of blocks increases at each step, this process must stop with a partition b ∼∗ p a for which dom(b) = codom∧(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We next apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='25 (or its dual) to all cases in which the involved blocks t1, t2 are transversal (note that this means that s is also transversal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Each such application will preserve the condition dom(·) = codom∧(·), as only transversal blocks will be merged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As this decreases the number of blocks, this process will stop with an element c ∼∗ p b ∼∗ p a such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' dom(c) = codom∧(c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' if s is a transversal c-block, A = s ∩ n, then A′ intersects at most one transversal c-block;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' if s is a transversal c-block, A′ = s ∩ n′, then A intersects at most one transversal c-block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will show that these conditions imply that c is a group element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let P be a block of ker(c) ∨ coker∧(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If P does not intersect any transversal block of c, then, by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=', neither does P ′ (and vice versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose that s = A∪B′ is a transversal c-block, and let P and Q be the blocks of ker(c)∨coker∧(c) such that A ⊆ P and B′ ⊆ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We claim that s = P ∪Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=', any block intersected by A′ must be transversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus, by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=', there exists a transversal c-block t such that A′ ⊆ C′, where C′ = t ∩ n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Applying the dual argument to C′ and using 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=', we obtain a transversal c-block w such that C ⊆ D, where D = w ∩ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since A′ ⊆ C′, we have A ⊆ C ⊆ D, so A ⊆ s ∩ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus, s = w, A = C = D, and A′ = C′ = D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will now prove that A = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let x ∈ P and select any y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since A ⊆ P, we have (y, x) ∈ ker(c) ∨ coker∧(c), and so there are y = z0, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , zk = x in n such that for every i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , k − 1}, either (zi, zi+1) ∈ ker(c) or (zi, zi+1) ∈ coker∧(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=', k − 1} and suppose that zi ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If (zi, zi+1) ∈ ker(c), then zi+1 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose (zi, zi+1) ∈ coker∧(c), that is, (z′ i, z′ i+1) ∈ coker(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then x′ i ∈ C′ (since A′ = C′), and so x′ i+1 ∈ C′ (since C′ ⊆ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus zi+1 ∈ C, and so zi+1 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since y = z0 ∈ A, it follows that x = zk ∈ A, and so P = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By a dual argument, B′ = Q′, and so s = P ∪ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence, c is a group element by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In Pn, ∼∗ p = ∼tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' That is, for a, b ∈ Pn, a ∼∗ p b if and only if aω+1 and bω+1 have the same cycle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a, b ∈ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose that a ∼tr b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='26, there are group elements c and d of Pn such that a ∼∗ p c and b ∼∗ p d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since ∼∗ p ⊆ ∼tr, we have c ∼tr a ∼tr b ∼tr d, and so c ∼tr d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By [4, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='15], as relations on the group elements of any semigroup, ∼p = ∼∗ p = ∼tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus, c ∼p d, and so a ∼∗ p c ∼p d ∼∗ p b, which implies a ∼∗ p b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have proved that ∼tr ⊆ ∼∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since ∼∗ p ⊆ ∼tr in any epigroup, ∼∗ p = ∼tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 38 Let a, b ∈ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We can check if a and b are p∗-conjugate (equivalently, tr-conjugate) in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We can calculate the successive positive powers of a and b until we obtain idempotents e and f, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then we check if ea (= aω+1) and fb (= bω+1) have the same cycle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Or, using Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='26 and Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='24 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='25, we calculate group elements c, d such that a ∼∗ p c and b ∼∗ p d, and we check if c and d have the same cycle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We now turn to BPn and Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a ∈ BPn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In this case, the partition u constructed in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='24 is an element of BPn as well, and therefore Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='24 and its dual also hold in BPn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We can now repeat the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='26, noting that the situations in which Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='25 or its dual are used cannot arise in BPn: if s is transversal, than A = s ∩ n is a singleton, so A′ cannot intersect different blocks t1, t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='27, we obtain: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In BPn, ∼∗ p = ∼tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' That is, for a, b ∈ BPn, a ∼∗ p b if and only if aω+1 and bω+1 have the same cycle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose that a ∈ Bn, {x, y} ⊆ n is a block of a, such that x′, y′ lie in (necessarily distinct) transversal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then a ∼p c, for some c ∈ Bn with lower rank than a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let {v, x′}, {w, y′} be the blocks containing x′, y′, and k the number of upper blocks of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As a is a partition in Bn, k is also the number of lower blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Consider u ∈ Bn with the following blocks: s and s′ for each upper block s of a, and {z, z′} for each z ∈ n that does not intersect an upper block of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It is straightforward to check that ua = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let c = au, so c ∼p a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The k upper blocks of a are also upper blocks of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In addition, {v, w} is an upper block of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' So c has more than k upper blocks, and hence also more than k lower blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that it has fewer transversal blocks than a, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Clearly, the dual version of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='29 holds as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a ∈ Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then there exists a group element c ∈ Bn such that a ∼∗ p c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Recall that ∼n ⊆ ∼∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a ∈ Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then a ∼n b (and hence a ∼∗ p b) for some b in n-normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose that there is a b-block {x, y} as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We can then use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='29 to obtain an element c such that b ∼∗ p c and c has a lower rank than b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If instead there is a b-block {x′, y′} such that x, y lie in transversal b-blocks, than we can find such c using the dual version of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We next obtain a partition a1 ∈ Bn in n-normal form satisfying c ∼n a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that c and a1 have the same rank as ∼n ⊆ D (by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have constructed an element a1 ∈ Bn in n-normal form such that a ∼∗ p a1 and a1 has a lower rank than a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We keep repeating this construction until we obtain a partition d ∈ Bn such that a ∼∗ p d, d is in n-normal form, and neither Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='29 nor its dual can be applied to d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (Note that d may be b if neither Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='29 nor its dual can be applied to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=') By Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='15, this means that {x, y} is an upper block of d if and only if {x′, y′} is a lower block of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence d is a group element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='27, we obtain: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In Bn, ∼∗ p = ∼tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' That is, for a, b ∈ Bn, a ∼∗ p b if and only if aω+1 and bω+1 have the same cycle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4 Conjugacies ∼o and ∼c in Pn, BPn, and Bn The conjugacy ∼o (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3) is the largest of the conjugacies considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In any semigroup, ∼n ⊆ ∼p ⊆ ∼∗ p ⊆ ∼o and ∼n ⊆ ∼c ⊆ ∼o [38, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In any epigroup, ∼n ⊆ ∼p ⊆ ∼∗ p ⊆ ∼tr ⊆ ∼o [4, Thm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Moreover, for any semigroup S, ∼o is the universal relation if S has a zero, and ∼o = ∼c if S has no zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It is known that ∼o is the identity relation on a semigroup S if and only if S is commutative and cancellative [4, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' There is no characterization of the semigroups (with no zero) in which ∼o is the universal relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In the finite partition monoids, which have no zero, ∼o is the universal relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In Pn, ∼o = Pn × Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 39 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let e = {{x, x′} : x ∈ [n]} be the identity in Pn and let a ∈ Pn be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We want to find g ∈ Pn such that ag = ge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Consider g ∈ Pn such that ker(g) = ker(aω), coker(g) = {{x′} : x′ ∈ [n′]}, and g does not have any transversal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then ker(ag) = ker(aaω) = ker(aω+1) = ker(aω) = ker(g), where the last but one equality follows from the fact that aω+1 H aω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since coker(g) is trivial and g has no transversal blocks, coker(ag) is also trivial and ag has no transversal blocks either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus ag = g = ge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Similarly, for h ∈ Pn such that coker(h) = coker(aω), ker(h) = {{x} : x ∈ [n]}, and h does not have any transversal blocks, we have ha = h = eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have proved that for every a ∈ Pn, a ∼o e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence ∼o = Pn × Pn since ∼o is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In the case that a ∈ BPn, the elements g and h constructed as above are in BPn as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence we immediately obtain the following classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In BPn, ∼o = BPn × BPn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We now consider ∼o for a Brauer moniod Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As ∼tr⊆∼o, it follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='23 that there is a partition Q of the set of available cycle types, such that a ∼o b if and only if the cycle types of aω+1 and bω+1 lie in the same part of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Moreover, as ∼n⊆∼o, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='16 shows that a has a ∼o-conjugate c in n-normal form (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will show below that this element can be chosen as a group element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The following lemma provides a description of such partitions, which follows directly from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='16 and Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose that c ∈ Bn is both a group element and in n-normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then there is a partition n = A∪B, such that A∪A′ contains all transversal b-blocks and B ∪B′ contains all non-transversal b-blocks (where we allow A = ∅ or B = ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Moreover, there is a parition of B into subsets Bi of size 2, such that Bi and B′ i are blocks for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We remark that |B| is even, and that we may identify cA with a permutation in SymA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a ∈ Bn be a group element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then there is a partition b ∈ Bn in n-normal form such that b is a group element with the same cycle type as a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let k be the number of blocks of ker(a)∨coker∧(a) that are used in the construction of the permutation corresponding to a (that is, the blocks of ker(a) ∨ coker∧(a) that intersect a transversal block of a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Pick a k-subset A of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Using only transversal blocks, we can construct a partition bA on A ∪ A′ that has the same cycle type as a (and which we might consider to be an element of SymA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In Bn, a block of ker(a) ∨ coker∧(a) that intersects one transversal of a has odd cardinality, while a block of ker(a) ∨ coker∧(a) that does not has even cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that |n \\ A| is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Partitioning B = n \\ A into 2-element sets Bi, we can extend ba to a partition b ∈ Bn by adding the blocks Bi, B′ i for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If the permutation associated with bA contains a cycle of size l, it is clear that we may identify a subset C of A such that bC represents this cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In the following, when we speak of such a representation, we will always assume that |C| = l (so unlike in the standard use of “cycle”, we do not allow any additional 1-cycles to be represented in C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a ∈ Bn be a group element in n-normal form, and suppose that C ⊆ n is such that aC represents a cycle of even length l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then there is a partition of C into 2-subsets Ci and b ∈ Bn such a ∼o b, b contains the blocks Ci, C′ i for all i, and aD = bD for D = n \\ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Order the elements of C as c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , cl, such that the a-blocks intersecting C are {cl, c′ 1} and {ci, c′ i+1} for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , l − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Partition C into blocks Ci = {ci, ci+l/2} for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' l/2, define g ∈ Bn with blocks Ci, C′ i and {z, z′} for z /∈ C, and set g = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It is straightforward to check that g, h witness a ∼o b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a ∈ Bn be a group element in n-normal form, and suppose that C, D ⊆ n, C ̸= D are such that aC, aD represents cycles of the same length l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then there is a partition of C ∪ D into 2-subsets Gi and b ∈ Bn such a ∼o b, b contains the blocks Gi, G′ i for all i, and aL = bL for L = n \\ (C ∪ D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 40 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose that C = {c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , cl}, D = {d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , dl} are ordered such that {cl, c′ 1}, {dl, d′ 1}, {ci, c′ i+1} and {di, d′ i+1}, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , l − 1, are the a-blocks intersecting C ∪ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Partition C ∪ D into blocks Gi = {ci, di} for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' l, define g ∈ Bn to have blocks Gi, G′ i and {z, z′} for z /∈ C ∪ D, and set g = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It is straightforward to check that g, h witness a ∼o b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a, b ∈ Bn, such that aω+1 and bω+1 have cycle types (k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , kn) and (l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , ln), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then a ∼o b if and only if ki ≡ li mod 2 for each odd i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose that ki ≡ li mod 2 for each odd i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Because ∼tr⊆∼o, and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='35, there exist partitions a′ ∼o a, b′ ∼o b, such that a′, b′ are both group elements in ∼n-normal form with the same cycle type as a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By repeated applications of the constructions from Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='36 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='37, we obtain partitions a′′ ∼o a′, b′′ ∼o b′, such that a′, b′ are both group elements in ∼n-normal form, and such the permutations corre- sponding to a′′, b′′ contain no even cycles and at most one j-cycle for each odd j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Moreover, a′′ [b′′] contains an odd j-cycle exactly if kj [lj] is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As we assumed that ki ≡ li mod 2 for each odd i, we see that a′′ and b′′ have the same cycle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that a′′ ∼tr b′′, thus a′′ ∼o b′′, and hence a ∼o b, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Assume now that ki ̸≡ li mod 2 for some odd i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a′′ ∼o a, b′′ ∼o b be constructed as in the first part, and construct a′′′ and b′′′ from a′′, b′′ by replacing all blocks of the form {x, y}, {x′, y′} with blocks {x, x′}, {y, y′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As this introduces an even number of 1-cycles, it follows that a′′′ ∼o a, b′′′ ∼o b by the first part of this proof, and moreover that the condition ki ̸≡ li mod 2 carries over to the cycle types of a′′′ and b′′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Moreover, a′′′, b′′′ are unit elements whose corresponding permutations only contains odd cycles with at most one j-cycle for j ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By abuse of notation, we will rename a′′′, b′′′ as a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Our aim os to show that a ̸∼o b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By way of contradiction, assume that g, h ∈ Bn witness a ∼o b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let Xa, Xb ⊆ n be the set of values z for which {z, z′} is a block of a or b, respectively (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' the values corresponding to 1-cycles of a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=') We claim that |Xa| = |Xb|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Consider z ∈ Xa, and assume that z lies in a transversal block {z, u′} of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then {z, u′} is a block of ag = gb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence {u, u′} is a block of b, and u ∈ Xb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' A dual argument shows that if z ∈ Xb and the g-block {z′, u} containing z′ is a transversal, then u ∈ Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence g induces a bijection between subsets Za ⊆ Xa, Zb ⊆ Xb, where Za, Z′ b consists of those elements of Xa, Xb that lie in transversal blocks of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that the elements of Xa \\ Za, and X′ b \\ Z′ b lie in non-transversal blocks of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As g ∈ Bn, it has the same number of upper and lower non-transversal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence to show the claim, it suffices to show that all non-transversal blocks of g lie in Xa or X′ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let {x, y} be an upper block of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then {xa−1, ya−1} is an upperblock of ag = gb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' As b is a unit, this is only possible if {xa−1, ya−1} is an upper g-block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Repeating this argument, we see that {xa−i, ya−i} is an upper g-block for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now suppose that x, y lie in some set C ⊆ n such that C sorresponds to one l-cycle of a with l ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that C is a union of upper blocks of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' However, l is odd, so this is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Assume instead that x ∈ C, y ∈ D, such that C, D represents a-cycles of different size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then there is an i such that, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='og.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' xa−i = x, ya−i ̸= y, contradicting that {x, y} is a g-block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that {x, y} ⊆ Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By a dual argument, if {x′, y′} is a lower block of g, then x, y ∈ Xb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The claim follows, and so |Xa| = |Xb| = k1 = l1, which also implies that i ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By replacing b with a conjugate of the form ubu−1 for a suitable unit u and g with gu−1, we may assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' that Xa = Xb (we once again abuse notation and name this new partitions b and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=') This process preserves the cycle type of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Applying the above considerations to our new value of g, we see that all g-blocks intersecting Xa ∪ X′ a are subsets of Xa ∪ X′ a, and that, moreover, all non-transversal g-blocks lie in Xa ∪ X′ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It follows that all g-blocks intersecting Y = n\\ Xa are transversal blocks and intersect n\\ X′ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence the induced subpartition gY is a unit element of BY , corresponding to a permutation of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Trivially, this is also true for aY , bY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Moreover the cycles types of aY , bY agree with those of a, b, except for the first position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In BY , we have aY gY = gY bY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Working in the unit group of By, we obtain that g−1 Y aY gY = bY , which is an equation of permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' However, this is not possible, as we assumed that ki ̸≡ li mod 2 for some 41 odd i, i ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By contradiction, a ̸∼o b, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since ∼c = ∼o in any semigroup that does not have a zero, we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The listed exceptional cases contain a zero and can be confirmed by direct calculation (See (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='5) for the definition of ∼c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=') Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In Pn, BPn, and Bn, ∼o = ∼c, except for P1, PB1, B2, where ∼c is equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' That is, on Pn and BPn, ∼c is the universal relation, except for P1, PB, where ∼c is equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If a, b ∈ Bn, n ̸= 2, such that aω+1 and bω+1 have cycle types (k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , kn) and (l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , ln), then a ∼c b if and only if ki ≡ li mod 2 for each odd i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' On B2, ∼c is equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 5 Conjugacy growth in polycyclic monoids The study of conjugation in polycyclic monoids was initiated in [3] by some of the authors of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Polycyclic monoids are inverse monoids with zero so ∼o is the universal relation and ∼i = ∼n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In [3] the notions of ∼p (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2), and ∼c (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='5) were characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In this section we intend to present a study on ∼n (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The conjugacy growth function of a finitely generated group G counts the number of conjugacy classes intersecting the ball of radius n in the Cayley graph of G centered at the identity, for all n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It has been studied for free groups [16,52,53], hyperbolic groups [17,18], solvable groups [9], linear groups in [10], acylindrically hyperbolic groups [1,36], certain branch groups [27], in the higher Heisenberg groups in [24], and several other classes of groups [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Given a notion of conjugation for monoids that is an equivalence relation, the conjugacy growth function of the groups can be extended to finitely presented monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In this section we will present the conjugacy growth functions of the polycyclic monoids, for the conjugations ∼n, ∼c, and ∼∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In the last few years, the conjugacy growth series (the generating series associated with the conjugacy growth functions) have been computed for several classes of groups based on the description of sets consisting of minimal length representatives from all conjugacy classes [1,11–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The paper [23] supports the conjecture that virtually abelian groups are the only ones with rational conjugacy series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Historically, one of the initial motivations for counting conjugacy classes of a given length came from counting closed geodesics of bounded length in compact Riemannian manifolds [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We first need some preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1 Characterization of the conjugacy relations in Pn Let n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Consider a set An = {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , pn}, and denote by A−1 n a disjoint copy {p−1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' , p−1 n }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let Σ = An ∪ A−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The polycyclic monoid Pn is the monoid with zero defined by the monoid presentation Pn = ⟨Σ0 | p−1 i pi = 1 and p−1 i pj = 0, i ̸= j}⟩, where Σ0 = Σ ∪ {0} and 0 is a symbol that is not in Σ that is interpreted as the zero of the monoid by what we consider implicit the multiplications by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Given x ∈ Σ, we define x−1 to be p−1 i if x = pi ∈ An, and to be pi if x = p−1 i ∈ A−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We define 1−1 = 1 and (xw)−1 = w−1x−1, for all x ∈ An and w ∈ A∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' It is well known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=', [45, subsection 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3]) that every nonzero element of Pn has a unique representation of the form yx−1 with y, x ∈ A∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Whenever we write a = yx−1, it will be understood that x, y ∈ A∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will identify nonzero elements of Pn with the words of this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The explicit multiplication is provided by the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We say that words x, v ∈ A∗ n prefix comparable if one is a prefix of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' ([3, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2]) Consider nonzero elements yx−1 and vu−1 of Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then: (1) yx−1 · vu−1 ̸= 0 iff x and v are prefix comparable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 42 (2) if yx−1 · vu−1 ̸= 0, then yx−1 · vu−1 = � yzu−1 if v = xz , y(uz)−1 if x = vz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (3) y = v in Pn iff y = v in A∗ n, and x−1 = u−1 in Pn iff x = u in A∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' A word w ∈ Pn is said to be cyclically reduced if w = 0 or w = yx−1, where x and y have no common prefix other than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Every nonzero element of Pn can be written in the form ryx−1r−1, with r ∈ A∗ n and yx−1 a cyclically reduced word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' From any a ∈ Pn, we compute a cyclically reduced word �a in the following way: if a = 0, we let �a be equal to 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' otherwise, a = ryx−1r−1 as above, so we let �a be the (possibly empty) cyclically reduced word yx−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will now characterize conjugacy ∼n in Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since Pn is an inverse monoid, we have ∼n=∼i by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='11, that is, for all a, b ∈ Pn, a ∼n b if and only if there exists g ∈ Pn such that g−1ag = b and gbg−1 = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a, b ∈ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then a ∼n b if and only if a = b = 0 or �a = �b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since [0]n = {0}, it remains to establish criteria for nonzero a, b ∈ Pn to be n-conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In the calculations below, it will be convenient to write a = yx−1 as a = a+a−1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a = a+a−1 − , b = b+b−1 − ∈ Pn with a ∼n b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then there exists g = g+g−1 − ∈ Pn such that g−g−1 + a+a−1 − g+g−1 − = b+b−1 − and g+g−1 − b+b−1 − g−g−1 + = a+a−1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='7) Since b+b−1 − ̸= 0, it follows by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1 that a− and g+ are prefix-comparable, g+ and a+ are also prefix comparable, and g−g−1 + a+a−1 − g+g−1 − = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 g−g−1 + a+rg−1 − if g+ = a−r, = � g−sg−1 − if a+r = g+s g−(g−s)−1 if g+ = a+rs g−g−1 + a+(g−r)−1 if a− = g+r, = � g−(g−rs)−1 if g+ = a+s g−s(g−r)−1 if a+ = g+s, where r, s ∈ A∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By these calculations, first equality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='7), and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1(4), we obtain: g−s = b+ and g− = b− if a+r = g+s and g+ = a−r, g− = b+ and g−s = b− if g+ = a+rs and g+ = a−r, g− = b+ and g−rs = b− if g+ = a+s and a− = g+r, g−s = b+ and g−r = b− if a+ = g+s and a− = g+r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Thus we have have four cases to consider, and in each case we can draw conclusions using the second equality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='7) and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' g−s = b+, g− = b−, a+r = g+s, g+ = a−r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then a+a−1 − = g+g−1 − b+b−1 − g−g−1 + = g+sg−1 + , so r = 1, and hence a = g+sg−1 + and b = g−sg−1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' g− = b+, g−s = b−, g+ = a+rs, g+ = a−r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then a+a−1 − = g+g−1 − b+b−1 − g−g−1 + = g+(g+s)−1, so s = r = 1, and hence a = g+g−1 + and b = g−g−1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' g− = b+, g−rs = b−, g+ = a+s, a− = g+r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then a+a−1 − = g+g−1 − b+b−1 − g−g−1 + = g+(g+rs)−1, so s = 1, and hence a = g+(g+r)−1 and b = g−(g−r)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' g−s = b+, g−r = b−, a+ = g+s, a− = g+r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then a+a−1 − = g+g−1 − b+b−1 − g−g−1 + = g+s(g+r)−1, and hence a = g+s(g+r)−1 and b = g−s(g−r)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that the forms of a and b deduced in Cases 1–3 are special cases of the forms deduced in Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Therefore, if a ∼n b, then a = g+s(g+r)−1 and b = g−s(g−r)−1 for some g+, g−, r, s ∈ A∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Conversely, if a = g+s(g+r)−1 and b = g−s(g−r)−1 for some g+, g−, r, s ∈ A∗ n, then it is straightforward to verify g−1ag = b and gbg−1 = a for g = g+g−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We have proved the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 43 Note that for any representative a ∈ Pn we have a ∼n �a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' This gives the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The set of cyclically reduced words is a set of representatives of minimal length of the partition Pn/∼n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For a nonzero representative a = yx−1 ∈ Pn, we denote by ρ(a) the representative word of x−1y in Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We also set ρ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Note that ρ(a) ∈ A∗ n ∪ (A−1 n )∗ ∪ {0}, for any representative a ∈ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Also note that ρ(a) = �a if and only if �a ∈ A∗ n ∪ (A−1 n )∗ ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let us recall the characterizations of ∼c and ∼p from [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' ([3, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='9]) Let a, b ∈ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then a ∼c b if and only if one of the following conditions is satisfied: (a) a = b = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (b) �a = �b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' or (c) �a,�b ∈ (A−1 n )∗ and �a ∼p �b in the free monoid (A−1 n )∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In particular, if an element of Pn is not in (A−1 n )∗ ∪ {0} then it is ∼c-conjugate to a unique element yx−1 such that y ̸= 1 and x and y have no common prefix other than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For a given alphabet X, let Lp(X) denote a set of representatives of minimal length of the partition resulting of the quotient of free monoid on X by the equivalence relation ∼p on X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The set of cyclically reduced words with a prefix in An ∪ {0} together with the set Lp(A−1 n ), is a set of representatives of minimal length of the partition Pn/∼c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Any two different a, b ∈ Pn such that a, b ∈ A∗ n or a, b ∈ (A−1 n )∗ are never n-conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' This shows that in Pn, conjugacy ∼n is strictly included in ∼c and ∼p (see [3, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' ([3, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6]) Let a, b ∈ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then a ∼p b if and only if one of the following conditions is satisfied: (a) a = ρ(b) = 0 or ρ(a) = b = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (b) ρ(a) = ρ(b) = 0 and �a = �b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (c) �a,�b ∈ A∗ n and �a ∼p �b in the free monoid A∗ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' or (d) �a,�b ∈ (A−1 n )∗ and �a ∼p �b in the free monoid (A−1 n )∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' From Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6 and other results in [3], we can deduce a characterization of ∼∗ p in Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let a, b ∈ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then a ∼∗ p b if and only if one of the following conditions is satisfied: (a) ρ(a) = ρ(b) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' (b) �a,�b ∈ A∗ n and �a ∼p �b in the free monoid A∗ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' or (c) �a,�b ∈ (A−1 n )∗ and �a ∼p �b in the free monoid (A−1 n )∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose a ∼∗ p b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then, by [3, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='7], either a ∼p b or a ∼p 0 ∼p b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In the former case, (a), (b), or (c) is satisfied by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose a ∼p 0 ∼p b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then ρ(a) = ρ(b) = 0 by [3, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4], and so (a) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Conversely, suppose that one of (a), (b), (c) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' If (b) or (c) holds, then a ∼p b by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='6, and so a ∼∗ p b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Suppose (a) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then, by [3, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4] again, a ∼p 0 ∼p b, and so a ∼∗ p b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' In particular, if a representative element of Pn is not in A∗ n ∪ (A−1 n )∗, then it is ∼∗ p-conjugate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The set Lp(An) ∪ Lp(A−1 n ) ∪ {0, 1}, is a set of representatives of minimal length of the partition Pn/∼∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 44 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2 Conjugacy growth functions in Pn Let M be a monoid generated by a finite set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then every element of M can be represented as a word in X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The length of an element a ∈ M is the minimum length of a word that represent y, written |a|X or just |a| if the context is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Since X is finite, for every integer m ≥ 0, there are only finitely many elements of M that are of length m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' This leads us to the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For a monoid M with finite generating set X, we define the strict growth function of M (with respect to X) respectively as σM,X(n) = #{a ∈ M : |a|X = n} for any n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Regarding the characterization of representatives of the polycyclic monoid given in the previous subsec- tion, we obtain the following result: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The polycyclic monoid on n generators Pn, has strict growth function given by σPn,Σ0(0) = 1, σPn,Σ0(1) = 2n + 1, and σPn,Σ0(m) = (m + 1)nm for m ≥ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let ∼j be a conjugacy in M that is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For a ∈ M, we denote by [a]∼j the ∼j- conjugacy class of a, and we write M/∼j for the set of ∼j-conjugacy classes in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For a ∈ M, we define the length of the conjugacy class [a]∼j by |[a]∼j|X = min{|b|X : b ∈ [a]∼j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For a monoid M with finite generating set X, and a conjugacy ∼j in M that is an equivalence relation, we define the strict conjugacy growth function of M relative to ∼j (with respect to X) respectively as ∼jσ M,X(n) = #{a ∈ M : |[a]∼j|X = n} for any n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We will now compute the conjugacy growth functions of the polycyclic monoids for the conjugacies ∼n, ∼c, and ∼∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The polycyclic monoid on n generators Pn, has strict conjugacy growth function relative to ∼n given by ∼n σ Pn,Σ0(0) = 1, ∼nσ Pn,Σ0(1) = 2n + 1, and ∼n σ Pn,Σ0(m) = 2nm + (m − 1)nm−1(n − 1), for m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We use Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3 to deduce the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The cases for m = 0 and m = 1 are easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For m ≥ 2, we can distinguish the case when the cyclically reduced word is in A∗ n ∪ (A−1 n )∗, for which we get 2nm ciclically reduced words of length m, from the cases where the cyclically reduced word of lenght m has the form yx−1, with x and y non-empty and with no common prefix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' To be able to compute the conjugacy growth functions of ∼c and ∼∗ p we need to compute the ∼p-conjugacy growth function of the free monoid on a given alphabet X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let X be an alphabet with |X| = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The ∼p-conjugacy growth function of the free monoid on X is ∼∗ pσ X∗,X(m) = � d|m � e|d µ �d e � ne d , m ≥ 1, where µ is the M¨obius function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 45 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The number of words in X∗ of length m is nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Given a word a in X of length m, a ∼p-conjugate word to a will be a cyclic permutation of a, that is, it will be some b ∈ X∗ with a = uv and b = vu, for some u, v ∈ X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' So, how many distinct cyclic permutations of a we may have?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We know that, a = uv = vu, with u, v ̸= 1, if and only if a = wk, for some w ̸= 1, and k > 1 [44, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' A word p is called primitive if whenever p = wk, for some w ∈ X∗, then k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The root of a word a, denoted √a, is the unique primitive word p such that a = pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence, a word a has |√a|X distinct cyclic permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Denote by f(d) the number of primitive words in X of length d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Then the number am of ∼p-conjugate elements in X∗ of length m is am = � d|m f(d) d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Now, the number of words in X∗ of length m can be given by nm = � d|m f(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Therefore, by the M¨obius inversion formula f(m) = � d|m µ �m d � nd, where µ is the M¨obius function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The polycyclic monoid on n generators Pn, has strict conjugacy growth function relative to ∼c given by ∼cσ Pn,Σ0(0) = 1, ∼cσ Pn,Σ0(1) = 2n+1, and ∼cσ Pn,Σ0(m) = nm +(m−1)nm−1(n−1)+ ∼∗ pσ A∗n,An(m), for m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We use Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='5 and the previous theorem to deduce the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The proof follows the same reasoning of the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The polycyclic monoid on n generators Pn, has strict conjugacy growth function relative to ∼∗ p given by ∼∗ pσ Pn,Σ0(0) = 1, ∼∗ pσ Pn,Σ0(1) = 2n + 1, and ∼∗ pσ Pn,Σ0(m) = 2 ∼∗ pσ A∗ n,An(m), for m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The result follows from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='8 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3 Conjugacy growth series of Pn In this subsection we describe the different growth series of the polyclyclic monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We begin by introducing the concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let M be a monoid generated by a finite set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The standard growth series of M is the following power series with indeterminate z: ΞM,X(z) = � m≥0 σM,X(m)zm, where σM,X is the strict growth function of M with respect to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let M be a monoid generated by a finite set X, and let ∼j be a conjugacy in M that is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The ∼j-conjugacy growth series of M is the following power series with indeterminate z: ∼j Ξ M,X(z) = � m≥0 ∼jσ M,X(m)zm, where ∼jσ M,X is the strict growth function of M with respect to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 46 Note that even if one cannot define in growth function for infinitely generated groups, the paper [6] gives the conjugacy growth series for some infinitely generated groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' From Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='13 we deduce the following: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Let X be an alphabet with |X| = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The ∼p-conjugacy growth series of the free monoid on X is ∼∗ p Ξ X∗,X(z) = � r,s≥1 nr rs ϕ (s) zrs, where ϕ is the totient Euler formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We can now give an explicit formula for the conjugacy growth series of the polycyclic monoids Pn for the conjugacies ∼n, ∼c and ∼∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The n-conjugacy growth series of Pn is ∼n Ξ Pn,Σ0(z) = 1 − nz2 (1 − nz2)2 + z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' According to Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3, we have to count the number of words sr−1, where r and s do not have a common prefix other than the empty word, plus the element 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The conjugacy class of 0 contributes z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We can do the former by counting all words yx−1 ∈ Pn, and then removing those for which x and y have at least one common beginning letter from An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' This gives z + 1 (1 − nz)2 − nz2 1 (1 − nz)2 , which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The ∼c-conjugacy growth series of Pn is given by ∼c Ξ Pn,Σ0(z) = 1 1 − nz + z + (n2 − n)z2 (1 − nz)2 + ∼∗ p Ξ A∗ n,An(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='5, we have to count the number of cyclically reduced words with a prefix in An ∪ {0} and the words in the set Lp(A−1 n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The conjugacy classes of the elements of A∗ n contribute 1 1−nz to the series, and the conjugacy class of 0 contributes z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Further, there are the conjugacy classes of the elements yx−1 such that both x and y are not empty and have no common prefix other than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' They contribute (nz)2 (1−nz)2 − nz2 (1−nz)2 to the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Finally, we have the conjugacy classes of the elements in (A−1 n )∗ \\ {1}, which contribute ∼∗ p Ξ A∗ n,An(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' For completeness, we present the analogous result for the ∼∗ p-conjugacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The ∼∗ p-conjugacy growth series of Pn is given by ∼∗ p Ξ Pn,Σ0(z) = 1 + z + 2 ∼∗ p Ξ A∗n,An(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' The conjugacy class of the empty word contributes 1 to the series, and the conjugacy class of 0 contributes z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Further, there are the conjugacy classes of the elements of A∗ n \\ {1} and the conjugacy classes of the elements in (A−1 n )∗ \\ {1}, which both contribute ∼∗ p Ξ A∗n,An(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' 47 6 Questions We characterized the conjugacy classes (for several different notions of conjugation) in the partition monoid and two of its friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Characterize the conjugacy relations for the other friends of the partition monoid (Planar, Jones, Kauffman, Martin, Temperley and Lieb, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Characterize the partial inner automorphisms for the partition monoid and its friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We know that there exist finitely generated groups for which the word problem is solvable, but the conjugacy problem is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence there exist semigroups for which the word problem is solvable, while (for various notions of conjugacy) the conjugacy problem is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' This leads us to the following question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Is there a finitely generated semigroup with solvable n-conjugacy problem and with unsolvable word problem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' We note that because of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='3, given a monoid with some nonidempotent elements, we cannot embed it injectively into a larger monoid such that all of its elements become n-conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Hence the construction in the proof of [3, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='2] will not work for n-conjugacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Can we identify the set of n-normal forms as a species in the sense of [8] in such a way to count the number of n-conjugacy classes in the partition monoid by the counting the isomorphism type series of this species?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Acknowledgements We thank Laura Ciobanu and Susan Hermiller for the idea of studying conjugacy growth in finitely generated groups, which extends naturally to finitely generated monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' JA, MK, AM and VM were supported by the Funda¸c˜ao para a Ciˆencia e a Tecnologia (Portuguese Founda- tion for Science and Technology) through the projects UIDB/MAT/00297/2020 and UIDP/MAT/00297/2020 (Centro de Matem´atica e Aplica¸c˜oes) and the FCT Project PTDC/MAT-PUR/31174/2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' MK was also supported by Simons Foundation Collaboration Grant 359872.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' VM was also supported by the FCT Project UID/MAT/00297/2019 (Centro de Matem´atica e Aplica¸c˜oes) and the FCT Project PTDC/MHC-FIL/2583/2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' References [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQf_wnL/content/2301.04252v1.pdf'} +page_content=' Antol´ın and L.' 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Let Fd +q be the d-dimensional vector space over the finite field Fq with q elements. +For each +non-zero r in Fq and E ⊂ Fd +q, we define W (r) as the number of quadruples (x, y, z, w) ∈ E4 such that +Q(x − y)/Q(z − w) = r, where Q is a non-degenerate quadratic form in d variables over Fq. When Q(α) = +�d +i=1 α2 +i with α = (α1, . . . , αd) ∈ Fd +q, Pham (2022) recently used the machinery of group actions and proved +that if E ⊂ F2 +q with q ≡ 3 (mod 4) and |E| ≥ Cq, then we have W (r) ≥ c|E|4/q for any non-zero square +number r ∈ Fq, where C is a sufficiently large constant, c is some number between 0 and 1, and |E| denotes +the cardinality of the set E. +In this article, we improve and extend Pham’s result in two dimensions to arbitrary dimensions with +general non-degenerate quadratic distances. +As a corollary of our results, we also generalize the sharp +results on the Falconer type problem for the quotient set of distance set due to the first two authors and +Parshall (2019). Furthermore, we provide improved constants for the size conditions of the underlying sets. +The key new ingredient is to relate the estimate of the W (r) to a quadratic homogeneous variety in +2d-dimensional vector space. This approach is fruitful because it allows us to take advantage of Gauss sums +which are more handleable than the Kloosterman sums appearing in the standard distance type problems. +1. Introduction +Let Fd +q, d ≥ 2, be the d-dimensional vector space over the finite field Fq with q elements. Throughout this +paper, we assume that q is a power of odd prime p. Given a set E in Fd +q, the distance set ∆(E) is defined by +∆(E) := {||x − y|| ∈ Fq : x, y ∈ E}, +where ||α|| = �d +i=1 α2 +i for α = (α1, . . . , αd) ∈ Fd +q. +In the finite field setting, the Falconer distance problem asks for the minimal exponent α > 0 such that +for any subset E of Fd +q with |E| ≥ Cqα, we have |∆(E)| ≥ cq. Here, and throughout this paper, C > 1 +denotes a sufficiently large constant, and 0 < c ≤ 1 denotes some constant independent of q and |E|, where +|E| denotes the cardinality of E. This problem was initially proposed by the first listed author and Rudnev +[20] as a finite field analogue of the Falconer distance problem in the Euclidean space. We notice that the +formulation of the finite field Falconer problem was also motivated on the Erd˝os distinct distances problem +over finite fields due to Bourgain, Katz and Tao [5]. Hence, the problem is also called the Erd˝os-Falconer +distance problem. We refer readers to [13, 18, 34, 16, 14, 15, 11, 12, 10] for precise definition, background +knowledge, and recent progress on the Erd˝os distinct distances problem and the Falconer distance problem +in the Euclidean setting. +As a strong version of the Erd˝os-Falconer distance problem, one can consider the Mattila-Sj¨olin distance +problem over finite fields which is to determine the smallest threshold β > 0 such that any subset E of Fd +q +with |E| ≥ Cqβ generates all possible distances, namely, ∆(E) = Fq. Using Fourier analytic machinery and +the Kloosterman sum estimate, the first listed author and Rudnev obtained the threshold (d + 1)/2 for all +dimensions d ≥ 2. +Theorem 1.1 (Iosevich and Rudnev, [20]). Suppose that E ⊂ Fd +q, d ≥ 2, and |E| > 2q(d+1)/2. Then we have +∆(E) = Fq. +The threshold (d + 1)/2 in Theorem 1.1 is the best currently known result on the Mattila-Sj¨olin distance +problem over finite fields for all dimensions d ≥ 2. It is considered as an extremely hard problem to improve +2010 Mathematics Subject Classification. 52C10, 42B05, 11T23. +Key words and phrases: Finite field, Quadratic distance, Quotient set +The research of the first listed author was supported in part by the National Science Foundation under grant no. HDR TRIPODS +-1934962 and by the NSF DMS-2154232. The second listed author was supported by Basic Science Research Programs through +National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1B07044469). +1 + +the (d+1)/2 result. Moreover, in the general setting of odd dimensions, it gives the optimal threshold, which +was proven in [19]. +However, in even dimensions, it has been believed that the exponent (d + 1)/2 can be improved but a +reasonable evidence or conjecture has not been stated in literature. In two dimensions, Murphy and Petridis +[27] showed that the threshold cannot be lower than 4/3 for the Mattila-Sj¨olin distance problem over finite +fields. +On the other hand, the first listed author and Rudnev [20] conjectured that the threshold (d + 1)/2 can +be lower to d/2 for the Erd˝os-Falconer distance problem in even dimensions, and in two dimensions the +threshold 4/3 was proven by the authors in [6] (see also [3]). Furthermore, when q is prime, the exponent +4/3 was improved to 5/4 by Murphy, Petridis, Pham, Rudnev, and Stevenson [28]. We also notice that the +threshold (d + 1)/2 cannot be improved for the Erd˝os-Falconer distance problem in general odd dimensions +(see also [19]). +The distance problems over finite fields have been extended in various directions. For example, Shparlinski +[32] studied the size of the distance set between two sets in Fd +q (see also [23, 25]). The generalized distance +problems with polynomial distances were investigated by the second listed author and Shen [24] and Vinh +[36]. +Covert, the second listed author, and Pi [8] studied the size of the k-distance set. +Hart and the +first listed author [9] extended the distance problem to the k-simplices problem over finite fields (see also +[2, 4, 29, 31, 35]). Shparlinski [33] addressed the Erd˝os-Falconer distance problem for the sum of the distance +set, namely ∆(E) + ∆(E) (see also [7, 22]). +Although lots of variants of the distance problems were extensively studied, the threshold d/2 for the set +E in Fd +q had not been addressed for any distance type problems until the first two authors and Parshall [21] +studied the following Mattila-Sj˝olin problem for the quotient set of the distance set over finite fields: +Problem 1.2 (The Mattila-Sj˝olin problem for the quotient set of the distance set). Given a set E in +Fd +q, d ≥ 2, the quotient set of the distance set, denoted by ∆(E) +∆(E), is defined by +∆(E) +∆(E) := +� ||x − y|| +||z − w|| : x, y, z, w ∈ E, ||z − w|| ̸= 0 +� +. +Determine the smallest exponent γ > 0 such that, for any set E ⊂ Fd +q with |E| ≥ Cqγ, we have +(1.1) +∆(E) +∆(E) = Fq. +The aforementioned authors obtained the threshold d/2 in even dimensions on the Mattila-Sj˝olin problem +for the quotient set of the distance set over finite fields. More precisely, they proved the following result. +Theorem 1.3 (Theorems 1.1, 1.2, [21]). Let E ⊂ Fd +q, d ≥ 2. Then the following statements hold: +(i) If d ≥ 2 is even and |E| ≥ 9qd/2, then ∆(E) +∆(E) = Fq. +(ii) If d ≥ 3 is odd and |E| ≥ 6qd/2, then ∆(E) +∆(E) ⊇ (Fq)2, where (Fq)2 := {a2 : a ∈ Fq}. +We shall write F∗ +q for the set of all non-zero elements in Fq. +As the main idea to deduce Theorem 1.3, the authors [21] first observed that for any r ∈ F∗ +q, we have +r ∈ ∆(E) +∆(E) +if +� +t∈Fq +v(t)v(rt) > v2(0), +where v(t) is the number of the pairs (x, y) ∈ E × E such that ||x − y|| = t, namely, +(1.2) +v(t) := +� +x,y∈E: +||x−y||=t +1. +Next, using the discrete Fourier analysis, they estimated a lower bound of � +t∈Fq v(t)v(rt) and an upper +bound of v2(0). Finally, the required size condition on the sets E was obtained by comparing them. Although +the method of the proof led to the optimal threshold result on the Mattila-Sj¨olin problem for the quotient +set of the distance set, it has two drawbacks below, as mentioned by Pham [30]. +2 + +• The proof is too sophisticated and it requires large amount of calculation. +• It is not clear from the proof that how many quadruples (x, y, z, w) ∈ E4 contribute to producing +each element r ∈ F∗ +q such that ||x−y|| +||z−w|| = r, namely, r ∈ ∆(E) +∆(E). +As a way to overcome the above issues, Pham [30] utilized the machinery of group actions in two dimensions +and obtained a lower bound of V (r) for any square number r in F∗ +q, where V (r) denotes the number of the +quadruples (x, y, z, w) ∈ E4 such that ||x−y|| +||z−w|| = r, namely, +(1.3) +V (r) := +� +x,y,z,w∈E: +||x−y|| +||z−w|| =r +1. +As a consequence, he provided a short proof to deduce the following lower bound of V (r) in two dimensions. +Theorem 1.4 (Theorem 1.2, [30]). Let E be a subset of F2 +q. Suppose that |E| ≥ Cq with q ≡ 3 (mod 4). +Then, for any non-zero square number r in F∗ +q, we have +V (r) ≥ c|E|4 +q +. +In particular, we have ∆(E) +∆(E) ⊇ (Fq)2 := {a2 : a ∈ Fq}. +Pham’s approach, based on the group action, is powerful in the sense that it gives a simple proof and an +information about a lower bound of V (r). However, his result, Theorem 1.4, is limited to two dimensions +with −1 square in Fq, and it gives us no information about V (r) for a non-square r in F∗ +q. +One of the main contributions of this paper is to improve and extend Pham’s result to all dimensions +for arbitrary finite fields. In particular, we will work on the problem with the non-degenerate quadratic +distances which generalize the usual distance. +The other contribution is to provide significantly improved explicit constants for size conditions on the +underlying sets. These are mainly based on our approach to view the problem for the set E in Fd +q as that for +the product set E × E associated with certain homogeneous variety of degree two in F2d +q +(see Section 5). As +a direct consequence of this method, we will also address a generalized version of Theorem 1.3, with more +accurate constants. +To precisely state our main results, let us set up some definitions and notations related to the quadratic +distance. +Definition 1.5 (Non-degenerate quadratic form). We identify a vector x = (x1, . . . , xd) ∈ Fd +q with a +d × 1 matrix over Fq. A non-degenerate quadratic form Q(x) is a homogeneous polynomial of degree 2 +in Fq[x1, . . . , xd], and is written by Q(x) = x⊤A x = +d� +i,j=1 +aijxixj for some d × d symmetric matrix A = [aij] +with det(A) ̸= 0, where x⊤ denotes the transpose of the column vector x and each entry aij of A belongs to +Fq. +Note that if A is the identity matrix, then Q(x) = ||x||. Hence, Q(x) generalizes the distance function +|| · ||. For E ⊂ Fd +q, the non-degenerate quadratic distance set ∆Q(E) is defined by +∆Q(E) := {Q(x − y) : x, y ∈ E} ⊂ Fq. +The quotient set of the quadratic distance set ∆Q(E) is defined as +∆Q(E) +∆Q(E) := +� b +a ∈ Fq : a, b ∈ ∆Q(E), a ̸= 0 +� +. +For r ∈ F∗ +q, and E ⊂ Fd +q, define W(r) as the number of quadruples (x, y, z, w) ∈ E4 such that Q(x−y) +Q(z−w) = r, +namely, +(1.4) +W(r) := +� +x,y,z,w∈E: +Q(x−y) +Q(z−w) =r +1. +Our main result is as follows. +3 + +Theorem 1.6. For E ⊂ Fd +q, d ≥ 2, let W(r) be defined as in (1.4). +(i) If d is even and |E| ≥ 4q +d +2 , then W(r) ≥ 5|E|4 +48q +for all r in F∗ +q. Moreover, the threshold d/2 is sharp +for any finite field Fq and all even dimensions d ≥ 2. +(ii) If d is odd and |E| ≥ 3q +d +2 , then W(r) ≥ 2|E|4 +45q +for all non-zero square numbers r in F∗ +q. Furthermore, +the threshold d/2 cannot be lower for any finite field Fq and all odd dimensions d ≥ 3. +(iii) If d is odd and |E| ≥ 11 +6 q +d+1 +2 , then W(r) ≥ 2|E|4 +363q for all r in F∗ +q. In addition, the threshold (d+1)/2 +cannot be improved for any finite field Fq and all odd dimensions d ≥ 3. +As a consequence of Theorem 1.6, we generalize Theorem 1.3 with accurate constants improved for the +cardinality of sets. +Corollary 1.7. Let E ⊂ Fd +q, d ≥ 2, and Q ∈ Fq[x1, . . . , xd] be a non-degenerate quadratic form. Then the +following statements are valid: +(i) If d is even and |E| ≥ 4qd/2, then ∆Q(E) +∆Q(E) = Fq. Moreover, the exponent d/2 cannot be improved for +arbitrary finite field Fq and all even dimensions d ≥ 2. +(ii) If d is odd and |E| ≥ 3qd/2, then ∆Q(E) +∆Q(E) ⊇ (Fq)2, where (Fq)2 := {a2 : a ∈ Fq}. In addition, the +exponent d/2 is optimal for arbitrary finite field Fq and all odd dimensions d ≥ 3. +(iii) If d is odd and |E| ≥ 11 +6 q(d+1)/2, then ∆Q(E) +∆Q(E) = Fq. Furthermore, the threshold (d + 1)/2 cannot be +lower for any finite field Fq and all odd dimensions d ≥ 3. +Proof. We claim that the statements (i), (ii), (iii) of the corollary directly follow from those (i), (ii),(iii) of +Theorem 1.6, respectively. To prove our claim, first observe that if r ∈ F∗ +q, then W(r) > 0 if and only if +r ∈ ∆Q(E) +∆Q(E). Combining this observation with Theorem 1.6 (i), (ii), (iii), the proof of the corollary is reduced to +showing that 0 ∈ ∆Q(E) +∆Q(E) under each assumption of Corollary (i), (ii), (iii). Since r ∈ ∆Q(E) +∆Q(E) for some non-zero +r in F∗ +q, we can choose x, y, z, w in E such that Q(x− y) ̸= 0 and Q(z − w) ̸= 0. Hence, 0 = Q(x−x) +Q(z−w) ∈ ∆Q(E) +∆Q(E), +as required. +□ +We have the following few comments on our results, Theorem 1.6 and Corollary 1.7. +• The results on Theorem 1.6 and Corollary 1.7 are independent of the choice of non-degenerate qua- +dratic forms Q. +• Theorem 1.6 (i) improves and extends Theorem 1.4 to all even dimensions and arbitrary finite fields +with the general non-degenerate quadratic distances. For instance, the theorem holds true and sharp +without any further assumption such as the condition q ≡ 3 (mod 4), which was necessary in proving +Theorem 1.4 by using the group actions. In addition, our theorem provides explicit constants with +small numbers, which are based on our different approaches to estimate the counting functions. +• Corollary 1.7 (i), (ii) clearly generalize Theorem 1.3 (i), (ii), respectively, with the improved con- +stants. Notice that when Q(x) = ||x||, the usual distance, the threshold (d + 1)/2 in Corollary 1.7 +(iii) follows immediately from Theorem 1.1. However, the constant 11/6 for the set size in Corollary +1.7 cannot be obtained from it since 2 > 11/6. +• In Section 4, we shall provide the sharpness examples for Theorem 1.6. In particular, we shall show +that the thresholds in the statements (i), (ii), (iii) of Theorem 1.6 are sharp for any finite field with- +out any further assumptions on the size of the ground field Fq. In [21], some sharpness examples for +Theorem 1.3 (i), (ii) were given but they work with some specific restriction to the size of Fq. +• In the proof of Theorem 1.6, to efficiently estimate a lower bound of W(r), r ̸= 0, we will work +on the product set E × E in 2d-dimensions instead of the set E ⊂ Fd +q, and reduce our problem to +4 + +the estimate of the explicit Fourier transform on quadratic homogeneous varieties in 2d-dimensional +vector spaces over Fq. This method enables us to obtain improved constants on the size conditions +of sets (see, for example, Section 5). +The rest of this paper will be organized to prove Theorem 1.6. +2. Equivalent forms of non-degenerate quadratic forms +Let Q(x) ∈ Fq[x1, . . . , xd] be a non-degenerate quadratic form. Then we can write Q(x) = x⊤A x for +some d × d matrix A with det(A) ̸= 0. Using a change of variables, by letting x = Cy for some non-singular +d × d matrix C, it follows that +Q(x) = Q(Cy) = (Cy)⊤A (Cy) = y⊤(C⊤A C) y. +Letting B = C⊤A C, we obtain a non-degenerate quadratic form Q′(y) := Q(Cy) = y⊤B y. In summary, after +a non-singular change of variables, the non-degenerate quadratic form Q(x) can be transformed to another +non-degenerate quadratic form Q′(x). In this case, we say that Q(x) is equivalent to Q′(x). Furthermore, it +is not hard to observe that +∆Q(E) = ∆Q′(E′), +where E ⊂ Fd +q and E′ := {C−1x : x ∈ E}. In addition, note that |E| = |E′|, and the size assumption of +any sets E is only considered as the main hypothesis for the distance type problems. Hence, without loss of +generality, in the proof of Theorem 1.6 we may choose any non-degenerate quadratic form Q′(x), which is +equivalent to Q(x), as a distance set. +Now we introduce a concrete quadratic form Q′(x) equivalent to any non-degenerate quadratic form Q(x). +Let η denote the quadratic character of F∗ +q, that is, η(t) = 1 if t is a non-zero square number of Fq, and +η(t) = −1 if t is a non-square number of Fq. +Lemma 2.1 ([1], Theorem 1 and [17], P.79). Let A be a d×d a symmetric matrix over Fq with det(A) ̸= 0 and +it is associated with a non-degenerate quadratic form Q(x) ∈ Fq[x1, . . . , xd]. Then the following statements +are true: +(i) If d is even, then the Q(x) is equivalent to +(2.1) +Q′(x) := +d−1 +� +i=1 +(−1)i+1x2 +i − εx2 +d = x2 +1 − x2 +2 + · · · + x2 +d−1 − εx2 +d, +where the ε is taken as any element in F∗ +q such that η((−1) +d +2 ε) = η(det(A)). +(ii) If d is odd, then the Q(x) is equivalent to +(2.2) +Q′(x) := +d−1 +� +i=1 +(−1)i+1x2 +i + εx2 +d = x2 +1 − x2 +2 + · · · + x2 +d−2 − x2 +d−1 + εx2 +d, +where the ε is taken as any element in F∗ +q such that η((−1) +d−1 +2 ε) = η(det(A)). +In the above lemma, note that if η(ε) = 1, we can simply choose ε = 1. On the other hand, if η(−ε) = 1, +then we can take −ε = 1, namely, ε = −1. +Example 2.2. When Q(x) = ||x||, it is clear that det(A) = 1. If d ≡ 2 (mod 4), then, by Lemma 2.1 (i), +η(−ε) = 1 and so Q(x) = ||x|| can be transformed to the following form: +x2 +1 − x2 +2 + · · · + x2 +d−1 + x2 +d. +If d ≡ 0 (mod 4), then, by Lemma 2.1 (i), η(ε) = 1 and so Q(x) = ||x|| can be transformed to the form +below: +x2 +1 − x2 +2 + · · · + x2 +d−1 − x2 +d. +On the other hand, when d is odd, we can apply Lemma 2.1 (ii) to conclude that Q(x) = ||x|| can be +transformed to the quadratic form x2 +1 − x2 +2 + · · · + x2 +d−2 − x2 +d−1 − x2 +d for d ≡ 3 (mod 4), and to the quadratic +form x2 +1 − x2 +2 + · · · + x2 +d−2 − x2 +d−1 + x2 +d for d ≡ 1 (mod 4). +For instance, we can consider the following simple, concrete example. +5 + +Example 2.3. Let Q(x) = ||x|| ∈ F3[x1, x2, x3]. Then, by the above example, we know that Q(x) = +x2 +1 + x2 +2 + x2 +3 is equivalent to x2 +1 − x2 +2 − x2 +3. Now let us prove this fact by a direct non-singular linear +substitution. To do this, we note that Q(x) = x⊤I3 x, where I3 denotes the 3 × 3 identity matrix. We use a +change of variables, by letting x = Cy, where C is the 3×3 non-singular symmetric matrix defined as below: +C = + + +1 +0 +0 +0 +1 +1 +0 +1 +−1 + + = C⊤. +It follows that +Q(x) = Q(Cy) = y⊤(C⊤I3C)y = y⊤(C⊤C)y = y⊤ + + +1 +0 +0 +0 +2 +0 +0 +0 +2 + + y. +Since 2 ≡ −1 (mod 3), we conclude that Q(x) = ||x|| is equivalent to x2 +1 − x2 +2 − x2 +3, as required. +As mentioned in the beginning of this section, we may assume that any non-degenerate quadratic form +Q(x) ∈ Fq[x1, . . . , xd] can be identified with Q′(x) defined as in Lemma 2.1 (i), (ii). Thus, from now on, we +fix the definition of the non-degenerate quadratic form Q(x), which we will use as the standard distance for +the non-degenerate quadratic form Q(x) ∈ Fq[x1, . . . , xd]. +Definition 2.4. (Standard distance functions and its dual functions) Let Q(x) ∈ Fq[x1, . . . , xd] be a non- +degenerate quadratic form and A be its associated matrix. Then we define the standard distance function +|| · ||Q and its dual function || · ||Q∗ on Fd +q as follows: +(i) If d is even, then +||x||Q := +d−1 +� +i=1 +(−1)i+1x2 +i − εx2 +d = x2 +1 − x2 +2 + · · · + x2 +d−1 − εx2 +d, +||m||Q∗ := +d−1 +� +i=1 +(−1)i+1m2 +i − ε−1m2 +d = m2 +1 − m2 +2 + · · · + m2 +d−1 − ε−1m2 +d, +where the ε can be taken as any element in F∗ +q such that η((−1) +d +2 ε) = η(det(A)). +(ii) If d is odd, then +||x||Q := +d−1 +� +i=1 +(−1)i+1x2 +i + εx2 +d = x2 +1 − x2 +2 + · · · + x2 +d−2 − x2 +d−1 + εx2 +d, +||m||Q∗ := +d−1 +� +i=1 +(−1)i+1m2 +i + ε−1m2 +d = m2 +1 − m2 +2 + · · · + m2 +d−2 − m2 +d−1 + ε−1m2 +d, +where the ε is taken as any element in F∗ +q such that η((−1) +d−1 +2 ε) = η(det(A)). +We also define the spheres with respect to the distances || · ||Q and || · ||Q∗. +Definition 2.5. Let t ∈ Fq and Q(x) ∈ Fq[x1, . . . , xd] be a non-degenerate quadratic form. We define +(SQ)t := {x ∈ Fd +q : ||x||Q = t} +and +(SQ∗)t := {m ∈ Fd +q : ||m||Q∗ = t}. +The variety (SQ)t can be regarded as a sphere of radius t with the standard distance function || · ||Q. The +variety (SQ∗)t can be called the dual sphere of (SQ)t. +3. Basics on the discrete Fourier analysis and related lemmas +The discrete Fourier analysis machinery will function as a main tool in proving Theorem 1.6. In this +section, we introduce some basics on it without proofs and review main properties of the Gauss sum. In +addition, we deduce a general counting lemma which will play a key role in the proof of our main theorem, +Theorem 1.6. +6 + +3.1. Discrete Fourier analysis and Gauss sums. We begin by defining the canonical additive character +of Fq. Let q be a power of prime p, say that q = pℓ. The absolute trace function Tr : Fq → Fp is well defined +as Tr(t) = t + tp + tp2 + · · · + tpℓ−1 (for example, see Section 3 of [26]). +Definition 3.1. (Canonical additive character and the quadratic character, [26]) The function χ defined by +χ(c) = e2πiTr(c)/p +for c ∈ Fq +is called the canonical additive character of Fq. On the other hand, the multiplicative character η is a function +F∗ +q → R, defined by +η(t) = +� 1 +if t is a square number of F∗ +q, +−1 +if t is not a square number of F∗ +q. +Recall that η(−1) = −1 ⇐⇒ q ≡ 3 (mod 4), and η(−1) = 1 ⇐⇒ q ≡ 1 (mod 4) (Remark 5.13, [26]). +Both the additive character χ and the multiplicative character ψ enjoy the orthogonality property: For +any m ∈ Fn +q , n ≥ 1, +� +x∈Fn +q +χ(m · x) = +� 0 +if m ̸= (0, . . . , 0) +qn +if m = (0, . . . , 0), +and +� +t∈F∗q +η(at) = 0 +if a ̸= 0. +where m · x denotes the usual dot product notation. +For a function f : Fn +q → C, the Fourier transform of f is defined as +�f(m) := q−n � +x∈Fn +q +χ(−m · x)f(x) +and the Fourier inversion theorem states that +f(x) = +� +m∈Fn +q +χ(m · x) �f(m). +The Plancherel theorem in this context says that +� +m∈Fn +q +| �f(m)|2 = 1 +qn +� +x∈Fn +q +|f(x)|2. +In particular, for any set E ⊂ Fn +q , we have +(3.1) +� +m∈Fn +q +| �E(m)|2 = |E| +qn = �E(0, . . . , 0). +Here, and throughout this paper, we identify the set E ⊂ Fd +q with the indicator function 1E of the set E. In +particular, when E = (0, . . . , 0), we write δ0(x) for the indicator function 1E(x). +We now introduce Gauss sums. +Definition 3.2. Let χ, η denote the canonical additive character and the quadratic character of F∗ +q. The +standard Gauss sum determined by χ and η is defined by +G = G(η, χ) := +� +t∈F∗q +η(t)χ(t). +The following estimate is well-known (see [26]): +|G| = √q. +The following theorem is very useful in finding an explicit value of the Fourier transform on the homoge- +neous variety. +Lemma 3.3. Let G denote the standard Gauss sum. Then we have +G2 = G(η, χ)2 = η(−1)q. +7 + +Proof. It is obvious that η = η and χ(t) = χ(−t). Therefore, it is seen by a change of variables that +G(η, χ) = η(−1)G(η, χ). +Hence, G(η, χ)2 = η(−1)|G(η, χ)|2. Since |G(η, χ)| = √q, we are done. +□ +It is not hard to note that for any a ∈ F∗ +q, +� +t∈Fq +χ(at2) = η(a)G. +Completing the square and using a simple change of variables, the above formula can be generalized to the +formula below: For any a ∈ F∗ +q and any b ∈ Fq, we have +(3.2) +� +t∈Fq +χ(at2 + bt) = η(a)χ +� b2 +−4a +� +G. +By a change of variables and properties of the quadratic character η, it is not difficult to note that for +b ̸= 0, we have +(3.3) +� +s∈F∗q +η(s)χ(bs−1) = +� +s∈F∗q +η(bs−1)χ(s) = +� +s∈F∗q +η(bs)χ(s) = η(b)G. +3.2. General counting lemma. In order to prove Theorem 1.1, the first listed author and Rudnev [20] +evaluated the values of the counting function v(t) in (1.2) by relating it to the Fourier transform on St := +{x ∈ Fd +q : ||x|| = t}, the sphere of radius t ∈ F∗ +q. In this subsection, we formulate their work in the general +setting, which will provide an initial step in estimating W(r) in (1.4). +Now let us work on Fn +q for an integer n ≥ 2. +Lemma 3.4 (General counting lemma). Let P(x) be a polynomial function on Fn +q . Consider an algebraic +variety V := {x ∈ Fn +q : P(x) = 0}. Then, for every set E ⊂ Fn +q , we have +� +x,y∈E: +P (x−y)=0 +1 = q2n � +m∈Fn +q +�V (m) |�E(m)|2. +Proof. The proof proceeds by modifying the method of the first listed author and Rudnev [20]. +It follows that +w(0) := +� +x,y∈E: +P (x−y)=0 +1 = +� +x,y∈Fn +q +V (x − y)E(x)E(y). +Applying the Fourier inversion theorem to the indicator function V (x − y), we get +w(0) = +� +x,y∈Fn +q +� +m∈Fn +q +�V (m)χ(m · (x − y))E(x)E(y). +Finally, applying the definition of the Fourier transform �E(m), the statement follows. +□ +The following is a direct consequence of the general counting lemma. +Corollary 3.5. Let Q(x) = ||x||Q for x ∈ Fd +q. Then, for every set E ⊂ Fd +q, we have +w(0) := +� +x,y∈E: +Q(x−y)=0 +1 = q2d � +m∈Fdq +� +(SQ)0(m) | �E(m)|2. +Proof. By Definition 2.5, recall that +(SQ)0 = {x ∈ Fd +q : Q(x) = 0}. +Then the statement of the corollary immediately follows, because it is a special case of Lemma 3.4 when +n = d, V = (SQ)0, E = E, and P = Q. +□ +8 + +4. Sharpness of Theorem 1.6 +In this section, we provide sharpness examples for the threshold results on Theorem 1.6 (i), (ii), (iii). In +order to construct such sets, it is enough to use the standard quadratic forms in Lemma 2.1 or Definition +2.4. This is because any non-degenerate quadratic forms that are equivalent yield the same result on the +distance type problem. +4.1. The proof of sharpness of Theorem 1.6 (i). Let the dimension d be even. When d ≥ 4, we define +a set E1 ⊂ Fd−2 +q +as +E1 := +� +(t1, t1, . . . , ti, ti, . . . , t(d−2)/2, t(d−2)/2) ∈ Fd−2 +q +: ti ∈ Fq, 1 ≤ i ≤ (d − 2)/2 +� +. +Now if d ≥ 4 is even, then define E = E1 × Fq × {0} = {(x′, xd−1, 0) ∈ Fd +q : x′ ∈ E1, xd−1 ∈ Fq}, and if d = 2, +then define E = Fq × {0}. Since d is even, the standard distance function is given by the form +Q(x) = ||x||Q = x2 +1 − x2 +2 + · · · + x2 +d−1 − εx2 +d. +Now notice that |E| = |E1||Fq| = qd/2 and ∆Q(E) = {(a − b)2 : a, b ∈ Fq} = (Fq)2. Since the ratio of +non-zero square numbers is also a square number in Fq, it follows that ∆Q(E) +∆Q(E) = (Fq)2. Since (Fq)2 ⊊ Fq, +there is r ∈ F∗ +q and a set E ⊂ Fd +q with |E| = qd/2 such that r /∈ ∆Q(E) +∆Q(E), namely, W(r) = 0. +In short, the set E provides the desired sharpness example. Notice that, in our construction of the set E, +we did not impose any specific assumptions on the underlying finite field Fq. +4.2. The proof of sharpness of Theorem 1.6 (ii). Since d ≥ 3 is odd, we adopt the following standard +distance function in Definition 2.4: +(4.1) +Q(x) = ||x||Q = x2 +1 − x2 +2 + · · · + x2 +d−2 − x2 +d−1 + εx2 +d. +We define a set H ⊂ Fd−1 +q +as +(4.2) +H := +� +(t1, t1, . . . , ti, ti, . . . , t(d−1)/2, t(d−1)/2) ∈ Fd−1 +q +: ti ∈ Fq, 1 ≤ i ≤ (d − 1)/2 +� +. +It is clear that |H| = q(d−1)/2 and the Cartesian product of two sets H and A ⊂ Fq is defined by H × A := +{(x′, xd) ∈ Fd +q : x′ ∈ H, xd ∈ A}. Letting E = H × A, we see that ∆Q(E) = {ε(a − b)2 : a, b ∈ A} and so +∆Q(E) +∆Q(E) = +� +(a−b)2 +(a′−b′)2 ∈ Fq : a, b, a′, b′ ∈ A, a′ ̸= b′� +⊂ (Fq)2. +Let q = pℓ for an integer ℓ ≥ 1 and an odd prime p. Then one can consider the finite field Fq as an +ℓ-dimensional vector space over Fp with a basis +� +1, θ, . . . , θℓ−1� +, where θ is an algebraic element of degree +ℓ over Fp. For every 0 < δ < 1/2, we can choose an arithmetic progression Bδ ⊂ Fp with |Bδ| ∼ p1/2−δ. +Let us define the set Aδ := +� +b0 + b1θ + · · · + bℓ−1θℓ−1 : b0, . . . , bℓ−1 ∈ Bδ +� +⊂ Fq and it is easy to verify that +|Aδ| = |Bδ|ℓ ∼ pℓ/2−δℓ = q1/2−δ. If we consider the difference set of Aδ which is defined as +Aδ − Aδ := +� +(b0 − b′ +0) + (b1 − b′ +1)θ + · · · + (bℓ−1 − b′ +ℓ−1)θℓ−1 : b0, b′ +0, . . . , bℓ−1, b′ +ℓ−1 ∈ Bδ +� +, +then |Aδ − Aδ| = |Bδ − Bδ|ℓ ∼ |Bδ|ℓ ∼ pℓ/2−δℓ. Here we have used the fact that |Bδ − Bδ| ∼ |Bδ| since Bδ +is an arithmetic progression. Therefore, we have shown that |Aδ − Aδ| ∼ |Aδ|. +Set Eδ = H×Aδ ⊂ Fd +q. Then |Eδ| = |H||Aδ| ∼ q +d +2 −δ. Observe that ∆Q(Eδ) = +� +ε(a − a′)2 ∈ Fq : a, a′ ∈ Aδ +� +. +Since |Aδ −Aδ| ∼ |Aδ|, we see that |∆Q(Eδ)| ∼ |Aδ|. Since +��� ∆Q(Eδ) +∆Q(Eδ) +��� ≤ |∆Q(Eδ)|2 ∼ q1−2δ and |(Fq)2| = q+1 +2 , +we conclude that ∆Q(Eδ) +∆Q(Eδ) ⊊ (Fq)2. Thus, for every 0 < δ < 1/2, there exists Eδ ⊂ Fd +q with |Eδ| ∼ q +d +2 −δ such +that W(r) = 0 for some non-zero r ∈ (Fq)2. Since δ can be taken as an arbitrary small positive number, the +threshold d/2 cannot be smaller. +4.3. The proof of sharpness of Theorem 1.6 (iii). Suppose that d is odd. Then, we can also use the +standard distance function in (4.1). Let H ⊂ Fd−1 +q +be the set defined as in (4.2). +Set E := H × Fq ⊂ Fd +q. Then it is not hard to see that |E| = q +d+1 +2 +and ∆Q(E) = {ε(a − a′)2 : a, a′ ∈ Fq}. +Also notice that +∆Q(E) +∆Q(E) = +�(a − a′)2 +(b − b′)2 : a, a′, b, b′ ∈ Fq, b ̸= b′ +� += (Fq)2. +9 + +Clearly, there is r ∈ F∗ +q such that r /∈ ∆Q(E) +∆Q(E), i.e., W(r) = 0. Thus, the set E guarantees the sharpness of +Theorem 1.6 (iii). +5. Proof of the main theorem (Theorem 1.6) +In this section we start proving Theorem 1.6. We may assume that Q(x) = ||x||Q and Q∗(m) = ||m||Q∗ +for x, m ∈ Fd +q, where ||x||Q and ||m||Q∗ are defined as in Definition 2.4. Let E ⊂ Fd +q. For each r ∈ F∗ +q, we +begin with estimating W(r) defined as in (1.4). First observe that W(r) can be written as +W(r) = +� +x,y,z,w∈E: +Q(x−y)=rQ(z−w) +1 − +� +x,y,z,w∈E: +Q(x−y)=0=Q(z−w) +1 =: M(r) − w2(0), +where w(0) is defined by +(5.1) +w(0) = +� +α,β∈E: +Q(α−β)=0 +1. +To estimate M(r), the first sum above, we write it as +M(r) = +� +(x,z),(y,w)∈E×E: +Q(x−y)−rQ(z−w)=0 +1. +We now relate M(r) to an estimate on a homogeneous variety of degree two in F2d +q . To this end, we need to +introduce the following definition. +Definition 5.1. Let r ∈ F∗ +q, Q(x) = ||x||Q, and Q∗(m) = ||m||Q∗ for x, m ∈ Fd +q. Then, for x, x′, m, m′ ∈ Fd +q, +we define +Qr(x, x′) := Q(x) − rQ(x′) +and +Q∗ +r(m, m′) := Q∗(m) − r−1Q∗(m′). +In addition, we define algebraic varieties VQr and VQ∗r lying in F2d +q +as follows: +VQr := {X ∈ F2d +q : Qr(X) = 0} +and +VQ∗r := {M ∈ F2d +q : Q∗ +r(M) = 0}. +Letting X = (x, z), Y = (y, w) ∈ E × E and using the notation in the above definition, we have +(5.2) +M(r) = +� +X,Y ∈E×E: +Qr(X−Y )=0 +1 = q4d � +M∈F2d +q +� +VQr(M)| � +E × E(M)|2, +where the last equality follows from the general counting lemma (see Lemma 3.4). +We will invoke the +following explicit formula for � +VQr(M), whose proof will be given in the following section. For a moment, we +accept the proposition below. +Proposition 5.2. For x = (x1, . . . , xd), m = (m1, . . . , md) ∈ Fd +q, let Q(x) = ||x||Q and Q∗(m) = ||m||Q∗, +where ||x||Q and ||m||Q∗ are defined as in Definition 2.4. Then, for each r ∈ F∗ +q and M ∈ F2d +q , the Fourier +transform � +VQr(M) of the indicator function of the variety VQr in F2d +q +is as follows: +(i) Let d be even. Then we have +� +VQr(M) = δ0(M) +q ++ VQ∗ +r(M) +qd +− +1 +qd+1 . +(ii) Let d be odd. Then we have +� +VQr(M) = δ0(M) +q ++ η(r)VQ∗r (M) +qd +− η(r) +qd+1 . +After replacing � +VQr(M) in (5.2) by its explicit value given in Proposition 5.2, we can easily get +M(r) = |E|4 +q ++ q3d +� +M∈VQ∗r +| � +E × E(M)|2 − qd−1|E|2 +for d ≥ 2 even, +10 + +and +M(r) = |E|4 +q ++ q3dη(r) +� +M∈VQ∗r +| � +E × E(M)|2 − qd−1η(r)|E|2 +for d ≥ 3 odd. +We also need the following proposition to estimate the quantity w(0) in (5.1). +Proposition 5.3. Let E ⊂ Fd +q. For t ∈ Fq, we define w(t) as the number of pairs (x, y) ∈ E × E such that +||x − y||Q = t, namely, +w(t) := +� +x,y∈E: +||x−y||Q=t +1, +where || · ||Q is the standard distance function in Definition 2.4. Then the following statements hold true: +(i) If d is even and η (ε) = 1, then we have +0 ≤ w(0) ≤ |E|2 +q ++ q +3d +2 +� +m∈(SQ∗)0 +| �E(m)|2. +(ii) If d is even and η (ε) = −1, then we have +0 ≤ w(0) ≤ |E|2 +q ++ q +d−2 +2 |E|. +(iii) If d is odd, then we have +0 ≤ w(0) ≤ |E|2 +q ++ q +d−1 +2 |E|. +We postpone the proof of Proposition 5.3 to the following section and let us accept it for a moment. +Since W(r) = M(r) − w2(0), invoking (i),(ii),(iii) of Proposition 5.3 together with the estimates of M(r), +lower bounds of W(r) are obtained as follows: +(Case A) If d ≥ 2 is even and η(ε) = 1, then +(5.3) +W(r) ≥ |E|4 +q ++ q3d +� +M∈VQ∗r +| � +E × E(M)|2 − qd−1|E|2 − + +|E|2 +q ++ q +3d +2 +� +m∈(SQ∗)0 +| �E(m)|2 + + +2 +. +(Case B) If d ≥ 2 is even and η(ε) = −1, then we have +(5.4) +W(r) ≥ |E|4 +q ++ q3d +� +M∈VQ∗r +| � +E × E(M)|2 − qd−1|E|2 − +�|E|2 +q ++ q +d−2 +2 |E| +�2 +. +(Case C) If d ≥ 3 is odd, then we have +(5.5) +W(r) ≥ |E|4 +q ++ q3dη(r) +� +M∈VQ∗r +| � +E × E(M)|2 − qd−1η(r)|E|2 − +�|E|2 +q ++ q +d−1 +2 |E| +�2 +. +In the following subsections, we will complete the proof of Theorem 1.6 (i), (ii), (iii), by assuming that +Propositions 5.2 and 5.3 hold true. +5.1. Proof of Theorem 1.6 (i). Let d ≥ 2 be even. Suppose that E ⊂ Fd +q with |E| ≥ 4qd/2. It is clear that +η(ε) is either 1 or −1. +First, we find a lower bound of W(r) under the assumption of (Case A). To do this, we expand the last +term in (5.3) and observe that +q3d +� +M∈VQ∗r +| � +E × E(M)|2 − + +q +3d +2 +� +m∈(SQ∗)0 +| �E(m)|2 + + +2 +≥ 0 +and +� +m∈(SQ∗)0 +| �E(m)|2 ≤ +� +m∈Fdq +| �E(m)|2 = |E| +qd . +11 + +Combining these observations and the inequality (5.3), we get +W(r) ≥ |E|4 +q +− qd−1|E|2 − |E|4 +q2 +− 2q +d−2 +2 |E|3. +Since 1 = +5 +48 + 3 +48 + 16 +48 + 24 +48, we can write |E|4 +q += +5 +48 +|E|4 +q ++ 3 +48 +|E|4 +q ++ 16 +48 +|E|4 +q ++ 24 +48 +|E|4 +q . Hence, the lower bound +of W(r) can be written as +W(r) ≥ 5 +48 +|E|4 +q ++ +� 3 +48 +|E|4 +q +− qd−1|E|2 +� ++ +�16 +48 +|E|4 +q +− |E|4 +q2 +� ++ +�24 +48 +|E|4 +q +− 2q +d−2 +2 |E|3 +� +. +Since q ≥ 3 and |E| ≥ 4qd/2, each value in parentheses above is non-negative. Hence, we obtain the desired +result: +W(r) ≥ 5 +48 +|E|4 +q +. +Next, let us find a lower bound of W(r) under the assumption of (Case B). We also expand the last term +in (5.4) and then observe that the sum in (5.4) can be ignored since it is positive. Thus, we get +W(r) ≥ |E|4 +q +− qd−1|E|2 − |E|4 +q2 +− qd−2|E|2 − 2q +d−4 +2 |E|3. +Since 1 = 20 +48 + 3 +48 + 16 +48 + 1 +48 + 8 +48, we can write +W(r) ≥ 20 +48 +|E|4 +q ++ +� 3 +48 +|E|4 +q +− qd−1|E|2 +� ++ +�16 +48 +|E|4 +q +− |E|4 +q2 +� ++ +� 1 +48 +|E|4 +q +− qd−2|E|2 +� ++ +� 8 +48 +|E|4 +q +− 2q +d−4 +2 |E|3 +� +. +Since q ≥ 3 and |E| ≥ 4qd/2, it is not hard to notice that each value in parentheses above is non-negative. +Hence, in this case we get a better lower bound: +W(r) ≥ 20 +48 +|E|4 +q +≥ 5 +48 +|E|4 +q +. +This completes the proof of the first part of Theorem 1.6. +5.2. Proof of Theorem 1.6 (ii). Let d ≥ 3 be odd and E ⊂ Fd +q. Assume that r ∈ F∗ +q is a non-zero square +number. Then η(r) = 1. Thus, it follows from (5.5) of (Case C) that +W(r) ≥ |E|4 +q ++ q3d +� +M∈VQ∗r +| � +E × E(M)|2 − qd−1|E|2 − +�|E|2 +q ++ q +d−1 +2 |E| +�2 +. +Expand the above square term and notice that +� +M∈VQ∗r +| � +E × E(M)|2 ≥ | � +E × E(0, . . . , 0)|2 = |E|4 +q4d . +Then we see that +(5.6) +W(r) ≥ |E|4 +q ++ |E|4 +qd +− qd−1|E|2 − |E|4 +q2 +− qd−1|E|2 − 2q +d−3 +2 |E|3. +We now prove the statement of Theorem 1.6 (ii). Suppose that |E| ≥ 3qd/2. Since the term |E|4 +qd +in the RHS +of (5.6) is positive, we can ignore it when we find a lower bound of W(r). More precisely, we have +(5.7) +W(r) ≥ |E|4 +q +− 2qd−1|E|2 − |E|4 +q2 +− 2q +d−3 +2 |E|3. +Since 1 = +2 +45 + 10 +45 + 15 +45 + 18 +45, we can write |E|4 +q += +2 +45 +|E|4 +q ++ 10 +45 +|E|4 +q ++ 15 +45 +|E|4 +q ++ 18 +45 +|E|4 +q . Therefore, (5.7) becomes +W(r) ≥ 2 +45 +|E|4 +q ++ +�10 +45 +|E|4 +q +− 2qd−1|E|2 +� ++ +�15 +45 +|E|4 +q +− |E|4 +q2 +� ++ +�18 +45 +|E|4 +q +− 2q +d−3 +2 |E|3 +� +. +Since q ≥ 3 and |E| ≥ 3qd/2, each value in parentheses above is non-negative. Thus, the required estimate +is obtained: +W(r) ≥ 2 +45 +|E|4 +q +. +12 + +5.3. Proof of Theorem 1.6 (iii). Suppose that d is odd and r ∈ F∗ +q is not a square number, namely, +η(r) = −1. By (5.5) of (Case C), +W(r) ≥ |E|4 +q +− q3d +� +M∈VQ∗r +| � +E × E(M)|2 + qd−1|E|2 − +�|E|2 +q ++ q +d−1 +2 |E| +�2 +. +Expand the last square term above and note that +0 ≤ +� +M∈VQ∗r +| � +E × E(M)|2 ≤ +� +M∈F2d +q +| � +E × E(M)|2 = |E|2 +q2d . +When r ∈ F∗ +q is non-square, we therefore get +(5.8) +W(r) ≥ |E|4 +q +− qd|E|2 + qd−1|E|2 − |E|4 +q2 +− qd−1|E|2 − 2q +d−3 +2 |E|3. +On the other hand, when r ∈ F∗ +q is a square number, we already know from (5.6) that +(5.9) +W(r) ≥ |E|4 +q ++ |E|4 +qd +− qd−1|E|2 − |E|4 +q2 +− qd−1|E|2 − 2q +d−3 +2 |E|3. +Now let us compare the above two lower bounds of W(r). It suffices to compare only the second and third +terms of them since the rest of the terms are equal. Since q ≥ 3, it is not hard to see that +−qd|E|2 + qd−1|E|2 < |E|4 +qd +− qd−1|E|2. +This implies that the lower bound of W(r) in (5.8) is much smaller than that in (5.9). Thus, the inequality +(5.8) holds for all r ∈ F∗ +q. In other words, for all r ∈ F∗ +q, +(5.10) +W(r) ≥ |E|4 +q +− qd|E|2 − |E|4 +q2 +− 2q +d−3 +2 |E|3. +Now we are ready to finish the proof of Theorem 1.6 (iii). Assume that |E| ≥ 11 +6 q +d+1 +2 . Since 1 = +2 +363 + 108 +363 + +121 +363 + 132 +363, we can write +|E|4 +q += +2 +363 +|E|4 +q ++ 108 +363 +|E|4 +q ++ 121 +363 +|E|4 +q ++ 132 +363 +|E|4 +q +. +Combining this with the above inequality (5.10), we get +W(r) ≥ +2 +363 +|E|4 +q ++ +�108 +363 +|E|4 +q +− qd|E|2 +� ++ +�121 +363 +|E|4 +q +− |E|4 +q2 +� ++ +�132 +363 +|E|4 +q +− 2q +d−3 +2 |E|3 +� +. +Since q ≥ 3 and |E| ≥ 11 +6 q(d+1)/2, we see from a direct computation that each value in parentheses above is +non-negative. Hence, we obtain that W(r) ≥ +2 +363 +|E|4 +q , which proves Theorem 1.6 (iii). +6. Proof of Propositions 5.2 and 5.3 +To efficiently prove the propositions, we begin by deducing a preliminary lemma, regarding the Fourier +transform on the diagonal homogeneous variety of degree two in Fn +q . Denote x = (x1, . . . , xn), m = +(m1, . . . , mn) ∈ Fn +q . Let a = (a1, . . . , an) ∈ Fn +q such that aj ̸= 0 for all j = 1, 2, . . . , n. Consider an al- +gebraic variety +Ha := + + +x ∈ Fn +q : +n +� +j=1 +ajx2 +j = 0 + + + . +We call this variety Ha the diagonal homogeneous variety of degree two with a coefficient vector a in Fn +q . +We also define the dual variety of Ha, denoted by Ha∗, as +Ha∗ := + + +m ∈ Fn +q : +n +� +j=1 +a−1 +j m2 +j = 0 + + + . +13 + +Lemma 6.1. Let Ha be the diagonal homogeneous variety of degree two with a coefficient vector a in Fn +q . +Then, for m ∈ Fn +q , the Fourier transform on Ha is given as follows: +(i) If n is even, then +� +Ha(m) = q−1δ0(m) + q− n +2 η + +(−1)n/2 +n +� +j=1 +aj + + Ha∗(m) − q−(n+2)/2η + +(−1)n/2 +n +� +j=1 +aj + + . +(ii) If n is odd, then +� +Ha(m) = + + + + + +q−1δ0(m) +if m ∈ Ha∗, +q− n+1 +2 η +� +(−1)(n+3)/2 +n� +j=1 +aj +� +η +� +n� +j=1 +a−1 +j m2 +j +� +if m /∈ Ha∗. +Proof. The proof uses the standard orthogonality argument of characters and Gauss sum estimates. +It +follows that +� +Ha(m) = q−n � +x∈Ha +χ(m · x) += q−n � +x∈Fn +q + +q−1 � +s∈Fq +χ +� +s(a1x2 +1 + · · · + anx2 +n) +� + + χ(m · x) += q−1δ0(m) + q−n−1 � +x∈Fn +q +� +s̸=0 +χ +� +s(a1x2 +1 + · · · + anx2 +n) +� +χ(m · x) += q−1δ0(m) + q−n−1 � +s̸=0 +n +� +j=1 +� +xj∈Fq +χ(sajx2 +j + mjxj). +Summing over xj ∈ Fq by the formula (3.2), we get +(6.1) +� +Ha(m) = q−1δ0(m) + q−n−1Gnη(a1 · · · an) +� +s̸=0 +ηn(s)χ +� +− 1 +4s +�m2 +1 +a1 ++ · · · + m2 +n +an +�� +. +(i) Suppose that n is even. Since ηn(s) = 1, the sum over s ̸= 0 is (qHa∗(m) − 1) by application of the +orthogonality of χ. By Lemma 3.3, note that Gn = (η(−1)q)n/2 = η((−1)n/2)qn/2. So we obtain that +� +Ha(m) = q−1δ0(m) + q− n+2 +2 η((−1)n/2a1 · · · an) (qHa∗(m) − 1) . +Hence, the statement of Theorem (i) is proven. +(ii) Suppose that n is odd. +Then ηn = η, and so, if +m2 +1 +a1 + · · · + m2 +n +an += 0, namely m ∈ Ha∗, then +� +Ha(m) = q−1δ0(m) by the orthogonality of η. On the other hand, when m /∈ Ha∗, note by (3.3) that +the sum over s ̸= 0 in (6.1) is equal to η +� +− 1 +4 +� +m2 +1 +a1 + · · · + m2 +n +an +�� +G. Also note that η(−4−1) = η(−1). +Then we get +� +Ha(m) = q−n−1Gn+1η(−1)η(a1 · · · an)η +�m2 +1 +a1 ++ · · · + m2 +n +an +� +. +Since G2 = η(−1)q by Lemma 3.3, we see that +Gn+1η(−1) = (η(−1)q)(n+1)/2η(−1) = η +� +(−1)(n+3)/2� +q(n+1)/2. +Inserting this into the above equation, we obtain the required value of � +Ha(m). +□ +Now we are ready to prove Proposition 5.2, which will be a special case of Lemma 6.1 (i). +14 + +6.1. Proof of Proposition 5.2. Let x, x′ ∈ Fd +q. Then we see that +VQr = {(x, x′) ∈ F2d +q : ||x||Q − r||x′||Q = 0}. +(i) Suppose that d is even. +Then using Definition 2.4 (i), the coefficient vector of ||x||Q is a′ := +(1, −1, . . ., 1, −ε) ∈ Fd +q and that of −r||x′||Q is a′′ := (−r, r, . . . , −r, rε) ∈ Fd +q. Hence, VQr is the +diagonal homogeneous variety of degree two with a coefficient vector a = (a′, a′′) in F2d +q . Thus, with +n = 2d even, we can apply the part (i) of Lemma 6.1, where m = M, Ha = VQr, Ha∗ = VQ∗r, and +η + + +n +� +j=1 +aj + + = η +� +(−1)drdε2� += 1. +Consequently we obtain that +� +VQr(M) = q−1δ0(M) + q−dη((−1)d)η +� +(−1)drdε2� +VQ∗r(M) − q−(d+1)η((−1)d)η +� +(−1)drdε2� +. +Since d is even, each value of the above quadratic characters η is 1 and we complete the proof of +Proposition 5.2 (i). +(ii) Suppose that d is odd. +By Definition 2.4 (ii), we see that a = (a′, a′′) ∈ F2d +q +such that a′ = +(1, −1, . . ., 1, −1, ε) ∈ Fd +q, and a′′ = (−r, r, . . . , −r, r, −rε) ∈ Fd +q. Hence, the proof is the same as the +case of (i) except that for odd d, +η + + +n +� +j=1 +aj + + = η +� +(−1)drdε2� += η(−r). +Thus, the conclusion of Proposition 5.2 (ii) follows. +6.2. Proof of Proposition 5.3. Since w(0) := +� +x,y∈E: +||x−y||Q=0 +1, it follows from the general counting lemma +(Lemma 3.4) that +(6.2) +w(0) = q2d � +m∈Fdq +� +(SQ)0(m)| �E(m)|2, +where (SQ)0 = {x ∈ Fd +q : ||x||Q = 0}. We will invoke the following explicit estimate on the Fourier transform +on (SQ)0, which can be proven by Lemma 6.1. +Corollary 6.2. Let m ∈ Fd +q. +(i) If d is even, then +� +(SQ)0(m) = q−1δ0(m) + q−d/2η(ε)(SQ∗)0(m) − q−(d+2)/2η(ε). +(ii) If d is odd, then +� +(SQ)0(m) = + + + + + +q−1δ0(m) +if m ∈ (SQ∗)0, +q− d+1 +2 η(ε)η +� +d−1 +� +j=1 +(−1)j+1m2 +j + ε−1m2 +d +� +if m /∈ (SQ∗)0. +Proof. Let a = (a1, . . . , ad) ∈ Fd +q be the coefficient vector of ||x||Q, x ∈ Fd +q. Then it is clear that (SQ)0 is the +diagonal homogeneous variety of degree two with a coefficient vector a in Fd +q. +(i) Assume that d is even. Then, by definition of ||x||Q, we see that a = (1, −1, . . . , 1, −ε) in Fd +q. Since +d� +j=1 +aj = (−1)d/2ε, the first part of the corollary follows by applying Lemma 6.1 (i). +(ii) Assume that d is odd. Then a = (1, −1, . . ., 1, −1, ε) in Fd +q and so +d +� +j=1 +aj = (−1)(d−1)/2ε. +Now applying Lemma 6.1 (ii), we obtain the statement of the second part of the corollary. +15 + +□ +We are now ready to complete the proof of Proposition 5.3. By the definition of w(0), it is clear that w(0) +is a non-negative integer, namely, w(0) ≥ 0. So it remains to find the required upper bounds for w(0). +6.2.1. Proof of Proposition 5.3-(i), (ii). Suppose that d is even. Then inserting the value of � +(SQ)0(m) in +Corollary 6.2 (i) into the formula (6.2), we see from a direct computation that +w(0) = |E|2 +q ++ q3d/2η (ε) +� +m∈(SQ∗)0 +| �E(m)|2 − q(d−2)/2|E|η (ε) , +where we also used the Plancherel theorem (3.1) with the simple fact that �E(0, . . . , 0) = q−d|E|. Notice +that the sign of the second term above is different from that of the third term since η (ε) = ±1 and +� +m∈(SQ∗)0 +| �E(m)|2 ≥ 0. So, to estimate an upper bound of w(0), we can ignore the negative term which is +exactly one of them. By this way, we easily obtain the required upper bounds in Proposition 5.3 (i), (ii). +6.2.2. Proof of Proposition 5.3-(iii). Suppose that d is odd. It is clear from (6.2) that +0 ≤ w(0) ≤ q2d � +m∈Fdq +���� +(SQ)0(m) +��� | �E(m)|2. +On the other hand, Corollary 6.2 (ii) implies that +���� +(SQ)0(m) +��� = +� q−1δ0(m) +if m ∈ (SQ∗)0, +q− d+1 +2 +if m /∈ (SQ∗)0. +Thus, combining the above facts, we obtain the desired estimate as follows: +w(0) ≤ q2d � +m∈Fdq +q−1δ0(m)| �E(m)|2 + q2d � +m∈Fdq +q− d+1 +2 | �E(m)|2 += q2d−1| �E(0, . . . , 0)|2 + q2dq− d+1 +2 +� +m∈Fdq +| �E(m)|2 += |E|2 +q ++ q +d−1 +2 |E|. +References +[1] O. Ahmadi and A. Mohammadian, Sets with many pairs of orthogonal vectors over finite fields, Finite Fields Appl. 37 +(2016), 179-192. +[2] M. Bennett, J. Chapman, D. Covert, D. Hart, A. Iosevich, and J. Pakianathan, Long paths in the distance graph over large +subsets of vector spaces over finite fields, J. Korean Math. Soc. 53 (2016), no.1, 115-126. +[3] M. Bennett, D. Hart, A. Iosevich, J. Pakianathan, and M. Rudnev, Group actions and geometric combinatorics in Fd +q, +Forum Math. 29 (2017), 91-110. +[4] M. 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Parshall, Simplices over finite fields, Proc. Amer. Math. Soc. 145 (2017), no.6, 2323-2334. +[30] T. Pham, Group action and L2-norm estimates of geometric problems, preprint (2022), arXiv:2208.04827. +[31] D. H. Pham, T. Pham, and L.A. Vinh, An improvement on the number of simplices in Fd +q, Discrete Appl. Math. 221 +(2017), 95-105. +[32] I. E. Shparlinski, On the set of distance between two sets over finite fields, Int. J. Math. Math. Sci. Volume 2006, Article +ID 59482, Pages 1-5. +[33] I.E. Shparlinski, On the additive energy of the distance set in finite fields, Finite Fields Appl. 42 (2016), 187–199. +[34] J. Solymosi and V. Vu, Near optimal bounds for the number of distinct distances in high dimensions, Combinatorica, 28, +no.1 (2008), 113-125. +[35] L. A. Vinh, On kaleidoscopic pseudo-randomness of finite Euclidean graphs, Discuss. Math. Graph Theory 32 (2012), +no.2, 279-287. +[36] L. A. Vinh, On the generalized Erd˝os-distance problems over finite fields, J. Number Theory, 133 (2013), no.9, 2939-2947. +Department of Mathematics, University of Rochester, Rochester, NY 14627 USA +Email address: iosevich@math.rochester.edu +Department of Mathematics, Chungbuk National University, Cheongju, Chungbuk 28644 Korea +Email address: koh131@chungbuk.ac.kr +Department of Mathematics, University of Rochester, Rochester, NY 14627 USA +Email address: frakhmon@ur.rochester.edu +17 + diff --git a/NdE3T4oBgHgl3EQfZAra/content/2301.04494v1.pdf b/NdE3T4oBgHgl3EQfZAra/content/2301.04494v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..247dfad332d97b69c76707135338d4d6458510d6 --- /dev/null +++ b/NdE3T4oBgHgl3EQfZAra/content/2301.04494v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5d685dd29d762cf74b79aa4b85007d23918263c9ead66d019dd8dc9469c25e19 +size 32477288 diff --git a/ONE3T4oBgHgl3EQfxAvE/content/tmp_files/2301.04708v1.pdf.txt b/ONE3T4oBgHgl3EQfxAvE/content/tmp_files/2301.04708v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..817fa7f98fa87b571490aa47e69ad58385125e52 --- /dev/null +++ b/ONE3T4oBgHgl3EQfxAvE/content/tmp_files/2301.04708v1.pdf.txt @@ -0,0 +1,16098 @@ +Two-Photon EXchange - TPEX +R. Alarcon,1 R. Beck,2 J.C. Bernauer,3, 4 M. Broering,5 E. Cline,3 B. Dongwi,6 I. Fernando,6 +M. Finger,7 M. Finger Jr.,7 I. Friˇsˇci´c,5 T. Gautam,6 D.K. Hasell,5 O. Hen,5 J. Holmes,1 T. Horn,8 +E. Ihloff,5 R. Johnston,5 J. Kelsey,5 M. Kohl,6 T. Kutz,9 I. Lavrukhin,10 S. Lee,5 W. Lorenzon,10 +F. Maas,11 H. Merkel,11 R.G. Milner,5 P. Moran,5 J. Nazeer,6 T. Patel,6 M. Rathnayake,6 +R. Raymond,10 R.P. Redwine,5 A. Schmidt,9 U. Schneekloth,12 D. Sokhan,13 M. Suresh,6 and C. Vidal5 +(The TPEX Collaboration) +1Arizona State University, Tempe, AZ, USA +2Friedrich Wilhelms Universit¨at, Bonn, Germany +3Stony Brook University, Stony Brook, NY, USA +4Riken BNL Research Center, Upton, NY, USA +5Massachusetts Institute of Technology, Cambridge, MA, USA +6Hampton University, Hampton, VA, USA +7Charles University, Prague, Czech Republic +8Catholic University of America, Washington, DC, USA +9The George Washington University, Washington, DC, USA +10University of Michigan, Ann Arbor, MI, USA +11Johannes Gutenberg Universit¨at, Mainz, Germany +12Deutsches Elektronen-Synchrotron, Hamburg, Germany +13University of Glasgow, Glasgow, Scotland +(Dated: January 13, 2023) +We propose a new measurement of the ratio of positron-proton to electron-proton, elastic scat- +tering at DESY to determine the contributions beyond single-photon exchange, which are essential +to the QED description of the most fundamental process in hadronic physics. A 20 cm long liq- +uid hydrogen target together with the extracted beam from the DESY synchrotron would yield an +average luminosity of 2.12 × 1035 cm−2·s−1·sr−1 (∼ 200 times the luminosity achieved by OLYM- +PUS). A commissioning run at 2 GeV followed by measurements at 3 GeV would provide new data +up to Q2 = 4.6 (GeV/c)2 (twice the range of current measurements). Lead tungstate calorime- +ters would be used to detect the scattered leptons at polar angles of 30°, 50°, 70°, 90°, and 110°. +The measurements could be scheduled to not interfere with the operation of PETRA. We present +rate estimates and simulations for the planned measurements including background considerations. +Initial measurements at the DESY test beam facility using prototype lead tungstate calorimeters +in 2019, 2021, and 2022 were made to check the Monte Carlo simulations and the performance of +the calorimeters. These tests also investigated different readout schemes (triggered and streaming). +Various upgrades are possible to shorten the running time and to make higher beam energies and +thus greater Q2 ranges accessible. +CONTENTS +1. Introduction +3 +2. DESY +6 +3. Proposed Experiment +7 +4. Liquid Hydrogen Target and Scattering Chamber +9 +4.1. Towards a functional LH2 Target for TPEX +12 +5. Lead Tungstate Calorimeters +12 +6. GEM Detectors +13 +7. Luminosity and Beam Alignment Monitor +13 +8. Beamdump / Faraday Cup +16 +9. Electronics and Readout System +17 +arXiv:2301.04708v1 [nucl-ex] 11 Jan 2023 + +2 +9.1. Trigger +17 +9.2. Front end electronics +17 +9.3. Baseline DAQ hardware and software +17 +9.4. Possible improvements +18 +10. Upgrades / Improvements to the Proposed Experiment +18 +11. Background Considerations +19 +11.1. Protons from e±p elastic scattering +19 +11.2. Møller and Bhabha scattering +20 +11.3. Pion Production +21 +12. Monte Carlo Simulations +23 +12.1. Electrons and Positrons +25 +12.2. Protons +26 +12.3. Neutrons +27 +12.4. π+ +28 +12.5. π0 +29 +13. Test Beam at DESY +30 +14. Conclusion +30 +A. Test Beam Results +31 +1. Calorimeter Setup and Tests +31 +2. Streaming and triggered readout +31 +3. Monte Carlo Simulation of Test Beam +33 +B. Monte Carlo Simulation for e− + p → e− + p + π0 at 2 GeV +36 +C. Monte Carlo Simulation for e− + p → e− + n + π+ at 2 GeV +37 +D. Monte Carlo Simulation for e+ + p → e+ + p + π0 at 2 GeV +38 +E. Monte Carlo Simulation for e+ + p → e+ + n + π+ at 2 GeV +39 +F. Monte Carlo Simulation for e− + p → e− + p + π0 at 3 GeV +40 +G. Monte Carlo Simulation for e− + p → e− + n + π+ at 3 GeV +41 +H. Monte Carlo Simulation for e+ + p → e+ + p + π0 at 3 GeV +42 +I. Monte Carlo Simulation for e+ + p → e+ + n + π+ at 3 GeV +43 +J. Hydrogen Properties +44 +K. Lead Tungstate, PbWO4, Properties +45 +L. Numbers Used for Calculations in this Proposal +45 +References +45 + +3 +1. +INTRODUCTION +Elastic lepton-proton scattering is a fundamental process that should be well described by QED. Understanding +this interaction is important to the scientific programs at FAIR, Jefferson Lab, and the future electron-ion collider +(EIC) planned for Brookhaven. It is described theoretically in the Standard Model by a perturbative expansion in +α = +1 +137 with radiative corrections. For more than half a century it has been assumed that the leading single-photon +exchange term adequately describes the scattering process and that higher-order terms are negligible. +However, +recent experiments at Jefferson Lab have been widely interpreted as evidence that higher order terms are significant +in elastic electron-proton scattering and must be included to correctly extract the proton elastic form factors. Recent +experiments, including the OLYMPUS experiment at DESY, show little evidence for significant contributions beyond +single photon exchange up to Q2 ≈ 2.3 (GeV/c)2. It is essential that the QED expansion be studied experimentally at +higher Q2 comparing the positron and electron scattering cross section to determine the contribution of higher order +terms not normally included in radiative corrections. +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +] +2 + [(GeV/c) +2 +Q +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +p +M + / G +p +E + G +p +µ +Unpolarized Measurements +Janssens 66 +Berger 71 +Litt 70 +Bartel 73 +Andivahis 94 +Walker 94 +Christy 04 +Qattan 05 +Polarization Measurements +Jones 00 +Pospischil 01 +Gayou 02 +Punjabi 05 +Crawford 07 +Puckett 10 +Ron 11 +Puckett 12 +FIG. 1: Proton form factor ratio measured using unpolarized [1–8] (blue) and polarized [9–16] (red) techniques. +The proton form factors, Gp +E and Gp +M, have historically been envisaged as very similar and are often modeled +by the same dipole form factor. Measurements over the past 50 years using the unpolarized Rosenbluth separation +technique yielded a ratio, µp Gp +E/Gp +M, close to unity over a broad range in Q2 shown by the blue data points in Fig. 1. +This supported the idea that Gp +E and Gp +M are similar. However, recent measurements using polarization techniques +revealed a completely different picture with the ratio decreasing rapidly with increasing Q2 as shown by the red data +points in Fig. 1. +The most commonly proposed explanation for this discrepancy is “hard” two-photon exchange contributions beyond +the standard radiative corrections to one-photon exchange. Two-photon exchange, TPE, (see Fig. 2) is generally ++ ++ ++ +1 +FIG. 2: Feynman diagrams for one- and two-photon exchange. Further diagrams for bremsstrahlung, vertex, +self-energy, and vacuum polarization radiative corrections are not shown but must also be included in calculations. +included as part of the radiative corrections when analyzing electron-proton scattering. However, it is usually only + +4 +included in the “soft” limit where one of the two photons, in the diagrams with two photons, is assumed to carry +negligible momentum and the intermediate hadronic state remains a proton. To include “hard” two-photon exchange, +a model for the off-shell, intermediate hadronic state must also be included, making the calculations difficult and +model dependent. +In the Born or single photon exchange approximation the elastic scattering cross section for leptons from protons +is given by the reduced Rosenbluth cross section, +dσe±p +dΩ += dσ +dΩ Mott +τGp +M +2 + ϵGp +E +2 +ϵ(1 + τ) +, +(1) +where: τ = +Q2 +4M 2 +p and ϵ = (1 + 2(1 + τ) tan2 θl +2 )−1. +To measure the “hard” two-photon contribution, one can measure the ratio R2γ = σe+p/σe−p at different values +of Q2 and ϵ. Note, the interference terms between one- and two-photon exchange change sign between positron and +electron scattering and this cross section ratio provides a measure of the two-photon exchange contribution. +The results from the OLYMPUS experiment [17] are shown in Fig. 3 together with various calculations. +The +0.97 +0.98 +0.99 +1 +1.01 +1.02 +1.03 +1.04 +1.05 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +R2γ +ϵ +Main spectrometer +12◦ telescopes +Correlated uncertainty +Blunden N only +Blunden N + ∆ +Bernauer +Tomalak +2.0 +1.5 +1.0 +0.5 +0.0 +Q2 +[(GeV/c)2] +FIG. 3: OLYMPUS results for R2γ as a function of ϵ. Inner error bars are statistical while the outer error bars +include uncorrelated systematic uncertainties added in quadrature. The gray band is correlated systematic +uncertainty. +deviation of the results from unity are small, on the order of 1%, and are below unity at large ϵ and rising with +decreasing ϵ. The dispersive calculations of Blunden [18] are systematically above the OLYMPUS results in this +energy regime. The results below unity cannot be explained by current QED calculations. The phenomenological +prediction from Bernauer [19] and the subtractive dispersion calculation from Tomalak [20] are in better agreement +with the OLYMPUS results but appear to rise too quickly as ϵ decreases. There is some indication that TPE increases +with decreasing ϵ or increasing Q2, suggesting that a significant “hard” two-photon contribution might be present at +lower ϵ or higher Q2. +Two other experiments, VEPP-3 [21] and CLAS [22], also measured the “hard” two-photon exchange contribution +to electron-proton elastic scattering. It is difficult to compare the results from the three experiments directly since the +measurements are at different points in the (ϵ, Q2) plane. To partially account for this, we can compare all the two- +photon exchange results by taking the difference with respect to a selected calculation evaluated at the correct (ϵ, Q2) +for each data point. This is shown in Fig. 4a for Blunden’s calculation and in Fig. 4b for Bernauer’s phenomenological +prediction, plotted versus Q2. In these views, the results from the three experiments are shown to be in reasonable +agreement supporting the previous conclusions. +The results from the three TPE experiments are all below Q2 = 2.3 (GeV/c)2. In this regime the discrepancy in +the form factor ratios is not obvious, so the small “hard” TPE contribution measured is consistent with the measured + +5 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +] +2 + [(GeV/c) +2 +Q +0.04 +− +0.03 +− +0.02 +− +0.01 +− +0 +0.01 +0.02 +0.03 +0.04 +0.05 +γ +2 +th +-R +γ +2 +exp +R +2 +Difference with respect to Blunden ND vs Q +OLYMPUS +CLAS +VEPP3 +(a) +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +] +2 + [(GeV/c) +2 +Q +0.04 +− +0.03 +− +0.02 +− +0.01 +− +0 +0.01 +0.02 +0.03 +0.04 +0.05 +γ +2 +th +-R +γ +2 +exp +R +2 +Difference with respect to Bernauer vs Q +OLYMPUS +CLAS +VEPP3 +(b) +FIG. 4: Difference between the results from the three recent experiments and (a) Blunden’s N+∆ calculation and +(b) Bernauer’s prediction. +. +form factor ratios. The suggested slope with ϵ indicates TPE may be important at smaller ϵ or higher Q2. But, since +this slope appears to deviate from Bernauer’s phenomenological prediction, which fits the observed discrepancy, it +may also suggest that “hard” TPE, while contributing, may not explain all of the observed form factor discrepancy. +Recently, the OLYMPUS data has also been analyzed to determine the charge-averaged yield for elastic lepton- +proton scattering [23]. The result is shown in Fig. 5. This measurement is insensitive to any charge-odd radiative +0.9 +0.95 +1 +1.05 +1.1 +1.15 +1.2 +1.25 +0 +0.5 +1 +1.5 +2 +2.5 +σe++σe− +2 +/ σdipole +Q2 [GeV2/c2] +Bernauer +Kelly +Arrington 03 +Arrington 07 +OLYMPUS +FIG. 5: The charge-averaged yield for elastic lepton-proton scattering from the OLYMPUS experiment [23]. +corrections including “hard” two-photon exchange and thus provides a better measure of the proton form factors. The +data shown covers an important range of Q2 where the GM form factor changes slope. The calculations by Kelly [24] +and Arrington [25, 26] appear to be in better agreement with the data, but Bernauer’s global fit [19] should be redone +to incorporate all the OLYMPUS data. +The two-photon exchange diagram in the QED expansion for electron scattering is an example of the more generic +electroweak photon-boson diagram (see Fig. 6 where V = Z0, W ±, or γ) which enters into a number of fundamental +processes in subatomic physics. The γ − Z box is a significant contribution to the asymmetry in parity-violating +electron scattering and the γ − W ± box is an important radiative correction in β−decay which must be implemented +to extract Vud of the Standard Model from 0+ → 0+ super-allowed nuclear β-decays. A workshop [27] was held + +6 +at the Amherst Center for Fundamental Interactions in September 2017, attended by physicists from these different +subfields, to discuss the Electroweak Box. A white paper is in preparation. +FIG. 6: More general electroweak box diagram that is important in many fundamental nuclear physics processes. +The proton form factors are fundamental to hadronic physics. Understanding the QED expansion, the role of +two-photon exchange, and the scale of radiative corrections at higher Q2 will be crucial in future studies at FAIR, +JLab, EIC, and elsewhere. The charge-averaged yield eliminates all charge-odd radiative corrections including the +leading terms of two-photon exchange, which cannot be calculated with current theories. Measuring the ratio of +positron-proton to electron-proton scattering is sensitive to the charge-odd radiative corrections and insensitive to +the charge-even radiative corrections. Together they help to study radiative corrections and unravel the proton form +factors. TPEX, like OLYMPUS, will provide both these measurements at higher Q2. +The discrepancy in the form factor ratio has not been resolved and the role played by two-photon exchange continues +to be widely discussed within the nuclear physics community [27–31]. Further measurements and theoretical work on +the role of two-photon exchange on the proton form factors are clearly needed. However, measurements at higher Q2 +and smaller ϵ, where the discrepancy is clear and TPE are expected to be larger, are difficult as the cross sections +decrease rapidly. In addition, there are not many laboratories capable of providing both electron and positron beams +with sufficient intensity. +The best, and for the foreseeable future only, opportunity is at DESY. This proposal outlines an experiment that +could measure R2γ at Q2 up to 4.6 (GeV/c)2 or higher, and ϵ below 0.1 where the form factor discrepancy is clear +(see Fig. 1). Such an experiment would overlap with the existing OLYMPUS data as a cross-check and would map +out the two-photon exchange contribution over a broad range in Q2 and ϵ to provide data to constrain theoretical +calculations. +The following sections describe the proposed site for the TPEX experiment at DESY, the experimental configuration +with its liquid hydrogen target, lead tungstate calorimeters, GEM detectors, luminosity monitor, beamdump/Faraday +cup, electronics and data acquisition, and possible improvements. Sources of background are considered together +with solutions and Monte Carlo simulations are presented. The appendices give more background plots, properties of +hydrogen and lead tungstate, some useful numbers for this proposal, and references. +2. +DESY +One of the primary requirements for measuring R2γ is high intensity positron and electron beams at energies of +several GeV available for nuclear and particle physics applications. DESY is effectively the only high energy physics +laboratory currently capable of such intense positron beams. The DESY II synchrotron can provide extracted beams +of up to 30 nA of positrons and up to 60 nA of electrons at energies between 0.5 and 6.3 GeV with a bunch frequency +of 12.5 Hz. +A proposal, currently under consideration at DESY for an extension to the present test beam facility [32], to +include an extracted lepton beam from DESY II provides a unique opportunity to investigate two-photon exchange. +The extracted beam would only be available when DESY-II is not needed for the operation of PETRA III. For our +purposes the electron and positron beams would be used directly at 2.0 GeV and 3.0 GeV with an option for higher +energies in the future. +The current operation of PETRA III uses only electrons. That would restrict the availability of positrons to times +when PETRA III is not operating due to scheduled maintenance or shutdown periods. +Hopefully this is not an +insurmountable problem and we believe our experiment can be successfully carried out in the shutdown periods. +Commissioning can be done with just electrons if necessary. If the storage ring PETRA III is running in “top up” +mode (fills every ∼ 30 s) we would not be able to run parasitically. For “non-top up” mode (fills every ∼ 240 s) it +might be possible to have the extracted beam for TPEX between fills for PETRA III. +If the modification to the test beam facility in Hall 2 provides a new, extracted beam area; this would allow a +left/right symmetric detector arrangement that is much preferred for this proposal to reduce systematics and to + +V ++ +V ++ +Y +V +Elastic +Inelastic +Born +Coulomb distortions +Dispersion corr.7 +increase count rate. +This proposal requires a significant effort from DESY: +1. The positron production target has been removed. This would need to be reinstalled. +2. A new, extracted beam area would have to be assembled. Two options are possible: +A - Hall 2 +- The floor space is currently occupied by another group that would have to be relocated. +- The “kicker” would have to be moved from its current location on DESY II to one suitable for providing +beams to the new area. +- The shielding wall around DESY II would have to be disassembled and reassembled with a beamline +incorporated to the new area. +B - R-Weg +- The transfer line previously used for DORIS would have to be re-established for a new experimental +area. +- A new area, possibly a specially designed experimental area would have to be developed. +3. For both options beamline elements (quadrupoles, steering magnets, vacuum pumps, valves, collimators, beam +dump, etc.) would have to be found or produced and then installed. +4. The new extracted beam area would need shielding walls, infrastructure services like power and water, an access +maze with interlocks, and a new counting hut. +5. Everything would need to be surveyed and aligned and then tests performed to satisfy all safety requirements. +In addition to enabling the TPEX experiment, an extracted beam facility at DESY would allow other experiments, +detector development, and material studies to be performed. Another interesting experiment would be Deeply Virtual +Compton Scattering, DVCS. This could also use the TPEX liquid hydrogen target and lead tungstate calorimeters +but with a different configuration to allow the scattered lepton and recoil proton to be detected in coincidence. Other +nuclear physics measurements could also benefit from comparing electron and positron scattering. +3. +PROPOSED EXPERIMENT +The proposed experimental configuration has ten 5 × 5 arrays of lead tungstate crystals at polar angles of 30°, 50°, +70°, 90°, and 110° left and right of the beam axis with the front face of the calorimeter modules at a radius of 1 m from +the target. Other configurations are possible and can be optimized with Monte Carlo studies. A simple schematic for +this arrangement is shown in Fig. 7. The electron or positron beam enters the scattering chamber along the beamline +(upper-right) and passes through the 20 cm long liquid hydrogen target before exiting the scattering chamber into +another section of beam line leading to the beamdump. At ±8 deg there are 3 m long beampipes connecting the +scattering chamber to the lead collimators before the Cherenkov detectors used to monitor the luminosity. These +beamlines are under vacuum and are used to reduce the multiple scattering for the relatively low energy (30–50 MeV) +Møller and Bhabha scattered leptons. +Using just the central 3 × 3 array of the 5 × 5 array of crystals to define the acceptance yields a solid angle of +3.6 msr at each angle. With a 20 cm long liquid hydrogen target the acceptance covers ±5.7° in polar and azimuthal +angle thus data is averaged over a small range in Q2 and ϵ. +We propose to commission the experiment using 2 GeV electrons. We do this to debug the electronics, detectors, +and data acquisition system taking advantage of the relatively high cross section at 2 GeV. We would require about +2 weeks of beam time for this commissioning after the experiment was installed and surveyed. We would also like a +brief run (few days) with positrons to verify that the beam alignment and performance do not change with positron +running. The commissioning run (including a few days with positrons) would also allow a crosscheck of the OLYMPUS +data at 30°, 50°, and 70° and give a modest extension in Q2 up to 2.7 (GeV/c)2. +Table I shows Q2, ϵ, differential cross section, and event rate expected for one day of running for the proposed +left/right symmetric configuration with 2 GeV lepton beams averaging 40 nA on a 20 cm liquid hydrogen target and +using just the central 3 × 3 array of crystals to calculate the acceptance area. +The TPEX experiment proper would run at 3.0 GeV and would require approximately 6 weeks (2 weeks with +electrons and 4 weeks with positrons in total) to collect the required statistics. Table II shows Q2, ϵ, differential cross +section, and event rate expected for one day of running for the proposed configuration with 3 GeV lepton beams. + +8 +FIG. 7: Geant4 simulation of a proposed TPEX target, scattering chamber, and detector configuration including the +luminosity monitors and beamlines. The lepton beam would enter through the beamline in the upper-right, traverse +the target cell, and scatter into the detectors or continue straight to the beamdump. +θ +Q2 +ϵ +dσ/dΩ +Events/day +(GeV/c)2 +fb +30° +0.834 +0.849 +2.41 × 107 +3.16 × 106 +50° +1.62 +0.611 +7.66 × 105 +1.01 × 105 +70° +2.19 +0.386 +1.00 × 105 +1.32 × 104 +90° +2.55 +0.224 +2.81 × 104 +3.70 × 103 +110° +2.78 +0.120 +1.22 × 104 +1.61 × 103 +TABLE I: Kinematics, cross section, and events expected in one day for an incident lepton beam of 2 GeV and +40 nA averaged current on a 20 cm liquid hydrogen target. +This would extend the measurements to Q2 = 4.57 (GeV/c)2 where the form factor ratio discrepancy is large. The +6 weeks could be divided into two three-week periods if that was more convenient. To minimize systematic we would +like to switch between positron and electron running as frequently as possible (e.g. 1 day positron, 1 day electron, +and 1 day positron repeating). +θ +Q2 +ϵ +dσ/dΩ +Events/day +(GeV/c)2 +fb +30° +1.69 +0.825 +2.41 × 106 +3.16 × 105 +50° +3.00 +0.554 +6.51 × 104 +8.55 × 103 +70° +3.82 +0.329 +8.94 × 103 +1.17 × 103 +90° +4.29 +0.184 +2.65 × 103 +3.48 × 102 +110° +4.57 +0.096 +1.20 × 103 +1.58 × 102 +TABLE II: Kinematics, cross section, and events expected in one day for an incident lepton beam of 3 GeV and +40 nA averaged current on a 20 cm liquid hydrogen target and 3.6 msr acceptance and a left/right symmetric +detector configuration. +The Q2 range that the proposed TPEX experiment would be capable of reaching is shown in Fig. 8 for the 2 and +3 GeV runs of this proposal. The reach with TPEX can be seen in relation to the discrepancy in the form factor +ratio. With additional crystals at back angles the 4 GeV runs would also be possible in a reasonable time frame. +The TPEX experiment at DESY would also measure the charge-averaged cross section just like the recent result +from OLYMPUS Fig. 5. +As mentioned above this cross section is insensitive to charge-odd radiative corrections +including “hard” two-photon exchange terms. Thus, it provides a more robust measure of the proton form factors. + +9 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +] +2 + [(GeV/c) +2 +Q +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +p +M + / G +p +E + G +p +µ +TPEX Range +2 GeV +3 GeV +4 GeV +Unpolarized Measurements +Janssens 66 +Berger 71 +Litt 70 +Bartel 73 +Andivahis 94 +Walker 94 +Christy 04 +Qattan 05 +Polarization Measurements +Jones 00 +Pospischil 01 +Gayou 02 +Punjabi 05 +Crawford 07 +Puckett 10 +Ron 11 +Puckett 12 +FIG. 8: Proton form factor ratio as before but also showing the Q2 range accessible with the proposed TPEX +configuration at 2 and 3 GeV. The 4 GeV range would be possible with additional crystals. +The expected charge-averaged cross section uncertainties (assuming dipole cross section) are shown in Fig. 9 for TPEX +assuming 6 days of running at 2 GeV and 6 weeks of running at 3 GeV with only 50% data collection efficiency. The +recent OLYMPUS results are also shown. +4. +LIQUID HYDROGEN TARGET AND SCATTERING CHAMBER +The OLYMPUS experiment that previously ran on the DORIS storage ring at DESY used an internal gas target with +typical areal density of 3 × 1015 atoms·cm−2. The lepton current averaged around 60 mA, yielding an instantaneous +luminosity about 1.12 × 10−6 fb−1· s−1. +For this new experiment we propose to build a liquid hydrogen target that will yield a luminosity about a factor of +200 times higher than that of the OLYMPUS experiment. This higher luminosity will greatly shorten the run time +needed at 2 GeV and help to make up for the lower cross section at higher beam energies. +TABLE III: Target system requirements. +Parameter +Performance Requirements +Liquid hydrogen +T≈20 K and P≥1 atm +Cool down time +< 3 hrs. +Exit windows +scattering into 25° – 120° and 7° – 9° allowed +Target Cell +end cap wall thickness tc ≤ 0.5 mm, +inner diameter 10 mm < i.d. < 20 mm, +wall thickness tw ≤ 1 mm, +20 cm in length +In order to satisfy the science needs for TPEX, and the safety requirements that always have to be taken into +consideration for liquid hydrogen targets, we propose to build a liquid hydrogen target system that is tailored for this +new experiment. The experimental requirements for the target system, detailed in Table III, include a single, 20-cm +long liquid hydrogen target with an areal density of 8.46×1023 atoms·cm−2 that can accommodate lepton currents up + +10 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +] +2 + [(GeV/c) +2 +Q +0.95 +1 +1.05 +1.1 +1.15 +1.2 +dipole +σ + / +averaged +σ +Charge Averaged Cross Section / Dipole +Measurements +Olympus +TPEX 2 GeV +TPEX 3 GeV +FIG. 9: Charge-averaged cross section divided by the dipole cross section from OLYMPUS and expected +uncertainties and coverage from TPEX at 2 and 3 GeV. +to 60 nA (30 nA for positrons and 60 nA for electrons); and long, thin scattering chamber windows to allow particles +to be accepted over the large solid angle subtended by the detectors. Thus, the TPEX cryotarget system requires +appropriate engineering and safety considerations. The Michigan group plans to work closely with the MIT-Bates +engineers and an external company, Creare, Inc., to design and fabricate the system. +Figure 10 presents the conceptual design of the target system. Panel (a) shows a schematic overview of the target +system, which consists of the scattering chamber, the cryo-cooler system, and the 20 cm long, and 2 cm wide single +target cell. Details of the target system are shown in panels (b) and (c). In order to maximize rigidity and withstand +the enormous force from atmospheric pressure, as well as to avoid welded and bolted joints, we propose to machine +the scattering chamber from a single piece of aluminum. +The dimensions of the scattering chamber windows shown in Fig. 10a are determined from the solid angle subtended +by the calorimeters. The two side exit windows cover the polar angles for the PbWO4 crystal calorimeters in the +range of 25° < θ < 120°. At the end of the 3 m long beampipes leading to the luminosity monitors are two tiny exit +windows cover a range of 7° < θ < 9°. The vertical dimensions of the two side exit windows cover an azimuthal angle +of φ = 0° ± 10°. +A schematic drawing of the 20 cm long target cell is shown in Fig. 11. At the time of this proposal, it had not yet +been decided if the target cell diameter should be 10 mm or 20 mm. This will in part depend on the lepton beam +properties. The general aim is to minimize the target cell diameter to restrict the amount of hydrogen present in the +target system, while minimizing heat load in the cell walls by the beam halo. The cell walls are made of 0.25 mm +thick, drawn aluminum tubes, similar to those used for cigar tube travel cases. It is expected that the entire target +system contains not more than 60 gas liters (or 75 ml LH2) of hydrogen gas. +The maximum beam heat load for the 3 GeV electron/positron beam impinging on the 20 cm long cell at 60 nA is +H = 60 nA · 0.070 g/cm3 · 20 cm · 30 MeV/(g/cm2) = 2.5 W. The thermal, or radiation heat load on the 20 cm long +and 20 mm diameter target cell is about 730 mW/(n + 1), where n is the number of superinsulation layers wrapped +loosely around the cell. So, for the expected 10 layers of superinsulation, the thermal heat load will be approximately +70 mW, which is much smaller than the beam heat load. We expect some bubble formation in the liquid H2 due to the +heat load from the lepton beams. The long, rectangular slot in the target cell, shown in Fig. 11, allows the bubbles +to escape into the top aluminum block. This helps to minimize density fluctuations and target thickness variations. +The entrance and exit cups of the target cell will be thinned by chemical etching to reduce the amount of material in +the beam, and thus the background caused by e± scattering off the target cell. + +11 +(a) +(b) +(c) +FIG. 10: Conceptual design of the TPEX target chamber: (a) shows the full chamber view with the lepton beam +entering from the left; (b) is a sectional drawing of the cryocooler system (1 – CH110-LT cryocooler, 2 – hydrogen +supply and exhaust lines, 3 – condenser with a cooling loop, 4 – target cell), and (c) is the top view of the target +chamber. +FIG. 11: Design overview of the 20 cm long target cell: 1 – top block with liquid hydrogen level sensor, 2 – target +cell, 3 – bottom block with temperature sensor and heater. + +0 +O +012 +Liquid hydrogen will be filled through a single fill tube that serves as the return tube for the boiled off hydrogen. +The fill/return tube connects the condenser, which is bolted to a cryo-cooler, with the top aluminum block also shown +in Fig. 11. This aluminum block also houses two liquid hydrogen level sensors (with one serving as a backup). Each +sensor is a 100 Ω Allen Bradley carbon resistor driven at 20 mA. One Lakeshore Cernox® thin film resistance cryogenic +temperature sensor and one (50 Ω, 50 W) cartridge heater are inside the bottom copper block. The temperature sensor, +the level sensors, and the heater are all monitored and controlled by a slow control system similar to that used in the +MUSE experiment [33]. +The cryo-cooler/condenser combination will closely follow the successful MUSE design [33]. We will therefore use +the CH110-LT single-stage cryo-cooler from Sumitomo Heavy Industries Ltd [34] for refrigeration. This cryo-cooler, +in combination with the Sumitomo F-70 compressor [35], was chosen for MUSE [36] over Cryomech partly because +Sumitomo has a service center in Darmstadt, Germany, while Cryomech does not have a service center in Europe. As +shown in Fig. 43, the cryo-cooler has a cooling power of 25 W at 20 K, which is more than sufficient to cool down and +fill the 70 ml LH2 target cell in approximately 2 hours [33]. +Geant4 Monte Carlo simulations will be performed for this conceptual design to verify that the experimental +requirements can be met. These simulations should tell us whether the basic cell design is acceptable, or whether +modifications to the scattering chamber exit windows are needed to reduce background. +4.1. +Towards a functional LH2 Target for TPEX +The Michigan group plans to work closely with the MIT-Bates engineers and an external company, Creare, Inc., +to design and fabricate the system. The U-M group will start with the current conceptual design, and improvements +informed by Geant4 simulations as well as the many lessons learned from building the cryogenic targets for the MUSE +and SeaQuest experiments, to complete the engineering design. This will be done in close collaboration with the MIT- +Bates engineers, and the external company, Creare1, who has performed the engineering design and the construction +of the MUSE target system. It is anticipated that this process will take about 6 months to allow sufficient time to +include periodic reviews by the DESY for compliance and safety issues. +Safety precautions and the lack of a fully developed slow control system at Creare will not allow a full-blown +cool-down test with LH2 to be performed before shipment to DESY. Instead a cool-down test with neon, which has a +similar boiling point as hydrogen but is not explosive, has to be performed. Once general cool-down performance and +target operation in vacuum, near 20 K, has been demonstrated, the target system will be shipped to DESY where the +neon test will be repeated in a staging area to ensure that all components are still functioning properly. A cool-down +test with about 5 cm3 of hydrogen, an amount small enough to be safe even if it exploded in the cryostat vessel, will +then be performed to start testing the slow control system and the safety procedures. Finally, a complete integration +test in the Hall 2 testbeam area to fully test all components, including slow controls and safety procedures will be +performed before starting the production run in the 2022. +5. +LEAD TUNGSTATE CALORIMETERS +For the proposed experiment we are leveraging the R&D experience [37, 38] from the CMS experiment and sub- +sequent applications by the Bonn and Mainz groups at CEBAF [39] and for PANDA [40]. We would start with ten +5 × 5 arrays of lead tungstate (PbWO4) crystals for a total of 250 crystals, some of which may be available from +Mainz. Other configurations are possible and will be investigated with more detailed Monte Carlo simulations. +Some properties for lead tungstate are provided in appendix K. We plan to use crystals 2×2×20 cm3. The density +is 8.3 g·cm−3 so each crystal weighs around 664 g, or 16.6 kg for a 5 × 5 array of crystals. Lead tungstate has a +radiation length X0 = 0.8904 cm, so these crystals are approximately 22.5 X0 for good longitudinal electromagnetic +shower confinement. The Moli`ere radius is 1.959 cm, so using just the central 3 × 3 array of crystals for acceptance, +the outer ring of crystals contains the transverse shower adequately. The nuclear interaction length for lead tungstate +is λI = 20.28 cm, so the crystals are roughly 0.986 λI. Again, other configurations are possible and further studies +and simulations are in progress. The energy resolution obtained with lead tungstate for the lepton energy range of +interest is approximately 2%. The Mainz Panda readout design uses Avalanche Photo-Diodes (APD). We are also +considering SiPM and PMT readout schemes. +1 Creare is a relatively large Small Business of approximately 150 people, including 60 engineers, 50 technicians, machinists and technical +specialists, and an in-house machine shop that is accustomed high-demanding high tolerance work. + +13 +Lead tungstate resolution varies with temperature. To achieve the best energy resolution, the crystal arrays should +be maintained at a constant temperature. The best energy resolution has been obtained at −25° C. This requires +refrigeration and complicates what would otherwise be very simple and compact calorimeter modules. Results from +the test runs at the DESY test beam facility on prototype lead tungstate calorimeters will be used to determine +whether or not such cooling is required or if adequate resolution can be obtained with more modest cooling to have +a stable temperature. +An alternative to lead tungstate is being investigated by T. Horn at Catholic University of America. +She is +developing high-density, ceramic glass crystals. These would be approximately 15% less dense than lead tungstate, so +a larger crystal might be required. But the ceramic glass is much easier to produce and would be 5–10 times cheaper. +In addition, the ceramic glass is not as sensitive to temperature, which would simplify the design. We will be testing +both lead tungstate and ceramic glass in the future. +Clearly, the lead tungstate crystals will be a crucial part of the TPEX experiment. +It will therefore be very +important to test and maintain the quality of the crystals whether they are produced by the firm Crytur in the Czech +Republic or obtained from existing supplies in Europe or America. The collaboration has colleagues from Charles +University in Prague, Czech Republic who have volunteered to take responsibility for testing the crystals, verifying +the quality and maintaining a database for tracking the crystals from delivery to the final calorimeter modules. +6. +GEM DETECTORS +It is proposed to stack two Gas Electron Multiplier (GEM) detector layers with two-dimensional readout in front +of each calorimeter. Thin absorbers will be placed between the target and the GEMs to stop low-energy Møller or +Bhabha leptons. The GEMs provide spatial information of the traversing charged particle at the 100 micrometer +precision level. The hits on two GEM elements are used to form a track segment providing directional information +between the impact point on the calorimeter and the event origin in the target. This serves to suppress charged- +particle backgrounds from regions other than the target. Also, the GEMs are insensitive to neutral particles, hence +they provide a veto against photons and neutrons. Using the calorimeter hit as a third tracking point will allow a +measure of the efficiency of each. +An active area of slightly more than 20x20 cm2 is required to fully cover the area of the calorimeter entrance. +A total of 20 elements is required to instrument ten calorimeter arms. The standard readout strip pitch of 0.4 mm +results in 500 channels per axis, or 1,000 channels per GEM element. The full experiment would have 20,000 channels. +Since the occupancy will be at the few percent level at most, zero suppression will reduce the amount of recorded +data substantially. +The Hampton group has developed GEM detectors for OLYMPUS, MUSE and DarkLight. Recently, the group +has established the novel scheme (NS2) of assembling GEM detectors without gluing, while stretching GEM foils +mechanically within a double frame structure, for the first time for nuclear physics applications. The scheme makes +the assembly fast and low risk, such that even a larger number of GEM elements can be produced fairly easily. +7. +LUMINOSITY AND BEAM ALIGNMENT MONITOR +The relative luminosity between the electron and positron running modes is the crucial normalization for the +proposed measurement. The luminosity could be monitored by a pair of small-angle detectors positioned downstream +on either side of the beamline. This approach was also used in the OLYMPUS experiment [41], and based on the +lessons learned from that experiment, could be improved substantially. Given the running conditions of the proposed +measurement, we favor a pair of quartz Cherenkov counters positioned 8° from the beamline to monitor the rates of +Møller and Bhabha scattering from atomic electrons in the target. +In OLYMPUS, the most accurate determination of the relative luminosity was obtained from the rates of multi- +interaction events—in which a Møller or Bhabha event occurred in the same bunch as a forward elastic e±p event [42]. +This method had an overall uncertainty of 0.36% and looked promising for future measurements. Unfortunately it +is not feasible for the proposed measurement because of the higher rate per bunch crossing, as seen in Fig. 12. The +multi-interaction event method requires that the event per bunch rate to be much less than unity. However, the +monitors for the proposed measurement will see approximately 104 Møller or Bhabha events per bunch crossing. +Instead, the proposed monitor can work by integrating the signal from all particles produced during each bunch. A +monitor placed at 8° has a number of advantages relative to the 1.3° placement of the OLYMPUS luminosity monitors. +First, at 8°, the Møller and Bhabha cross sections are only a few percent different, whereas for the OLYMPUS monitors, +which covered the symmetric angle (90° in the center-of-mass frame), the two cross sections differed by over 50%, +with significant angular dependence. Second, the Møller/Bhabha rate completely dwarfs the e±p elastic scattering + +14 +10−8 +10−7 +10−6 +10−5 +10−4 +10−3 +10−2 +2◦ +4◦ +6◦ +8◦ +10◦ +12◦ +14◦ +100 +101 +102 +103 +104 +105 +2 GeV +3 GeV +TPEX monitors +OLYMPUS +monitors +OLYMPUS +TPEX +Scattering angle +Counts per bunch +ep elastic +Møller +Bhabha (total) +FIG. 12: Whereas the forward monitors in OLYMPUS had an event per bunch rate well below 1, the TPEX +monitors will see 104 Møller or Bhabha events per bunch crossing. +PMT +PMT +Target +3 m +Collimator +Quartz Cherenkov +Radiator +1 cm radius +aperture +(Not to scale) +FIG. 13: Schematic for the proposed luminosity monitor, consisting of two quartz Cherenkov detectors with an +acceptance defined by 1 cm radius apertures in high-Z collimators +rate, meaning that it is really only sensitive to QED processes. No form factors or any other hadronic corrections2 +are needed to calculate the Møller and Bhabha cross sections. Third, the sensitivity to alignment scales as 1/ sin θ, +meaning the monitor will be much more robust to small misalignments, which were a significant problem for the +OLYMPUS luminosity monitor. +To test the feasibility of the proposed luminosity monitor, we have developed a preliminary design, and run a +Monte Carlo simulation to test the sensitivity to misalignments and beam position shifts, which were the dominant +systematic errors for the OLYMPUS luminosity determination [42]. A schematic of the design is shown in Fig. 13. +The monitor consists of two quartz Cherenkov detectors, which act as independent monitors. Cherenkov detectors +were chosen because they are widely used for monitoring in high-rate applications, such as in parity-violating electron +scattering [43, 44]. +The two detectors operate independently and can cross-check each other, helping to reduce +systematic errors from beam alignment. In this design, the monitors are placed 3 m away from the center of the +target, along the 8° scattering angle. The acceptance is defined by a collimator with a circular aperture with a radius +of 1 cm. +2 other than the radiative correction from vacuum polarization + +15 +−0.6% +−0.4% +−0.2% +0% +0.2% +0.4% +0.6% +−800 +−600 +−400 +−200 +0 +200 +400 +600 +800 +Bias: 0.21 ± 0.01 %/mm +Bias in Le+/Le− +Asymmetric Beam Offset [µm] +Single Monitor +(a) +−0.6% +−0.4% +−0.2% +0% +0.2% +0.4% +0.6% +−800 +−600 +−400 +−200 +0 +200 +400 +600 +800 +Bias: 0 ± 0.009 %/mm +Bias in Le+/Le− +Asymmetric Beam Offset [µm] +With Both Monitors +(b) +FIG. 14: The potential bias from a charge-asymmetric beam misalignment is a mere 0.2%/mm in a single monitor +(a), and this is completely eliminated by using a pair of monitors (b). +−0.8% +−0.6% +−0.4% +−0.2% +0% +0.2% +0.4% +0.6% +0.8% +−20 +−10 +0 +10 +20 +Bias: 0.0235 ± 0.0004 %/mm +Bias in Le+/Le− +Error in Collimator Position [mm] +Single Monitor +FIG. 15: Errors in the positioning of the collimator aperture also have a minimal impact on the relative luminosity +determination. +Fig. 14 shows the effect on the luminosity determination from beam misalignment. The most pernicious misalign- +ment would be one that is asymmetric between electron and positron modes, and so this was the focus of this study. +When using a single monitor, an asymmetric misalignment would cause a mere 0.21 ± 0.1 %/mm bias in the determi- +nation of the relative luminosity. When using the combination of both a left and right monitor, this bias is completely +eliminated to the uncertainty of this simulation. For comparison, the OLYMPUS luminosity monitor was sensitive +to asymmetric misalignments at the level of 5.7 %/mm, and it was estimated that the beam position monitors could +control the asymmetric misalignment to within 20 µm [42]. Such control will probably not be possible in the proposed +measurement due to the much smaller beam current. However the simulation demonstrates that such control is not +needed. This is largely due to the flatness of the Møller and Bhabha cross sections at 8° (see Fig. 12). One downside +of this robustness is that the proposed monitors are not very sensitive as beam alignment monitors. +Fig. 15 shows the simulation results for the effect on the luminosity determination if one of the collimator apertures +were to be positioned in a different place than expected. The maximum effect occurs when the collimator is shifted to +a larger or smaller scattering angle, and so this was the focus of the study. For a single monitor, the effect is a mere +0.02 %/mm. In practice, both monitors can have positioning errors, and these effects could end up adding or partially +canceling. Regardless, because of the positioning at 8◦, the bias is minimal. For comparison, the effect on OLYMPUS + +16 +was approximately 0.13 %/mm, with a survey accuracy of approximately 0.5 mm. Based on the experience gained +from OLYMPUS, we can make improvements in the collimator positioning. One obvious improvement is to include +integral survey marks on the collimator itself, since the aperture defines the monitor acceptance. +The simulation shows that two of the major systematic limitations of the OLYMPUS luminosity monitor will be +minimal for the proposed design. The third major systematic, stemming from the residual magnetic field along the +beamline, will be irrelevant. The proposed design does have other systematic limitations. The biggest concern will be +the amplitude stability of the photomultiplier. With about 104 particles passing through the aperture every bunch +crossing, there is no way to calibrate the light yield from the data itself. To guard against gain drifts, an external +calibration source will be vital. A pulsed light source, e.g. a UV laser, coupled to a fiber-optic distribution system can +be used to monitor the gain of both photomultipliers throughout the experiment, while an independent photodiode +can be used to cross check that the laser intensity itself does not drift. Several of us have experience with laser +calibration systems used in previous experiments [45, 46]. Sub-percent level accuracy should be achievable, though +this will almost certainly be the limiting systematic effect. +8. +BEAMDUMP / FARADAY CUP +A new extracted beam facility from DESY II will need a beamdump. M. Schmitz (DESY) and C. Tschalar (MIT) +have looked into the requirements and proposed very similar configurations with an aluminum core and a copper shell. +M. Schmitz’s design was an aluminum cylinder 10 cm in diameter and 50 cm long embedded in a copper shell 22 cm +in diameter and 65 cm long and had water cooling. C. Tschalar’s design was larger with 20 cm diameter and 50 cm +long aluminum in a 32 cm diameter and 75 cm long copper shell but was air cooled. Both recommended that the +beamdump be surrounded by neutron absorbing material like cement blocks or borated polyethylene. +Assuming a maximum current of 100 nA and a beam energy of 7 GeV the maximum power to be handled is 700 W. +To contain the showering you want order of 5 Moli´ere radii laterally and order of 25 radiation lengths longitudinally. +To be conservative we have selected C. Tschalar’s numbers as a starting point Fig. 16. To augment the luminosity +Beam Dump and Faraday cup +Copper +Aluminium +Insulator +Negative Voltage +secondary emission +Window +Vacuum +Water cooling +FIG. 16: Schematic of a possible beamdump / Faraday cup for TPEX +measurement proposed above we thought it might be useful to modify the beamdump to function as a Faraday cup as + +17 +well to integrate the charge that passes through the target. Then, assuming the length of the target cell and density +of liquid hydrogen are known we can get a quick measure of the luminosity. As shown in the figure an insulated ring +held at negative voltage of a few hundred volts is needed to suppress secondary emission from back scattering out of +the Faraday cup. The beamdump / Faraday cup is under vacuum but this need only be roughing vacuum pressure. +9. +ELECTRONICS AND READOUT SYSTEM +The requirements for data acquisition are comparatively modest. Less than 300 channels have to be read out, +including the luminosity monitor. With a readout at a low and fixed frequency, no busy logic is required. +9.1. +Trigger +The beam has a very low and fixed bunch frequency of 12.5 Hz, allowing us to trigger on the bunch clock instead of +a trigger detector. The only complication is that for proper gate alignment, the beam bunch signal has to be shifted +by up to 80 ms, with stability on the 10 ns level. This can be achieved with an FPGA, which can also generate the +required gates. A V2495 module from CAEN is available in the collaboration and would be adequate. +9.2. +Front end electronics +For the calorimeter and luminosity monitors, all proposed readout devices require the acquisition of a time-integrated +current pulse with a QDC. However, they differ in the required front end electronics: +• PMTs require only a base for the high-voltage distribution. Active bases can minimize power losses and improve +stability. Such bases are available commercially, or can be manufactured by the collaboration. Circuit designs +are readily available and can be adapted to fit the required form factor. No further signal conditioning (except +attenuation in the case of the luminosity monitors) is required for interfacing with standard QDCs. +• APDs and SiPMs require driver and preamplifier circuits. In the case of APDs, we would copy the tested design +for the PANDA detector [47] from Mainz. For SiPMs, the MUSE collaboration produced an amplifier design +which could be adapted [48]. +9.3. +Baseline DAQ hardware and software +In addition to the V2495 for trigger generation, the baseline configuration for 250+2 detectors would require eight +32-channel QDC modules like CAEN’s V792 or Mesytec’s MQDC-32. This can be housed in a single VME crate and +read out via a single board computer (SBC) as the VME controller. +The data rates are low, with about 6 kB/s for the readout of 252 channels at 12.5 Hz. This makes it possible to +store all experiment data to a single server outside of the experimental area via standard network file systems. This +rate requires less than 4 GB per week. +The SBU group already developed the DAQ software for the test beams, which will be basis for the DAQ solution +of the actual experiment. +For the GEMs, multiple readout solutions are possible: +• APV based readout, based on the MPD-4 readout boards already used for SBS@JLAB and MUSE [36]. While +APVs are out of production, these collaborations have a significant number of APV readout cards and MPDs +available, and experience in operating these components. +• SAMPA based readout. This is currently in development at JLab for TDIS and other projects. Compared to +GEMs, the signal quality is better and the wave form can be sampled as well. There has been a lot of progress +on the testing of the chips, which will be in production for the foreseeable future. However, a switch would +require the procurement of new hardware. +• VMM based readout. The VMM chip is considered to be the successor to the APV chip in the Scalable Readout +System (SRS). This new development is cost-effective and scalable, and has been adopted and recommended +by RD-51 in the framework of the SRS readout scheme. The Mainz MAGIX collaboration recently decided + +18 +to start using VMM for their GEM readout at MESA. The VMM offers readout with time and pulse shape +digitization directly on the front-end card. Two VMM chips are housed on one front-end card to process 128 +readout channels. +The data rate from the GEMs is considerably higher, but still manageable, particularly with zero suppression. With +1,000 channels per detector and 20 detectors, the estimated rate is between 500 kB/s (1 sample per event and channel) +to 5MB/s (10 samples per event and channel), resulting in about 1.5 TB per week of beam time. With zero suppression +this could be reduced to a level of 300 GB per week. +9.4. +Possible improvements +We are evaluating multiple improvements over this baseline design: +• Higher trigger rate: Instead of triggering just on the beam clock, the FPGA can generate additional gates +before and after each trigger, spaced so that all conversion and data transmission can happen before the real +gate opens. This triples the trigger and thus data rate—easily handled by the proposed system—but would +allow for baseline and background monitoring. +• Instead of QDCs, which only give information about the integrated charge, the signal wave forms could be +digitized with high speed ADCs. This would allow even better baseline control, but would increase the bandwidth +considerably. For example, sampling the signals for 1 µs with 250 MHz at 14 bit would result in about 6 kB/s +per channel, less than 1.6 MB/s in total. These data rates are still readily managed by the system outlined +above, and about 1 TB of storage per week. Commercial solutions for these digitizers exist, but are about +factor four more costly than QDCs. Cost-effective alternatives are the 12-ch WaveBoard 2.0 designed by INFN +Roma/Genova or commercial boards using the DRS4 chip [49], like CAEN’s V1742. The DRS4 chip realizes an +analog buffer to allow for the cost effective and high-speed (multi-GSamples/s) sampling of events. The trade-off +is considerable dead-time for the conversion; however this is completely hidden in the proposed experiment by +the comparably low trigger rate. The digitization of the waveform would provide additional insights into the +detected particle and it’s timing, allowing us to improve background rejection. +The decision on these improvements will be based on our experience with these options in test beams planned for the +near future. +10. +UPGRADES / IMPROVEMENTS TO THE PROPOSED EXPERIMENT +While the configuration proposed so far is possible and would allow the two-photon exchange contribution to be +investigated in a region where the observed form factor discrepancy is clear; a number of upgrades are possible. +1. The current configuration assumes that 250 lead tungstate crystals can be obtained. Clearly if less or more +crystals are possible the configuration would change. Adding more crystals to the back angle calorimeters would +increase the acceptance in a region of low count rate. With an additional 5 × 5 array above and below the +current modules the solid angle would be increased from 3.6 msr to 15.6 msr an increase of 4.3. +2. If the showering in the 5 × 5 arrays of PbWO4 is well understood; it may be possible to accept a larger area +of the calorimeter, say an effective area of 6.4 msr rather than the 3.6 msr using just the central 3 × 3 array. +This would increase the acceptance by 1.78. Placing a tracking detector, e.g. GEM, immediately before the +calorimeter would help to define the acceptance. This needs to be investigated with test beam studies with a +5 × 5 calorimeter. +3. Move the back angle calorimeters closer to the target. Going to a radius of 0.5 m would increase the count rate +by a factor of four, though increasing the angular range subtended and thus reduce the Q2 resolution. Addition +of GEM tracking may help this but needs further Monte Carlo simulation and study. +These options could increase the count rate significantly, making even higher beam energies accessible. Table IV +shows the kinematic reach and differential cross section for just the back angle, 110°, for various lepton beam en- +ergies. A measurement at lepton beam energy of 4 GeV would extend the two-photon exchange measurements to +6.39 (GeV/c)2, and only requires an improvement by a factor of five to be comparable to the proposed measurement +rate at 3 GeV. + +19 +Ebeam +θ +Q2 +ϵ +dσ/dΩ +GeV +(GeV/c)2 +fb +2.0 +110° +2.78 +0.120 +1.22 × 104 +3.0 +110° +4.57 +0.096 +1.20 × 103 +4.0 +110° +6.39 +0.080 +2.23 × 102 +5.0 +110° +8.23 +0.068 +5.92 × 101 +6.0 +110° +10.1 +0.060 +2.00 × 101 +TABLE IV: Kinematics and cross section for measurements at 110° for lepton beam energies possible with DESY-II +11. +BACKGROUND CONSIDERATIONS +11.1. +Protons from e±p elastic scattering +As proposed, the experiment does not measure the scattered lepton and proton in coincidence. While this would have +some benefits it would also require detectors at far forward angles where the event rates from elastic lepton scattering, +Møller and Bhabha scattering, and pion production would be problematic. Nevertheless, protons from elastic lepton- +proton scattering will strike the proposed detectors and will be a source of background for the measurement. +20 +30 +40 +50 +60 +70 +80 +90 +100 +110 +120 + [deg] +lepton +θ +0 +10 +20 +30 +40 +50 +60 +70 + [deg] +proton +θ +ep Elastic Kinematics +beam +E +2.0 GeV +3.0 GeV +(a) +20 +30 +40 +50 +60 +70 +80 +90 +100 +110 +120 + [deg] +θ +0 +0.5 +1 +1.5 +2 +2.5 +3 +P [GeV/c] +ep Elastic Kinematics +Lepton momentum (2.0 GeV) +Lepton momentum (3.0 GeV) +Proton momentum (2.0 GeV) +Proton momentum (3.0 GeV) +(b) +FIG. 17: Kinematics for ep elastic scattering. (a) Proton scattering angle as a function of the lepton scattering +angle. (b) Lepton and proton momenta as a function of their respective scatting angles. +Fig. 17a shows the relation between the proton polar scattering angle and the lepton scattering angle. Fig. 17b +shows the momentum for the lepton and proton as a function of their scattering angle. The proton momentum is +greater than that of the lepton at forward angles but drops more rapidly as its scattering angle increases. Protons +will also be detected in the calorimeters and will have to be identified and corrected for on an event by event basis. +The calorimeter modules at over 22 X0 will adequately contain the electromagnetic showers and detect most of the +lepton energy, but the proton will not deposit its full energy as the calorimeter is only around one nuclear interaction +length in depth. Monte Carlo studies (presented below) indicate that the proton will deposit at most 300 to 400 MeV +in the calorimeters at 30° and 50° and significantly less at larger angles, particularly if an absorber shield is placed in +front of the calorimeters. This will allow the lepton signal to be clearly resolved from the proton signal. +At 30° the proton rate will be an occasional nuisance as it is significantly less than the lepton rate as shown in +Fig. 18a. However, at 50° the proton rate will be 10–100 times that for the lepton. This is still not a problem, as +the rate is manageable (approximately once every three beam spills) plus the deposited proton energy will be more +than 700 MeV lower than the lepton’s. But, at 70° the rate for protons is 104 − 105 greater than that for leptons +resulting in multiple protons from every beam spill. However, as discussed in the Monte Carlo section, with a suitable +absorber the protons can be stopped before the calorimeter without significantly affecting the lepton signal. It may +also be possible to eliminate this if the calorimeter timing and readout is sufficiently fast. The β value for the proton +is shown in Fig. 18b and is around 0.5 at 70°. That means the protons will arrive around 3 ns after the lepton possibly +allowing a timing window to exclude them. At 90° the proton rate is even higher but the energy is much lower and + +20 +20 +30 +40 +50 +60 +70 +80 +90 +100 +110 +120 + [deg] +θ +4 +10 +5 +10 +6 +10 +7 +10 +8 +10 +9 +10 +10 +10 + [fb] +σ +ep Elastic Kinematics +Lepton cross section (2.0 GeV) +Lepton cross section (3.0 GeV) +Proton cross section (2.0 GeV) +Proton cross section (3.0 GeV) +(a) +20 +30 +40 +50 +60 +70 +80 +90 +100 +110 +120 + [deg] +proton +θ +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +proton +β +ep Elastic Kinematics + (2.0 GeV) +β +Proton + (3.0 GeV) +β +Proton +(b) +FIG. 18: Kinematics for ep elastic scattering. (a) Cross sections for lepton and proton as a function of their polar +angle. (b) Beta for the proton as a function of its polar angle. +can be handled by an absorber and/or timing. +11.2. +Møller and Bhabha scattering +The cross sections for Møller and Bhabha scattering into the detector angles being considered in this proposal are +given in Table V. For an average lepton current of 40 nA incident on a 20 cm liquid hydrogen target the luminosity +is 2.11 × 10−4 fb−1· s−1. If we consider the face of the entire 5 × 5 array of each calorimeter module this corresponds +to 10 msr so the luminosity factor becomes 2.11 × 10−6. Multiplying this factor through the cross sections in Table V +yields rates ranging from around 4 to 2 × 109 events per second. +θ +Møller +Bhabha e+ +Bhabha e− +fb +fb +fb +2.0 GeV +30° +1.223 × 1014 +2.863 × 108 +1.219 × 1014 +50° +2.991 × 1014 +3.866 × 107 +2.989 × 1014 +70° +1.986 × 1015 +9.089 × 106 +1.985 × 1015 +90° +diverges +0 +diverges +110° +0 +0 +0 +3.0 GeV +30° +1.223 × 1014 +1.274 × 108 +1.220 × 1014 +50° +2.991 × 1014 +1.719 × 107 +2.989 × 1014 +70° +1.985 × 1015 +4.041 × 106 +1.985 × 1015 +90° +diverges +0 +diverges +110° +0 +0 +0 +TABLE V: Cross section for Møller and Bhabha scattering as a function of the polar scattering angle. +The energies of these Møller and Bhabha scattered leptons are low, less than 4 MeV at 30° and even lower at the +larger angles. For the most part they are still relativistic so a timing cut is not possible except at 90° and possibly at +70° if the calorimeter electronics are fast. To sweep these leptons away would require a magnetic field around 400 G. +However, Monte Carlo studies show that a simple 10 mm aluminum absorber before the calorimeter modules will +stop these particles from producing any signal in the calorimeter without degrading the response to the higher energy +leptons of interest. Since a 10 mm aluminum plate over the front face of the calorimeter array would work well as +part of the cooling system; the high rate of Møller and Bhabha scattered leptons is not a problem. + +21 +11.3. +Pion Production +Another source of background comes from pion production. There are four reactions to consider: +e− + p → e− + p + π0 +(2) +e− + p → e− + n + π+ +(3) +e+ + p → e+ + p + π0 +(4) +e+ + p → e+ + n + π+ +(5) +A Monte Carlo pion event generator was used to simulate these four reactions at 2 and 3 GeV. The calorimeter +modules in the proposed configuration will be struck by the leptons (electrons or positrons), baryons (protons or +neutrons), and pions (π0 or π+) from the various pion production reactions. In the case of π0 production the most +likely decay to two photons must also be considered. +The event rate per day and the momentum distribution of the leptons, baryons, and pions incident on the 5 × 5 +calorimeter array at 30° for the reactions e− + p → e− + p + π0 and → e− + n + π+ for an incident electron beam +energy of 2 and 3 GeV are given in Fig. 19. (N.B. No accounting for π0 decay or the energy actually deposited in +the calorimeters has been made. A more complete Monte Carlo simulation is in progress and further plots for pion +production are provided in the Appendix.) +The total event rate for electrons from pion production at 2 GeV striking the 5 × 5 face of the calorimeter array +at 30° is 2.07 × 106 per day. This is comparable to the 7.92 × 105 events per day expected in the central 3 × 3 array +for the elastic scattering events we wish to detect. However, the elastic events are peaked around a momentum of +1.56 GeV/c while the lepton momentum from pion production has a small peak around 1.35 GeV/c and a long tail +to much lower momenta. With PbWO4’s excellent energy resolution this difference should be easily resolved. The +rates at this angle are such that we can expect one of these pion events every beam spill. This background must be +detected and corrected on an event by event basis. The deposited energies of the baryons and pions are significantly +lower but will also contribute to the background and will need to be handled in the analysis. +θ +e− + p + π0 +e− + n + π+ +e+ + p + π0 +e+ + n + π+ +2.0 GeV +30° +2.08 × 106 +2.06 × 106 +2.06 × 106 +2.08 × 106 +50° +2.64 × 105 +2.54 × 105 +2.60 × 105 +2.56 × 105 +70° +7.22 × 104 +7.14 × 104 +7.28 × 104 +7.16 × 104 +90° +2.86 × 104 +2.90 × 104 +2.84 × 104 +2.90 × 104 +110° +1.40 × 104 +1.45 × 104 +1.41 × 104 +1.38 × 104 +3.0 GeV +30° +4.02 × 105 +4.08 × 105 +4.08 × 105 +4.06 × 105 +50° +6.60 × 104 +6.60 × 104 +6.64 × 104 +6.62 × 104 +70° +6.58 × 103 +6.54 × 103 +6.64 × 103 +6.72 × 103 +90° +2.24 × 103 +2.26 × 103 +2.20 × 103 +2.26 × 103 +110° +9.22 × 102 +9.10 × 102 +9.50 × 102 +9.32 × 102 +TABLE VI: Event rates per day for leptons from pion production striking the 5 × 5 calorimeter detector arrays. +Fig. 19 shows the number of particles detected per day at the calorimeter positioned at 30° for each particle produced +in pion production from e−p at 2 and 3 GeV. The plots for e+p are similar and plots for all detector angles are given +in the appendix. At higher angles the lepton rates fall quickly and are broadly distributed in momentum. The rates +for protons and neutrons also fall quickly plus they deposit little energy in the calorimeters. Pion rates remain fairly +uniform with angle but are at a low momentum and as Monte Carlo studies indicate the deposited energy is even +less. The π0 decay to two photons however will be a rather uniform, low energy background. Further Monte Carlo +studies are needed to verify that the background events can be cleanly resolved from the lepton signals that need to +be measured. +Table VI gives the daily event rates for the leptons from pion production striking the 5×5 array of each calorimeter. +These rates should be compared with those in Table I and Table II that give the rates for the events of interest striking +the central 3 × 3 array. The lepton rate from pion production is generally higher than the elastic scattered events of +interest. However, the lepton events of interest, arising from elastic ep scattering, are peaked at significantly higher +energies while the lepton energies from pion production are lower in energy and distributed over a broad range. + +22 +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +Momentum [MeV/c] +0 +50 +100 +150 +200 +250 +300 +350 +400 +3 +10 +× +p and 30 degrees +- + production at 2 GeV e +0 +π +30 deg, 2 GeV e- +-e +p +0 +π +p and 30 degrees +- + production at 2 GeV e +0 +π +(a) +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +Momentum [MeV/c] +0 +50 +100 +150 +200 +250 +3 +10 +× +p and 30 degrees +- + production at 2 GeV e ++ +π +30 deg, 2 GeV e- +-e +n ++ +π +p and 30 degrees +- + production at 2 GeV e ++ +π +(b) +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +2000 +2200 +Momentum [MeV/c] +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +3 +10 +× +p and 30 degrees +- + production at 3 GeV e +0 +π +30 deg, 3 GeV e- +-e +p +0 +π +p and 30 degrees +- + production at 3 GeV e +0 +π +(c) +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +2000 +2200 +Momentum [MeV/c] +0 +20 +40 +60 +80 +100 +120 +140 +160 +3 +10 +× +p and 30 degrees +- + production at 3 GeV e ++ +π +30 deg, 3 GeV e- +-e +n ++ +π +p and 30 degrees +- + production at 3 GeV e ++ +π +(d) +FIG. 19: Number of particles directed towards the 5 × 5 calorimeter array situated at 30° from the reactions (a) +e− + p → e− + p + π0 and (b) e− + p → e− + p + π+ at 2 GeV and (c) e− + p → e− + p + π0 and (d) +e− + p → e− + p + π+ at 3 GeV during one day of running at the nominal luminosity. +θ +e− + p + π0 +e− + n + π+ +e+ + p + π0 +e+ + n + π+ +2.0 GeV +30° +6.96 × 106 +6.04 × 106 +5.08 × 106 +6.64 × 106 +50° +7.14 × 106 +7.70 × 106 +7.16 × 106 +7.06 × 106 +70° +5.14 × 105 +9.74 × 105 +5.30 × 105 +8.86 × 105 +90° +0 +0 +0 +0 +110° +0 +0 +0 +0 +3.0 GeV +30° +2.42 × 106 +2.46 × 106 +2.62 × 106 +3.26 × 106 +50° +3.76 × 106 +3.56 × 106 +3.38 × 106 +3.86 × 106 +70° +7.02 × 104 +6.56 × 104 +7.26 × 104 +9.30 × 104 +90° +0 +0 +0 +0 +110° +0 +0 +0 +0 +TABLE VII: Event rates per day for baryons from pion production striking the 5 × 5 calorimeter detector arrays. + +23 +θ +e− + p + π0 +e− + n + π+ +e+ + p + π0 +e+ + n + π+ +2.0 GeV +30° +2.78 × 106 +2.18 × 106 +2.02 × 106 +2.54 × 106 +50° +4.50 × 106 +4.04 × 106 +4.36 × 106 +5.18 × 106 +70° +3.36 × 106 +4.44 × 106 +4.10 × 106 +3.94 × 106 +90° +2.90 × 106 +2.52 × 106 +2.50 × 106 +2.86 × 106 +110° +1.68 × 106 +1.81 × 106 +1.21 × 106 +1.95 × 106 +3.0 GeV +30° +1.04 × 106 +1.34 × 106 +1.35 × 106 +1.48 × 106 +50° +2.10 × 106 +1.96 × 106 +1.46 × 106 +1.60 × 106 +70° +1.53 × 105 +1.39 × 106 +1.81 × 106 +1.34 × 106 +90° +8.16 × 105 +7.58 × 105 +6.10 × 105 +6.68 × 105 +110° +4.20 × 105 +3.18 × 105 +3.22 × 105 +2.80 × 105 +TABLE VIII: Event rates per day for pions from pion production striking the 5 × 5 calorimeter detector arrays. +Table VII and Table VIII give the corresponding daily rates for the baryons and pions striking the 5 × 5 array +of each calorimeter. While these rates by themselves are comparable to the elastic ep events of interest the energy +actually deposited in the calorimeter will be significantly less and should be readily distinguished from the elastic +lepton signal. Of course more detailed Monte Carlo simulations are necessary and these are in progress. +12. +MONTE CARLO SIMULATIONS +In order to study the energy deposited in the 5×5 calorimeter arrays proposed in this document a simple GEANT4 +Monte Carlo [50] simulation was developed. Beams of electrons, protons, and pions (π+ and π0) were directed through +1 m of air at the center of a 5×5 calorimeter array at normal incidence. Initial momenta of 100 MeV/c to 2500 MeV/c +in 100 MeV/c steps were studied. +Four combinations of absorbers (none, 10 mm Al, 10 mm Al + 10 mm Pb, and 10 mm Al + 20 mm Pb) were placed +at the front face of the calorimeter array to study the effect this would have. The 10 mm aluminum plate would +naturally form part of the cooling system needed to obtain a stable energy resolution from the PbWO4 crystals. The +various thicknesses of lead were introduced to study how this could be used to reduce background from protons and +pions and the effect this would have on the lepton signal. +Fig. 20 illustrates the Monte Carlo studies performed. With the simulations, details of the longitudinal and trans- +verse energy distributions can be studied though in an actual experiment only the energy deposited in the individual +crystals are available. However, these studies show that the electron shower is effectively contained longitudinally and +the transverse distribution is narrow. Further studies will investigate the reconstruction of position and angle from +the energy deposited in the crystals alone and also unfolding events with multiple incident particles. +Results for various incident particles and absorbers are presented as a function of the particle momentum and +absorber thicknesses in the following sections. + +24 +E_vs_Z +Entries 2.182026e+07 +Mean +49.34 +− + +Std Dev + 31.69 +100 +− +80 +− +60 +− +40 +− +20 +− +0 +20 +40 +60 +80 +100 +0 +2 +4 +6 +8 +10 +12 +14 +E_vs_Z +Entries 2.182026e+07 +Mean +49.34 +− + +Std Dev + 31.69 +Z Energy Distribution +(a) +5 +− +4 +− +3 +− +2 +− +1 +− +0 +1 +2 +3 +4 +5 +5 +− +4 +− +3 +− +2 +− +1 +− +0 +1 +2 +3 +4 +50 +100 +200 +300 +400 +500 +600 +700 +800 +E_Crystal +Entries 2.182026e+07 +Mean x + 0.001336 +Mean y +0.001096 +− + +Std Dev x + 0.9165 +Std Dev y + 0.9169 +E_Crystal +Entries 2.182026e+07 +Mean x + 0.001336 +Mean y +0.001096 +− + +Std Dev x + 0.9165 +Std Dev y + 0.9169 +Crystal Energy +(b) +50 +− +40 +− +30 +− +20 +− +10 +− +0 +10 +20 +30 +40 +50 +50 +− +40 +− +30 +− +20 +− +10 +−0 +10 +20 +30 +40 +500 +10 +20 +30 +40 +50 +E_vs_XY +Entries 2.182026e+07 +Mean x + 0.01195 +Mean y + 0.005381 +Std Dev x + 8.874 +Std Dev y + 8.879 +E_vs_XY +Entries 2.182026e+07 +Mean x + 0.01195 +Mean y + 0.005381 +Std Dev x + 8.874 +Std Dev y + 8.879 +XY Energy Distribution +(c) +FIG. 20: Monte Carlo studies of electron showering in a +5 × 5 PbWO4 calorimeter. Incident electron momentum +was 1000 MeV and a 10 mm aluminum absorber was +placed before the crystals. (a) Longitudinal energy +distribution. (b) Total energy detected by each crystal. +(c) Transverse energy distribution in the calorimeter. + +25 +12.1. +Electrons and Positrons +As shown in Fig. 21 a lepton incident on the central crystal of the calorimeter array deposits almost all its energy +in the calorimeter. Most of that energy (∼ 80%) is in the central crystal. The shower width is also quite narrow +(∼ 10 mm). Note that the 10 mm aluminum absorber has almost no effect on the lepton shower. The lead absorber +increases the transverse width of the shower significantly at low momenta and to a lesser degree at higher momenta +resulting in some losses in total energy and the percentage deposited in the central crystal. +Not surprisingly positrons have a virtually identical behavior and are therefore not plotted separately here. +0 +500 +1000 +1500 +2000 +2500 +Electron Momenta [MeV/c] +0 +500 +1000 +1500 +2000 +2500 +Energy Deposited [MeV] +Electron Energy Deposition +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +(a) +0 +500 +1000 +1500 +2000 +2500 +Electron Momenta [MeV/c] +0 +500 +1000 +1500 +2000 +2500 +Energy Deposited [MeV] +Electron Energy Deposition +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Electron Momenta [MeV/c] +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Percentage Deposited in Peak [%] +Electron Energy Deposition in Peak +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +(b) +0 +500 +1000 +1500 +2000 +2500 +Electron Momenta [MeV/c] +0 +500 +1000 +1500 +2000 +2500 +Energy Deposited [MeV] +Electron Energy Deposition +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Electron Momenta [MeV/c] +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Percentage Deposited in Peak [%] +Electron Energy Deposition in Peak +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Electron Momenta [MeV/c] +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +MC RMS Shower Width [mm] +Electron Monte Carlo Shower Width +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +(c) +FIG. 21: Electron showering in a 5 × 5 PbWO4 +calorimeter array as a function of incident electron +momentum with different absorbers. (a) Sum of +energies in all 25 crystals, (b) Percentage of energy in +the central crystal, and (c) RMS width of transverse +shower development. + +26 +12.2. +Protons +Fig. 22 shows the results for proton incident on the calorimeter array. The total energy deposited in the calorimeter +is significantly less than the incident energies and for the most part is a third for momenta below 1000 MeV/c and +between 300 and 400 MeV/c for higher momenta. This is consistent with the calorimeter being only one nuclear +interaction length in depth so the proton has a tendency to pass straight through depositing only a fraction of its +energy. Most of the energy deposited (∼ 70%) is in the struck crystal. The absorbers have little effect except at low +incident momenta where the proton can be completely absorbed. This may be useful in stopping the large number of +lower energy protons produced at backward angles as well as low energy protons from pion production. +0 +500 +1000 +1500 +2000 +2500 +Electron Momenta [MeV/c] +0 +500 +1000 +1500 +2000 +2500 +Energy Deposited [MeV] +Electron Energy Deposition +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Electron Momenta [MeV/c] +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Percentage Deposited in Peak [%] +Electron Energy Deposition in Peak +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Electron Momenta [MeV/c] +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +MC RMS Shower Width [mm] +Electron Monte Carlo Shower Width +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Positron Momenta [MeV/c] +0 +500 +1000 +1500 +2000 +2500 +Energy Deposited [MeV] +Positron Energy Deposition +Absorber +None +10 mm Al +10 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+Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Proton Momenta [MeV/c] +0 +100 +200 +300 +400 +500 +600 +Energy Deposited [MeV] +Proton Energy Deposition +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Proton Momenta [MeV/c] +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Percentage Deposited in Peak [%] +Proton Energy Deposition in Peak +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +(b) +0 +500 +1000 +1500 +2000 +2500 +Electron Momenta [MeV/c] +0 +500 +1000 +1500 +2000 +2500 +Energy Deposited [MeV] +Electron Energy Deposition +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Electron Momenta [MeV/c] +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Percentage Deposited in Peak [%] +Electron Energy Deposition in Peak +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Electron Momenta [MeV/c] +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +MC RMS Shower Width [mm] +Electron Monte Carlo Shower Width +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Positron Momenta [MeV/c] +0 +500 +1000 +1500 +2000 +2500 +Energy Deposited [MeV] +Positron Energy Deposition +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Positron Momenta [MeV/c] +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Percentage Deposited in Peak [%] +Positron Energy Deposition in Peak +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Positron Momenta [MeV/c] +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +MC RMS Shower Width [mm] +Positron Monte Carlo Shower Width +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Proton Momenta [MeV/c] +0 +100 +200 +300 +400 +500 +600 +Energy Deposited [MeV] +Proton Energy Deposition +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Proton Momenta [MeV/c] +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Percentage Deposited in Peak [%] +Proton Energy Deposition in Peak +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Proton Momenta [MeV/c] +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +MC RMS Shower Width [mm] +Proton Monte Carlo Shower Width +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +(c) +FIG. 22: Proton showering in a 5 × 5 PbWO4 +calorimeter array as a function of incident proton +momentum and different absorbers. (a) Sum of +energies deposited in all 25 crystals, (b) Percentage of +energy in the central crystal, and (c) RMS width of +transverse shower development. + +27 +12.3. +Neutrons +Fig. 23 shows the results for neutrons incident on the calorimeter array. The total energy deposited in the calorimeter +just 5%–15% of the incident energies. About 50% of the energy deposited is in the struck crystal. The absorbers have +little effect. +0 +500 +1000 +1500 +2000 +2500 +Electron Momenta [MeV/c] +0 +500 +1000 +1500 +2000 +2500 +Energy Deposited [MeV] +Electron Energy Deposition +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Electron Momenta [MeV/c] +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Percentage Deposited in Peak [%] +Electron Energy Deposition in Peak +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Electron Momenta [MeV/c] +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +MC RMS Shower Width [mm] +Electron Monte Carlo Shower Width +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Positron Momenta [MeV/c] +0 +500 +1000 +1500 +2000 +2500 +Energy Deposited [MeV] +Positron Energy Deposition +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Positron Momenta [MeV/c] +0 +10 +20 +30 +40 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+Electron Momenta [MeV/c] +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Percentage Deposited in Peak [%] +Electron Energy Deposition in Peak +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Electron Momenta [MeV/c] +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +MC RMS Shower Width [mm] +Electron Monte Carlo Shower Width +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Positron Momenta [MeV/c] +0 +500 +1000 +1500 +2000 +2500 +Energy Deposited [MeV] +Positron Energy Deposition +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Positron Momenta [MeV/c] +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Percentage Deposited in Peak [%] +Positron Energy Deposition in Peak +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Positron Momenta [MeV/c] +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +MC RMS Shower Width [mm] +Positron Monte Carlo Shower Width +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Proton Momenta [MeV/c] +0 +100 +200 +300 +400 +500 +600 +Energy Deposited [MeV] +Proton Energy Deposition +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Proton Momenta [MeV/c] +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Percentage Deposited in Peak [%] +Proton Energy Deposition in Peak +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Proton Momenta [MeV/c] +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +MC RMS Shower Width [mm] +Proton Monte Carlo Shower Width +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Neutron Momenta [MeV/c] +0 +50 +100 +150 +200 +250 +300 +350 +400 +Energy Deposited [MeV] +Neutron Energy Deposition +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Neutron Momenta [MeV/c] +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Percentage Deposited in Peak [%] +Neutron Energy Deposition in Peak +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Neutron Momenta [MeV/c] +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +MC RMS Shower Width [mm] +Neutron Monte Carlo Shower Width +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +(c) +FIG. 23: Neutron showering in a 5 × 5 PbWO4 +calorimeter array as a function of incident neutron +momentum and different absorbers. (a) Sum of +energies deposited in all 25 crystals, (b) Percentage of +energy in the central crystal, and (c) RMS width of +transverse shower development. + +28 +12.4. +π+ +Fig. 24 shows the calorimeter response to incident π+ mesons. The total energy deposited in the calorimeter array +varies from around 50% of the incident momenta below 500 MeV to 25% at higher momenta. The various absorber +thicknesses have a small effect. 50%–60% is deposited in the central crystal and the RMS width for the transverse +shower development is fairly constant around 14 mm. This reduced signal from π+ will aid in distinguishing them +from the leptons of interest. Response with π− mesons is similar. +0 +500 +1000 +1500 +2000 +2500 +Electron Momenta [MeV/c] +0 +500 +1000 +1500 +2000 +2500 +Energy Deposited [MeV] +Electron Energy Deposition +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Electron Momenta [MeV/c] +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Percentage Deposited in Peak [%] +Electron Energy Deposition in Peak +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Electron Momenta [MeV/c] +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +MC RMS Shower Width [mm] +Electron Monte Carlo Shower Width +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Positron Momenta [MeV/c] +0 +500 +1000 +1500 +2000 +2500 +Energy Deposited [MeV] +Positron Energy Deposition +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Positron Momenta [MeV/c] +0 +10 +20 +30 +40 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mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Pi+ Momenta [MeV/c] +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Percentage Deposited in Peak [%] +Pi+ Energy Deposition in Peak +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Pi+ Momenta [MeV/c] +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +MC RMS Shower Width [mm] +Pi+ Monte Carlo Shower Width +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +(c) +FIG. 24: π+ showering in a 5 × 5 PbWO4 calorimeter +array as a function of incident pion momentum with +different absorbers. (a) Sum of energies deposited in +all 25 crystals, (b) Percentage of energy in the central +crystal, and (c) RMS width of transverse shower +development. + +29 +12.5. +π0 +Fig. 25 shows the calorimeter response to π0 mesons originating at the target 1 m away. The π0s primarily decay +isotropically to two photons that may or may not strike the calorimeter. For low energies the probability is small and +very little energy is deposited in the calorimeter. At higher energies the two photons are boosted in the direction of +the calorimeter and deposit a more significant fraction of their original energy. 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+1000 +1500 +2000 +2500 +Pi0 Momenta [MeV/c] +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +2000 +Energy Deposited [MeV] +Pi0 Energy Deposition +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Pi0 Momenta [MeV/c] +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +Percentage Deposited in Peak [%] +Pi0 Energy Deposition in Peak +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +0 +500 +1000 +1500 +2000 +2500 +Pi0 Momenta [MeV/c] +0 +5 +10 +15 +20 +25 +30 +35 +40 +MC RMS Shower Width [mm] +Pi0 Monte Carlo Shower Width +Absorber +None +10 mm Al +10 mm Al + 10 mm Pb +10 mm Al + 20 mm Pb +(c) +FIG. 25: π0 showering in a 5 × 5 PbWO4 calorimeter +array as a function of incident pion momentum with +different absorbers. (a) Sum of energies deposited in +all 25 crystals, (b) Percentage of energy in the central +crystal, and (c) RMS width of transverse shower +development. + +30 +13. +TEST BEAM AT DESY +The Monte Carlo studies discussed in the previous section are encouraging. The proposed experiment can make +a significant and direct measurement of the two-photon contribution in a region of Q2 and ϵ where the discrepancy +is clear. However, the Monte Carlo studies must be verified. It is therefore important that the performance of the +calorimeter modules be studied in a test beam. +An initial prototype calorimeter was tested at the DESY test beam facility in the fall, 2019. The results are reported +here in appendix A. These initial tests with a small 3× 3 prototype design are encouraging with good agreement with +the Monte Carlo but further tests are needed. +We propose to perform these measurements at the DESY test beam facility [32] using a 5×5 calorimeter array. The +purpose of the test beam activity will be to measure the performance of the PbWO4 calorimeter array and to verify +the Monte Carlo simulations. We would use various energies, with and without the absorber plates, and incident at +various positions and angles across the calorimeter array. Monte Carlo simulations can suggest and be used to train +reconstruction algorithms but these need to be verified with actual measurements therefore the proposed test beam +studies are very important. +If the ceramic glass crystals being developed by Tanja Horn (CUA) are available we would also test these in the +prototype calorimeters. This clearly has links with efforts underway in Europe and the United States for future +detectors for the proposed Electron-Ion Collider. +14. +CONCLUSION +The observed discrepancy in the proton form factor ratio is a fundamental problem in nuclear physics and possibly in +quantum electrodynamics. Why are the leading order QED radiative corrections insufficient to resolve the discrepancy? +Are higher order corrections necessary or are more detailed models for the intermediate hadronic state needed? Or is +some other process responsible? +An extracted positron and electron beam facility at DESY would provide a unique opportunity to measure the +two-photon exchange contribution to elastic lepton-proton scattering over a kinematic range where the observed +discrepancy is clearly evident. The above proposal outlines an initial plan for an experimental configuration that +could help resolve this issue and provide insight to the radiative corrections needed to understand the proton form +factors at higher momentum transfers. + +31 +Appendix A: Test Beam Results +Over two weeks in September-October, 2019, a calorimeter consisting of a 3×3 array of PbWO4 crystals was studied +at the DESY test beam facility [32]. Tests were made scanning the electron beam across the face of the calorimeter +and with different thicknesses of absorber plates. We also compared, in parallel, a traditional, triggered readout with +a streaming readout scheme. Further test beam studies were made in the fall of 2021 and spring of 2022 using a more +realistic 5 × 5 array of lead tungstate crystals. This calorimeter was also cooled and measured at 25°, 10°, -10°, and +-25°C. A small number of high-density ceramic glass crystals were also tested. The analysis of the 2022 test run is +ongoing. +1. +Calorimeter Setup and Tests +The calorimeter used nine 2×2×20 cm3 lead tungstate crystals read out using Hamamatsu R1166 PMTs attached +to one end of each crystal. The crystals were wrapped with one layer of white Tyvek (0.4 mm thick) and an outer +layer of opaque aluminum foil (0.09 mm thick). The crystal-PMT assemblies were placed inside a black anodized +aluminum housing. See Fig. 26. Copper tubes for water cooling were installed on the outside of the aluminum box. +The calorimeter assembly was mounted on an XY translation table but was electrically isolated from the table. A +collimator and a set of four thin scintillators upstream of the calorimeter were using in the triggered readout. +FIG. 26: Photo of 3x3 lead tungstate calorimeter prototype and trigger detectors used in initial test run at DESY. +High voltage for the PMTs was provided by LeCroy 1461N modules. +Signals from the PMTs were divided by a 50 Ω splitter. One side of each splitter output was connected through a +100 ns delay cable to CAEN V792 QDC. The signals from the four thin scintillators were combined in a coincidence +unit requiring a triple coincidence that was used to trigger the QDC. The other splitter output was connected to a +CAEN V1725 digitizer. Since the digitizer had only 8 channels a decision was made to read out crystals 1 to 7, and +use channel 0 to record the trigger signal in parallel. +The gain from each crystal and PMT was matched using a 5.2 GeV beam incident on the center of the crystal and +the HV adjusted to give a common value close to the end of the QDC range. +Data were collected at beam energies of 2, 3, 4, and 5 GeV and with 2 × 2 mm2 and 8 × 8 mm2 collimators. For all +conditions scans were made over the face of the calorimeter and using 0, 1, and 2 cm thick lead absorber plates before +the calorimeter. Typical energy spectra for all four beam energies can be seen in Fig. 27. The left figure shows the +QDC spectra (with pedestal subtraction) and the right figure the digitizer spectra for events which are in coincidence +with the trigger signal. +Fig. 28 shows the sum of all 25 signals with a 5 GeV beam centered on the central crystal. The root analysis tool +functions “gaus” and “crystalball” were used to fit the spectra to determine peak position. Linearity of the peak +position with incident energy is shown in Fig. 29. +2. +Streaming and triggered readout +In the triggered readout scheme all channels of the QDC are read out together after receiving a trigger signal. This +takes some time during which the QDC is unable to record new events (deadtime). On the other hand the streaming +readout scheme using the digitizer continuously records events in all channels. Thus, the streaming system records + +Trigger +3x3 detector +detectors +assembly32 +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +QDC [a.u.] +0 +50 +100 +150 +200 +250 +Counts + = 5 GeV +0 +E + = 4 GeV +0 +E + = 3 GeV +0 +E + = 2 GeV +0 +E +Crystal 4 +0 +2000 +4000 +6000 +8000 +10000 +DIGI [a.u.] +50 +100 +150 +200 +250 +300 +Counts + = 5 GeV +0 +E + = 4 GeV +0 +E + = 3 GeV +0 +E + = 2 GeV +0 +E +Crystal 4 coincident with channel 0 +FIG. 27: Deposited energy in central crystal recorded by QDC (left) and digitizer (right). The digitizer spectra also +required a coincidence with a trigger signal in digitizer channel 0. +FIG. 28: Sum of energies deposited in all crystals recorded by QDC (left) and digitizer. The digitizer spectrum also +required a coincidence with the trigger signal in channel 0. +FIG. 29: Energy dependence of the peak position in QDC (left) and digitizer (right). +more events in individual channels though many signals may be uncorrelated from cosmic rays, noise, or background +events. To make sense of the large amount of data collected by the digitizer it is necessary to determine the relative +timing of all channels. Then signals at a common time can be reasonably assumed to arise from the same event, like +corresponding to showering in the calorimeter (see Fig. 30). Similarly, the relative timing to the trigger signal used for +the QDC (connected to channel 0 of the digitizer) can be determined and used to compare the same event collected +by the QDC with that recorded by the digitizer. + +QDC SUM +220 +Entries +8627 +200 +Mean +9227 +Std Dev +1679 +180 +160 +140 +ounts +- data +120 +- gaus fit +C +100 +- crystalball fit +80 +60 +40 +20 +0 +0 +2000 +4000 +6000 +8000 +10000 +12000 +14000 +16000 +QDC [a.u.]DIGLSUM7 DET +Entries +35119 +250 +Mean +2.605e+04 +Std Dev +3280 +200 +Counts +data +150 + gaus fit +- crystalball fit +100 +50 +0 +0 +5000 +10000 +15000 +20000 +25000 +30000 +35000 +40000 +DIGI[a.u.]10,000 +Mean Channel Sum +5,000 +Data +LinearFit +A·X + B +A=1438.3± 0.5 +B = 4247 ± 2 +0 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +5.5 +Beam Energy [GeV]30,000 +Data +Ax + B +A=3322.9±0.7 +Sum +B = 11087 ± 2 +LinearFit1 +25.000 +Mean Channel +20,000 +15,000 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +5.5 +BeamEnergy[GeV]33 +DIGI CH4 +Entries + 21452 +Mean + 7182 +Std Dev + 454.5 +0 +2000 +4000 +6000 +8000 +10000 +12000 +DIGI [a.u.] +0 +100 +200 +300 +400 +500 +Counts +DIGI CH4 +Entries + 21452 +Mean + 7182 +Std Dev + 454.5 +DIGI CH4 +FIG. 30: Deposited energy in central detector (channel 4) with coincidence signals in at least 6 neighboring crystals. +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● ● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● ● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +●● +●● +● +● +● ● +● +● +● +● +● +●● +● +● +● +● +● +● +● +● +● +● +● +● +● +●● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +●● +● +●● +● +● +● +● +●● +● +● +● +● +● +● +● +● +●● +● +● +● +● +● +● +● +● +● +●● +● +● +● +● +● +●● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +●● +● +●● 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+● +● +● +● +● +● +● +● +● +●●● +● +● ● ● +● +●● +● +● +● +● +● +● +● +● +● +● +● +●● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +●● +●● +● ● +● +● +● +● +● ● +● +● +● +● +● +● +● +● +● +● +●●● +● +● +● +● +● +● +● +● +● +● +●● +● +●● +● +●● +● +● +● +● +● +● +●● +● +● +● +● +● +● +●● +● +● +● +● +●● +● ● +● +● +●● +● +●● +● +● +● +● +● +● +● +● +● +● ● +● +● +● +● +● +● +● +● +● +● +● +● +● +● ● +●● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +●● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +●● +● +● +● +● +● +● +● +● +● +● +● +● +● +●●● +● +● +● +● +● +● +● +● +● +● +● ● +● +● +● +● +● +● +●● +● +● +●● +● +●● +● ● +●● +● +● +●● +● +● +● +●● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● ● +● +● +●● +● +● +● +● +●● +● +● +● +● +● +● +● +● +● +● +● +● +● +●●● +● +● +● +● +● +● +● +● +● +● +● ● +●● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +●●● +● +● +● +● ● +● +● +● +● +●● +● +● +● ● +● +● +● +● +●● +● +● +● +● +● +● +●● +● ● +● +● +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +0 +500 +1000 +1500 +2000 +2500 +3000 +DIGI [a.u.] +QDC [a.u.] +FIG. 31: Left figure shows the energy deposited in the digitizer versus that for the QDC. The right figure is the 3D +histogram plot of the same data. +3. +Monte Carlo Simulation of Test Beam +A Monte Carlo simulation of the test beam was developed in Geant4 [50]. We use the FTFP BERT physics list +provided by Geant to simulate the showers and energy loss processes in the crystals. We reproduced the TB24/1 area +from the available drawings and technical details provided by DESY. This included the calorimeter, absorber plates, +trigger scintillators, collimator, and the origin of the beam source at the DESY II ring. This last item was found +to be very important to account for the significant energy straggling observed in the measured spectra. The Geant4 +visualization of the front face of the 3 × 3 calorimeter array is shown in Fig. 32. +Fig. 33 shows a comparison between the measured QDC data and the simulation for the central crystal for a 2 GeV +incident beam. +Fig. 34 shows a similar comparison between simulation and the digitizer data. The discrepancies between simulation +and data can be ascribed to an incomplete model of the experimental hall, specifically any material causing energy +loss upstream of the hall. We believe with improved modeling the agreement could be better. + +600 +$400 +200 +0 +3000 +2000 +2000 +4000 +QDC [a.u.] +6000 +1000 +DIGI [a.u.] +800034 +FIG. 32: The Geant4 simulation view of the front face of the calorimeter. +0 +500 +1000 +1500 +Energy Deposited (AU) +1 +10 +2 +10 +3 +10 +Counts +Geant4 Simulation +QDC Data +0 +500 +1000 +1500 +Energy Deposited (AU) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +2.2 +3 +10 +× +Counts +Geant4 Simulation +QDC Data +Geant4 Simulation +QDC Data +FIG. 33: Comparison between Geant4 simulation and data from the QDCs using logarithmic (left) and linear (right) +scales. The simulation is scaled to the height of the data. + +35 +0 +500 +1000 +1500 +Energy Deposited (AU) +1 +10 +2 +10 +3 +10 +Counts +Geant4 Simulation +Digitizer Data +0 +500 +1000 +1500 +Energy Deposited (AU) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +3 +10 +× +Counts +Geant4 Simulation +Digitizer Data +Geant4 Simulation +Digitizer Data +FIG. 34: Comparison between Geant4 simulation and data from the digitizers. The simulation is scaled to the +height of the data. + +36 +Appendix B: Monte Carlo Simulation for e− + p → e− + p + π0 at 2 GeV +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +Momentum [MeV/c] +0 +50 +100 +150 +200 +250 +300 +350 +400 +3 +10 +× +p and 30 degrees +- + production at 2 GeV e +0 +π +30 deg, 2 GeV e- +-e +p +0 +π +p and 30 degrees +- + production at 2 GeV e +0 +π +(a) +0 +200 +400 +600 +800 +1000 +1200 +Momentum [MeV/c] +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +3 +10 +× +p and 50 degrees +- + production at 2 GeV e +0 +π +50 deg, 2 GeV e- +-e +p +0 +π +p and 50 degrees +- + production at 2 GeV e +0 +π +(b) +0 +100 +200 +300 +400 +500 +600 +700 +800 +Momentum [MeV/c] +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +3 +10 +× +p and 70 degrees +- + production at 2 GeV e +0 +π +70 deg, 2 GeV e- +-e +p +0 +π +p and 70 degrees +- + production at 2 GeV e +0 +π +(c) +0 +100 +200 +300 +400 +500 +600 +Momentum [MeV/c] +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +220 +3 +10 +× +p and 90 degrees +- + production at 2 GeV e +0 +π +90 deg, 2 GeV e- +-e +p +0 +π +p and 90 degrees +- + production at 2 GeV e +0 +π +(d) +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Momentum [MeV/c] +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +3 +10 +× +p and 110 degrees +- + production at 2 GeV e +0 +π +110 deg, 2 GeV e- +-e +p +0 +π +p and 110 degrees +- + production at 2 GeV e +0 +π +(e) +FIG. 35: Number of electrons, protons, and π0 +directed towards the 5 × 5 calorimeter arrays at 30°, +50°, 70°, 90°, and 110° during one day of running at +the nominal luminosity for the reaction +e− + p → e− + p + π0 at 2 GeV. + +37 +Appendix C: Monte Carlo Simulation for e− + p → e− + n + π+ at 2 GeV +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +Momentum [MeV/c] +0 +50 +100 +150 +200 +250 +3 +10 +× +p and 30 degrees +- + production at 2 GeV e ++ +π +30 deg, 2 GeV e- +-e +n ++ +π +p and 30 degrees +- + production at 2 GeV e ++ +π +(a) +0 +200 +400 +600 +800 +1000 +1200 +Momentum [MeV/c] +0 +50 +100 +150 +200 +250 +300 +350 +3 +10 +× +p and 50 degrees +- + production at 2 GeV e ++ +π +50 deg, 2 GeV e- +-e +n ++ +π +p and 50 degrees +- + production at 2 GeV e ++ +π +(b) +0 +100 +200 +300 +400 +500 +600 +700 +800 +Momentum [MeV/c] +0 +50 +100 +150 +200 +250 +3 +10 +× +p and 70 degrees +- + production at 2 GeV e ++ +π +70 deg, 2 GeV e- +-e +n ++ +π +p and 70 degrees +- + production at 2 GeV e ++ +π +(c) +0 +100 +200 +300 +400 +500 +600 +Momentum [MeV/c] +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +220 +3 +10 +× +p and 90 degrees +- + production at 2 GeV e ++ +π +90 deg, 2 GeV e- +-e +n ++ +π +p and 90 degrees +- + production at 2 GeV e ++ +π +(d) +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Momentum [MeV/c] +0 +20 +40 +60 +80 +100 +120 +140 +3 +10 +× +p and 110 degrees +- + production at 2 GeV e ++ +π +110 deg, 2 GeV e- +-e +n ++ +π +p and 110 degrees +- + production at 2 GeV e ++ +π +(e) +FIG. 36: Number of electrons, neutrons, and π+ +directed towards the 5 × 5 calorimeter arrays at 30°, +50°, 70°, 90°, and 110° during one day of running at +the nominal luminosity for the reaction +e− + p → e− + n + π+ at 2 GeV. + +38 +Appendix D: Monte Carlo Simulation for e+ + p → e+ + p + π0 at 2 GeV +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +Momentum [MeV/c] +0 +20 +40 +60 +80 +100 +120 +140 +160 +3 +10 +× +p and 30 degrees ++ + production at 2 GeV e +0 +π +30 deg, 2 GeV e+ ++ +e +p +0 +π +p and 30 degrees ++ + production at 2 GeV e +0 +π +(a) +0 +200 +400 +600 +800 +1000 +1200 +Momentum [MeV/c] +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +3 +10 +× +p and 50 degrees +- + production at 2 GeV e +0 +π +50 deg, 2 GeV e- +-e +p +0 +π +p and 50 degrees +- + production at 2 GeV e +0 +π +(b) +0 +100 +200 +300 +400 +500 +600 +700 +800 +Momentum [MeV/c] +0 +50 +100 +150 +200 +250 +3 +10 +× +p and 70 degrees ++ + production at 2 GeV e +0 +π +70 deg, 2 GeV e+ ++ +e +p +0 +π +p and 70 degrees ++ + production at 2 GeV e +0 +π +(c) +0 +100 +200 +300 +400 +500 +600 +Momentum [MeV/c] +0 +20 +40 +60 +80 +100 +120 +140 +3 +10 +× +p and 90 degrees ++ + production at 2 GeV e +0 +π +90 deg, 2 GeV e+ ++ +e +p +0 +π +p and 90 degrees ++ + production at 2 GeV e +0 +π +(d) +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Momentum [MeV/c] +0 +20 +40 +60 +80 +100 +120 +3 +10 +× +p and 110 degrees ++ + production at 2 GeV e +0 +π +110 deg, 2 GeV e+ ++ +e +p +0 +π +p and 110 degrees ++ + production at 2 GeV e +0 +π +(e) +FIG. 37: Number of positrons, protons, and π0 +directed towards the 5 × 5 calorimeter arrays at 30°, +50°, 70°, 90°, and 110° during one day of running at +the nominal luminosity for the reaction +e+ + p → e+ + p + π0 at 2 GeV. + +39 +Appendix E: Monte Carlo Simulation for e+ + p → e+ + n + π+ at 2 GeV +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +Momentum [MeV/c] +0 +50 +100 +150 +200 +250 +3 +10 +× +p and 30 degrees ++ + production at 2 GeV e ++ +π +30 deg, 2 GeV e+ ++ +e +n ++ +π +p and 30 degrees ++ + production at 2 GeV e ++ +π +(a) +0 +200 +400 +600 +800 +1000 +1200 +Momentum [MeV/c] +0 +50 +100 +150 +200 +250 +300 +3 +10 +× +p and 50 degrees ++ + production at 2 GeV e ++ +π +50 deg, 2 GeV e+ ++ +e +n ++ +π +p and 50 degrees ++ + production at 2 GeV e ++ +π +(b) +0 +100 +200 +300 +400 +500 +600 +700 +800 +Momentum [MeV/c] +0 +50 +100 +150 +200 +250 +3 +10 +× +p and 70 degrees ++ + production at 2 GeV e ++ +π +70 deg, 2 GeV e+ ++ +e +n ++ +π +p and 70 degrees ++ + production at 2 GeV e ++ +π +(c) +0 +100 +200 +300 +400 +500 +600 +Momentum [MeV/c] +0 +50 +100 +150 +200 +250 +3 +10 +× +p and 90 degrees ++ + production at 2 GeV e ++ +π +90 deg, 2 GeV e+ ++ +e +n ++ +π +p and 90 degrees ++ + production at 2 GeV e ++ +π +(d) +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Momentum [MeV/c] +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +3 +10 +× +p and 110 degrees ++ + production at 2 GeV e ++ +π +110 deg, 2 GeV e+ ++ +e +n ++ +π +p and 110 degrees ++ + production at 2 GeV e ++ +π +(e) +FIG. 38: Number of positrons, neutrons, and π+ +directed towards the 5 × 5 calorimeter arrays at 30°, +50°, 70°, 90°, and 110° during one day of running at +the nominal luminosity for the reaction +e+ + p → e+ + n + π+ at 2 GeV. + +40 +Appendix F: Monte Carlo Simulation for e− + p → e− + p + π0 at 3 GeV +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +2000 +2200 +Momentum [MeV/c] +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +3 +10 +× +p and 30 degrees +- + production at 3 GeV e +0 +π +30 deg, 3 GeV e- +-e +p +0 +π +p and 30 degrees +- + production at 3 GeV e +0 +π +(a) +0 +200 +400 +600 +800 +1000 +1200 +1400 +Momentum [MeV/c] +0 +50 +100 +150 +200 +250 +3 +10 +× +p and 50 degrees +- + production at 3 GeV e +0 +π +50 deg, 3 GeV e- +-e +p +0 +π +p and 50 degrees +- + production at 3 GeV e +0 +π +(b) +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +Momentum [MeV/c] +0 +10000 +20000 +30000 +40000 +50000 +60000 +70000 +80000 +90000 +p and 70 degrees +- + production at 3 GeV e +0 +π +70 deg, 3 GeV e- +-e +p +0 +π +p and 70 degrees +- + production at 3 GeV e +0 +π +(c) +0 +100 +200 +300 +400 +500 +600 +700 +Momentum [MeV/c] +0 +10000 +20000 +30000 +40000 +50000 +60000 +p and 90 degrees +- + production at 3 GeV e +0 +π +90 deg, 3 GeV e- +-e +p +0 +π +p and 90 degrees +- + production at 3 GeV e +0 +π +(d) +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Momentum [MeV/c] +0 +5000 +10000 +15000 +20000 +25000 +30000 +35000 +40000 +45000 +p and 110 degrees +- + production at 3 GeV e +0 +π +110 deg, 3 GeV e- +-e +p +0 +π +p and 110 degrees +- + production at 3 GeV e +0 +π +(e) +FIG. 39: Number of electrons, protons, and π0 +directed towards the 5 × 5 calorimeter arrays at 30°, +50°, 70°, 90°, and 110° during one day of running at +the nominal luminosity for the reaction +e− + p → e− + p + π0 at 3 GeV. + +41 +Appendix G: Monte Carlo Simulation for e− + p → e− + n + π+ at 3 GeV +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +2000 +2200 +Momentum [MeV/c] +0 +20 +40 +60 +80 +100 +120 +140 +160 +3 +10 +× +p and 30 degrees +- + production at 3 GeV e ++ +π +30 deg, 3 GeV e- +-e +n ++ +π +p and 30 degrees +- + production at 3 GeV e ++ +π +(a) +0 +200 +400 +600 +800 +1000 +1200 +1400 +Momentum [MeV/c] +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +3 +10 +× +p and 50 degrees +- + production at 3 GeV e ++ +π +50 deg, 3 GeV e- +-e +n ++ +π +p and 50 degrees +- + production at 3 GeV e ++ +π +(b) +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +Momentum [MeV/c] +0 +10000 +20000 +30000 +40000 +50000 +60000 +70000 +p and 70 degrees +- + production at 3 GeV e ++ +π +70 deg, 3 GeV e- +-e +n ++ +π +p and 70 degrees +- + production at 3 GeV e ++ +π +(c) +0 +100 +200 +300 +400 +500 +600 +700 +Momentum [MeV/c] +0 +10000 +20000 +30000 +40000 +50000 +p and 90 degrees +- + production at 3 GeV e ++ +π +90 deg, 3 GeV e- +-e +n ++ +π +p and 90 degrees +- + production at 3 GeV e ++ +π +(d) +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Momentum [MeV/c] +0 +2000 +4000 +6000 +8000 +10000 +12000 +14000 +16000 +18000 +p and 110 degrees +- + production at 3 GeV e ++ +π +110 deg, 3 GeV e- +-e +n ++ +π +p and 110 degrees +- + production at 3 GeV e ++ +π +(e) +FIG. 40: Number of electrons, neutrons, and π+ +directed towards the 5 × 5 calorimeter arrays at 30°, +50°, 70°, 90°, and 110° during one day of running at +the nominal luminosity for the reaction +e− + p → e− + n + π+ at 3 GeV. + +42 +Appendix H: Monte Carlo Simulation for e+ + p → e+ + p + π0 at 3 GeV +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +2000 +2200 +Momentum [MeV/c] +0 +20 +40 +60 +80 +100 +120 +3 +10 +× +p and 30 degrees ++ + production at 3 GeV e +0 +π +30 deg, 3 GeV e+ ++ +e +p +0 +π +p and 30 degrees ++ + production at 3 GeV e +0 +π +(a) +0 +200 +400 +600 +800 +1000 +1200 +1400 +Momentum [MeV/c] +0 +50 +100 +150 +200 +250 +3 +10 +× +p and 50 degrees +- + production at 3 GeV e +0 +π +50 deg, 3 GeV e- +-e +p +0 +π +p and 50 degrees +- + production at 3 GeV e +0 +π +(b) +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +Momentum [MeV/c] +0 +20 +40 +60 +80 +100 +120 +3 +10 +× +p and 70 degrees ++ + production at 3 GeV e +0 +π +70 deg, 3 GeV e+ ++ +e +p +0 +π +p and 70 degrees ++ + production at 3 GeV e +0 +π +(c) +0 +100 +200 +300 +400 +500 +600 +700 +Momentum [MeV/c] +0 +5000 +10000 +15000 +20000 +25000 +30000 +p and 90 degrees ++ + production at 3 GeV e +0 +π +90 deg, 3 GeV e+ ++ +e +p +0 +π +p and 90 degrees ++ + production at 3 GeV e +0 +π +(d) +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Momentum [MeV/c] +0 +5000 +10000 +15000 +20000 +25000 +p and 110 degrees ++ + production at 3 GeV e +0 +π +110 deg, 3 GeV e+ ++ +e +p +0 +π +p and 110 degrees ++ + production at 3 GeV e +0 +π +(e) +FIG. 41: Number of positrons, protons, and π0 +directed towards the 5 × 5 calorimeter arrays at 30°, +50°, 70°, 90°, and 110° during one day of running at +the nominal luminosity for the reaction +e+ + p → e+ + p + π0 at 3 GeV. + +43 +Appendix I: Monte Carlo Simulation for e+ + p → e+ + n + π+ at 3 GeV +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +2000 +2200 +Momentum [MeV/c] +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +3 +10 +× +p and 30 degrees ++ + production at 3 GeV e ++ +π +30 deg, 3 GeV e+ ++ +e +n ++ +π +p and 30 degrees ++ + production at 3 GeV e ++ +π +(a) +0 +200 +400 +600 +800 +1000 +1200 +1400 +Momentum [MeV/c] +0 +20 +40 +60 +80 +100 +120 +140 +160 +3 +10 +× +p and 50 degrees ++ + production at 3 GeV e ++ +π +50 deg, 3 GeV e+ ++ +e +n ++ +π +p and 50 degrees ++ + production at 3 GeV e ++ +π +(b) +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +Momentum [MeV/c] +0 +10000 +20000 +30000 +40000 +50000 +60000 +p and 70 degrees ++ + production at 3 GeV e ++ +π +70 deg, 3 GeV e+ ++ +e +n ++ +π +p and 70 degrees ++ + production at 3 GeV e ++ +π +(c) +0 +100 +200 +300 +400 +500 +600 +700 +Momentum [MeV/c] +0 +5000 +10000 +15000 +20000 +25000 +30000 +35000 +40000 +p and 90 degrees ++ + production at 3 GeV e ++ +π +90 deg, 3 GeV e+ ++ +e +n ++ +π +p and 90 degrees ++ + production at 3 GeV e ++ +π +(d) +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Momentum [MeV/c] +0 +10000 +20000 +30000 +40000 +50000 +110 deg, 3 GeV e+ ++ +e +n ++ +π +p and 110 degrees ++ + production at 3 GeV e ++ +π +(e) +FIG. 42: Number of positrons, neutrons, and π+ +directed towards the 5 × 5 calorimeter arrays at 30°, +50°, 70°, 90°, and 110° during one day of running at +the nominal luminosity for the reaction +e+ + p → e+ + n + π+ at 3 GeV. + +44 +Appendix J: Hydrogen Properties +Hydrogen normally exists as a diatomic molecule, H2. The molecule occurs in two forms or allotropes: orthohy- +drogen, where the nuclear spins of the two atoms are parallel (J = 0, 2, 4, . . . ); and parahydrogen, where the nuclear +spins are anti-parallel (J = 1, 3, 5, . . . ). +The concentrations of the two allotropes vary with temperature. At 80 K the concentration of each is roughly +the same. At room temperature and above it is generally 75% orthohydrogen. At 19 K it is 99.75% parahydrogen. +Various parameters are given in Table IX. +Para-Equilibrium +Normal +Critical point +Temperature +32.976 K +33.19 K +Pressure +1.2928 MPa (12.759 atm) +1.315 MPa (12.98 atm) +Density +31.43 kg/m3 (15.59 mol/L) +30.12 kg/m3 (14.94 mol/L) +Normal boiling point +Temperature +20.268 K +20.39 K +Pressure +0.101325 MPa (1 atm) +0.101325 MPa (1 atm) +Density (liquid) +70.78 kg/m3 (35.11 mol/L) +71.0 kg/m3 (35.2 mol/L) +Density (vapor) +1.338 kg/m3 (0. 6636 mol/L) 1.331 kg/m3 (0.6604 mol/L) +Triple point +Temperature +13.803 K +13.957 K +Pressure +0.00704 MPa (0.0695 atm) +0.00720 MPa (0.0711 atm) +Density (solid) +86.50 kg/m3 (42.91 mol/L) +86.71 kg/m3 (43.01 mol/L) +Density (liquid) +77.03 kg/m3 (38.21 mol/L) +77.2 kg/m3 (38.3 mol/L) +Density (vapor) +0.126 kg/m3 (0.0623 mol/L) 0.130 kg/m3 (0.0644 mol/L) +Molecular Weight +2.01588 +TABLE IX: Parameters for hydrogen +FIG. 43: Cooling power of the cold head being considered for the TPEX liquid hydrogen target system. + +CH-110 System +SHI +Cryogenics Group +Performance +Low Temperature Version +CH11oLT Cold Head Capacity Map Using F-70 Compressor at 60 Hz +300 +275 +250 +225 +200 +175 +[W] +150 +HeatLoad +125 +100 +75 +50 +25 +0 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Temperature [K]45 +Appendix K: Lead Tungstate, PbWO4, Properties +From the 2020 Particle Data Group Atomic and Nuclear Properties of Materials: +Density PbWO4 += 8.300 g·cm−3 +2 × 2 × 20 cm3 crystal += 664.0 g +Moli`ere radius += 1.959 cm +Nuclear Interaction Length λI = 168.3 g·cm−2 += 20.28 cm +Radiation Length X0 += 7.39 g·cm−2 += 0.8904 cm +Energy Loss dE/dx += 1.229 MeV·g−1·cm2 += 10.2 MeV·cm−1 +Appendix L: Numbers Used for Calculations in this Proposal +Density LH2 += 0.07078 g·cm−3 += 4.2289×1022 atoms·cm−3 +20 cm LH2 target += 8.4578×1023 atoms·cm−2 +40 nA on 20 cm LH2 target = 2.1116×1035 cm−2·s−1·sr−1 += 2.1116×10−4 fb−1·s−1·sr−1 +1.0 × 107 fb += 7.6018 events·s−1 into 3.6 msr +[1] T. 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Republic 8Catholic University of America,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' DC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' USA 9The George Washington University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' DC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' USA 10University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Ann Arbor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' MI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' USA 11Johannes Gutenberg Universit¨at,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Mainz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Germany 12Deutsches Elektronen-Synchrotron,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Hamburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Germany 13University of Glasgow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Glasgow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Scotland (Dated: January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2023) We propose a new measurement of the ratio of positron-proton to electron-proton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' elastic scat- tering at DESY to determine the contributions beyond single-photon exchange,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' which are essential to the QED description of the most fundamental process in hadronic physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A 20 cm long liq- uid hydrogen target together with the extracted beam from the DESY synchrotron would yield an average luminosity of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 × 1035 cm−2·s−1·sr−1 (∼ 200 times the luminosity achieved by OLYM- PUS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A commissioning run at 2 GeV followed by measurements at 3 GeV would provide new data up to Q2 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 (GeV/c)2 (twice the range of current measurements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Lead tungstate calorime- ters would be used to detect the scattered leptons at polar angles of 30°, 50°, 70°, 90°, and 110°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The measurements could be scheduled to not interfere with the operation of PETRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' We present rate estimates and simulations for the planned measurements including background considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Initial measurements at the DESY test beam facility using prototype lead tungstate calorimeters in 2019, 2021, and 2022 were made to check the Monte Carlo simulations and the performance of the calorimeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' These tests also investigated different readout schemes (triggered and streaming).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Various upgrades are possible to shorten the running time and to make higher beam energies and thus greater Q2 ranges accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' CONTENTS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Introduction 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' DESY 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Proposed Experiment 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Liquid Hydrogen Target and Scattering Chamber 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Towards a functional LH2 Target for TPEX 12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Lead Tungstate Calorimeters 12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' GEM Detectors 13 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Luminosity and Beam Alignment Monitor 13 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Beamdump / Faraday Cup 16 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Electronics and Readout System 17 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='04708v1 [nucl-ex] 11 Jan 2023 2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Trigger 17 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Front end electronics 17 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Baseline DAQ hardware and software 17 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Possible improvements 18 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Upgrades / Improvements to the Proposed Experiment 18 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Background Considerations 19 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Protons from e±p elastic scattering 19 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Møller and Bhabha scattering 20 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Pion Production 21 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Monte Carlo Simulations 23 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Electrons and Positrons 25 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Protons 26 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Neutrons 27 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' π+ 28 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' π0 29 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Test Beam at DESY 30 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Conclusion 30 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Test Beam Results 31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Calorimeter Setup and Tests 31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Streaming and triggered readout 31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Monte Carlo Simulation of Test Beam 33 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Monte Carlo Simulation for e− + p → e− + p + π0 at 2 GeV 36 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Monte Carlo Simulation for e− + p → e− + n + π+ at 2 GeV 37 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Monte Carlo Simulation for e+ + p → e+ + p + π0 at 2 GeV 38 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Monte Carlo Simulation for e+ + p → e+ + n + π+ at 2 GeV 39 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Monte Carlo Simulation for e− + p → e− + p + π0 at 3 GeV 40 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Monte Carlo Simulation for e− + p → e− + n + π+ at 3 GeV 41 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Monte Carlo Simulation for e+ + p → e+ + p + π0 at 3 GeV 42 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Monte Carlo Simulation for e+ + p → e+ + n + π+ at 3 GeV 43 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Hydrogen Properties 44 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Lead Tungstate, PbWO4, Properties 45 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Numbers Used for Calculations in this Proposal 45 References 45 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' INTRODUCTION Elastic lepton-proton scattering is a fundamental process that should be well described by QED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Understanding this interaction is important to the scientific programs at FAIR, Jefferson Lab, and the future electron-ion collider (EIC) planned for Brookhaven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' It is described theoretically in the Standard Model by a perturbative expansion in α = 1 137 with radiative corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' For more than half a century it has been assumed that the leading single-photon exchange term adequately describes the scattering process and that higher-order terms are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' However, recent experiments at Jefferson Lab have been widely interpreted as evidence that higher order terms are significant in elastic electron-proton scattering and must be included to correctly extract the proton elastic form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Recent experiments, including the OLYMPUS experiment at DESY, show little evidence for significant contributions beyond single photon exchange up to Q2 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='3 (GeV/c)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' It is essential that the QED expansion be studied experimentally at higher Q2 comparing the positron and electron scattering cross section to determine the contribution of higher order terms not normally included in radiative corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 0 1 2 3 4 5 6 7 8 9 ] 2 [(GeV/c) 2 Q 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 2 p M / G p E G p µ Unpolarized Measurements Janssens 66 Berger 71 Litt 70 Bartel 73 Andivahis 94 Walker 94 Christy 04 Qattan 05 Polarization Measurements Jones 00 Pospischil 01 Gayou 02 Punjabi 05 Crawford 07 Puckett 10 Ron 11 Puckett 12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 1: Proton form factor ratio measured using unpolarized [1–8] (blue) and polarized [9–16] (red) techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The proton form factors, Gp E and Gp M, have historically been envisaged as very similar and are often modeled by the same dipole form factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Measurements over the past 50 years using the unpolarized Rosenbluth separation technique yielded a ratio, µp Gp E/Gp M, close to unity over a broad range in Q2 shown by the blue data points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This supported the idea that Gp E and Gp M are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' However, recent measurements using polarization techniques revealed a completely different picture with the ratio decreasing rapidly with increasing Q2 as shown by the red data points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The most commonly proposed explanation for this discrepancy is “hard” two-photon exchange contributions beyond the standard radiative corrections to one-photon exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Two-photon exchange, TPE, (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2) is generally + + + 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2: Feynman diagrams for one- and two-photon exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Further diagrams for bremsstrahlung, vertex, self-energy, and vacuum polarization radiative corrections are not shown but must also be included in calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' included as part of the radiative corrections when analyzing electron-proton scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' However, it is usually only 4 included in the “soft” limit where one of the two photons, in the diagrams with two photons, is assumed to carry negligible momentum and the intermediate hadronic state remains a proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' To include “hard” two-photon exchange, a model for the off-shell, intermediate hadronic state must also be included, making the calculations difficult and model dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' In the Born or single photon exchange approximation the elastic scattering cross section for leptons from protons is given by the reduced Rosenbluth cross section, dσe±p dΩ = dσ dΩ Mott τGp M 2 + ϵGp E 2 ϵ(1 + τ) , (1) where: τ = Q2 4M 2 p and ϵ = (1 + 2(1 + τ) tan2 θl 2 )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' To measure the “hard” two-photon contribution, one can measure the ratio R2γ = σe+p/σe−p at different values of Q2 and ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Note, the interference terms between one- and two-photon exchange change sign between positron and electron scattering and this cross section ratio provides a measure of the two-photon exchange contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The results from the OLYMPUS experiment [17] are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 together with various calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='99 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='9 1 R2γ ϵ Main spectrometer 12◦ telescopes Correlated uncertainty Blunden N only Blunden N + ∆ Bernauer Tomalak 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 Q2 [(GeV/c)2] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3: OLYMPUS results for R2γ as a function of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Inner error bars are statistical while the outer error bars include uncorrelated systematic uncertainties added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The gray band is correlated systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' deviation of the results from unity are small, on the order of 1%, and are below unity at large ϵ and rising with decreasing ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The dispersive calculations of Blunden [18] are systematically above the OLYMPUS results in this energy regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The results below unity cannot be explained by current QED calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The phenomenological prediction from Bernauer [19] and the subtractive dispersion calculation from Tomalak [20] are in better agreement with the OLYMPUS results but appear to rise too quickly as ϵ decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' There is some indication that TPE increases with decreasing ϵ or increasing Q2, suggesting that a significant “hard” two-photon contribution might be present at lower ϵ or higher Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Two other experiments, VEPP-3 [21] and CLAS [22], also measured the “hard” two-photon exchange contribution to electron-proton elastic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' It is difficult to compare the results from the three experiments directly since the measurements are at different points in the (ϵ, Q2) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' To partially account for this, we can compare all the two- photon exchange results by taking the difference with respect to a selected calculation evaluated at the correct (ϵ, Q2) for each data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 4a for Blunden’s calculation and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 4b for Bernauer’s phenomenological prediction, plotted versus Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' In these views, the results from the three experiments are shown to be in reasonable agreement supporting the previous conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The results from the three TPE experiments are all below Q2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='3 (GeV/c)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' In this regime the discrepancy in the form factor ratios is not obvious, so the small “hard” TPE contribution measured is consistent with the measured 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 2 ] 2 [(GeV/c) 2 Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='04 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='03 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='02 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='01 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='05 γ 2 th R γ 2 exp R 2 Difference with respect to Blunden ND vs Q OLYMPUS CLAS VEPP3 (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 2 ] 2 [(GeV/c) 2 Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='04 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='03 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='02 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='01 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='05 γ 2 th R γ 2 exp R 2 Difference with respect to Bernauer vs Q OLYMPUS CLAS VEPP3 (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 4: Difference between the results from the three recent experiments and (a) Blunden’s N+∆ calculation and (b) Bernauer’s prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' form factor ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The suggested slope with ϵ indicates TPE may be important at smaller ϵ or higher Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' But, since this slope appears to deviate from Bernauer’s phenomenological prediction, which fits the observed discrepancy, it may also suggest that “hard” TPE, while contributing, may not explain all of the observed form factor discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Recently, the OLYMPUS data has also been analyzed to determine the charge-averaged yield for elastic lepton- proton scattering [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The result is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This measurement is insensitive to any charge-odd radiative 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 σe++σe− 2 / σdipole Q2 [GeV2/c2] Bernauer Kelly Arrington 03 Arrington 07 OLYMPUS FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 5: The charge-averaged yield for elastic lepton-proton scattering from the OLYMPUS experiment [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' corrections including “hard” two-photon exchange and thus provides a better measure of the proton form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The data shown covers an important range of Q2 where the GM form factor changes slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The calculations by Kelly [24] and Arrington [25, 26] appear to be in better agreement with the data, but Bernauer’s global fit [19] should be redone to incorporate all the OLYMPUS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The two-photon exchange diagram in the QED expansion for electron scattering is an example of the more generic electroweak photon-boson diagram (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 6 where V = Z0, W ±, or γ) which enters into a number of fundamental processes in subatomic physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The γ − Z box is a significant contribution to the asymmetry in parity-violating electron scattering and the γ − W ± box is an important radiative correction in β−decay which must be implemented to extract Vud of the Standard Model from 0+ → 0+ super-allowed nuclear β-decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A workshop [27] was held 6 at the Amherst Center for Fundamental Interactions in September 2017, attended by physicists from these different subfields, to discuss the Electroweak Box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A white paper is in preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 6: More general electroweak box diagram that is important in many fundamental nuclear physics processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The proton form factors are fundamental to hadronic physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Understanding the QED expansion, the role of two-photon exchange, and the scale of radiative corrections at higher Q2 will be crucial in future studies at FAIR, JLab, EIC, and elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The charge-averaged yield eliminates all charge-odd radiative corrections including the leading terms of two-photon exchange, which cannot be calculated with current theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Measuring the ratio of positron-proton to electron-proton scattering is sensitive to the charge-odd radiative corrections and insensitive to the charge-even radiative corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Together they help to study radiative corrections and unravel the proton form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' TPEX, like OLYMPUS, will provide both these measurements at higher Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The discrepancy in the form factor ratio has not been resolved and the role played by two-photon exchange continues to be widely discussed within the nuclear physics community [27–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Further measurements and theoretical work on the role of two-photon exchange on the proton form factors are clearly needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' However, measurements at higher Q2 and smaller ϵ, where the discrepancy is clear and TPE are expected to be larger, are difficult as the cross sections decrease rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' In addition, there are not many laboratories capable of providing both electron and positron beams with sufficient intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The best, and for the foreseeable future only, opportunity is at DESY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This proposal outlines an experiment that could measure R2γ at Q2 up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 (GeV/c)2 or higher, and ϵ below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1 where the form factor discrepancy is clear (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Such an experiment would overlap with the existing OLYMPUS data as a cross-check and would map out the two-photon exchange contribution over a broad range in Q2 and ϵ to provide data to constrain theoretical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The following sections describe the proposed site for the TPEX experiment at DESY, the experimental configuration with its liquid hydrogen target, lead tungstate calorimeters, GEM detectors, luminosity monitor, beamdump/Faraday cup, electronics and data acquisition, and possible improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Sources of background are considered together with solutions and Monte Carlo simulations are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The appendices give more background plots, properties of hydrogen and lead tungstate, some useful numbers for this proposal, and references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' DESY One of the primary requirements for measuring R2γ is high intensity positron and electron beams at energies of several GeV available for nuclear and particle physics applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' DESY is effectively the only high energy physics laboratory currently capable of such intense positron beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The DESY II synchrotron can provide extracted beams of up to 30 nA of positrons and up to 60 nA of electrons at energies between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='3 GeV with a bunch frequency of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A proposal, currently under consideration at DESY for an extension to the present test beam facility [32], to include an extracted lepton beam from DESY II provides a unique opportunity to investigate two-photon exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The extracted beam would only be available when DESY-II is not needed for the operation of PETRA III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' For our purposes the electron and positron beams would be used directly at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV with an option for higher energies in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The current operation of PETRA III uses only electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' That would restrict the availability of positrons to times when PETRA III is not operating due to scheduled maintenance or shutdown periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Hopefully this is not an insurmountable problem and we believe our experiment can be successfully carried out in the shutdown periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Commissioning can be done with just electrons if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' If the storage ring PETRA III is running in “top up” mode (fills every ∼ 30 s) we would not be able to run parasitically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' For “non-top up” mode (fills every ∼ 240 s) it might be possible to have the extracted beam for TPEX between fills for PETRA III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' If the modification to the test beam facility in Hall 2 provides a new, extracted beam area;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' this would allow a left/right symmetric detector arrangement that is much preferred for this proposal to reduce systematics and to V + V + Y V Elastic Inelastic Born Coulomb distortions Dispersion corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='7 increase count rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This proposal requires a significant effort from DESY: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The positron production target has been removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This would need to be reinstalled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A new, extracted beam area would have to be assembled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Two options are possible: A - Hall 2 The floor space is currently occupied by another group that would have to be relocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The “kicker” would have to be moved from its current location on DESY II to one suitable for providing beams to the new area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The shielding wall around DESY II would have to be disassembled and reassembled with a beamline incorporated to the new area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' B - R-Weg The transfer line previously used for DORIS would have to be re-established for a new experimental area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A new area, possibly a specially designed experimental area would have to be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' For both options beamline elements (quadrupoles, steering magnets, vacuum pumps, valves, collimators, beam dump, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=') would have to be found or produced and then installed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The new extracted beam area would need shielding walls, infrastructure services like power and water, an access maze with interlocks, and a new counting hut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Everything would need to be surveyed and aligned and then tests performed to satisfy all safety requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' In addition to enabling the TPEX experiment, an extracted beam facility at DESY would allow other experiments, detector development, and material studies to be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Another interesting experiment would be Deeply Virtual Compton Scattering, DVCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This could also use the TPEX liquid hydrogen target and lead tungstate calorimeters but with a different configuration to allow the scattered lepton and recoil proton to be detected in coincidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Other nuclear physics measurements could also benefit from comparing electron and positron scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' PROPOSED EXPERIMENT The proposed experimental configuration has ten 5 × 5 arrays of lead tungstate crystals at polar angles of 30°, 50°, 70°, 90°, and 110° left and right of the beam axis with the front face of the calorimeter modules at a radius of 1 m from the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Other configurations are possible and can be optimized with Monte Carlo studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A simple schematic for this arrangement is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The electron or positron beam enters the scattering chamber along the beamline (upper-right) and passes through the 20 cm long liquid hydrogen target before exiting the scattering chamber into another section of beam line leading to the beamdump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' At ±8 deg there are 3 m long beampipes connecting the scattering chamber to the lead collimators before the Cherenkov detectors used to monitor the luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' These beamlines are under vacuum and are used to reduce the multiple scattering for the relatively low energy (30–50 MeV) Møller and Bhabha scattered leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Using just the central 3 × 3 array of the 5 × 5 array of crystals to define the acceptance yields a solid angle of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 msr at each angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' With a 20 cm long liquid hydrogen target the acceptance covers ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='7° in polar and azimuthal angle thus data is averaged over a small range in Q2 and ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' We propose to commission the experiment using 2 GeV electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' We do this to debug the electronics, detectors, and data acquisition system taking advantage of the relatively high cross section at 2 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' We would require about 2 weeks of beam time for this commissioning after the experiment was installed and surveyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' We would also like a brief run (few days) with positrons to verify that the beam alignment and performance do not change with positron running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The commissioning run (including a few days with positrons) would also allow a crosscheck of the OLYMPUS data at 30°, 50°, and 70° and give a modest extension in Q2 up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='7 (GeV/c)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Table I shows Q2, ϵ, differential cross section, and event rate expected for one day of running for the proposed left/right symmetric configuration with 2 GeV lepton beams averaging 40 nA on a 20 cm liquid hydrogen target and using just the central 3 × 3 array of crystals to calculate the acceptance area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The TPEX experiment proper would run at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV and would require approximately 6 weeks (2 weeks with electrons and 4 weeks with positrons in total) to collect the required statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Table II shows Q2, ϵ, differential cross section, and event rate expected for one day of running for the proposed configuration with 3 GeV lepton beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 7: Geant4 simulation of a proposed TPEX target, scattering chamber, and detector configuration including the luminosity monitors and beamlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The lepton beam would enter through the beamline in the upper-right, traverse the target cell, and scatter into the detectors or continue straight to the beamdump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' θ Q2 ϵ dσ/dΩ Events/day (GeV/c)2 fb 30° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='834 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='849 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='41 × 107 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 × 106 50° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='611 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='66 × 105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='01 × 105 70° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='386 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='00 × 105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='32 × 104 90° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='224 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='81 × 104 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 × 103 110° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='120 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='22 × 104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='61 × 103 TABLE I: Kinematics, cross section, and events expected in one day for an incident lepton beam of 2 GeV and 40 nA averaged current on a 20 cm liquid hydrogen target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This would extend the measurements to Q2 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='57 (GeV/c)2 where the form factor ratio discrepancy is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The 6 weeks could be divided into two three-week periods if that was more convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' To minimize systematic we would like to switch between positron and electron running as frequently as possible (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 1 day positron, 1 day electron, and 1 day positron repeating).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' θ Q2 ϵ dσ/dΩ Events/day (GeV/c)2 fb 30° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='825 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='41 × 106 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 × 105 50° 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='554 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='51 × 104 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='55 × 103 70° 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='329 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='94 × 103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='17 × 103 90° 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='184 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='65 × 103 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='48 × 102 110° 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='096 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 × 103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='58 × 102 TABLE II: Kinematics, cross section, and events expected in one day for an incident lepton beam of 3 GeV and 40 nA averaged current on a 20 cm liquid hydrogen target and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 msr acceptance and a left/right symmetric detector configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The Q2 range that the proposed TPEX experiment would be capable of reaching is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 8 for the 2 and 3 GeV runs of this proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The reach with TPEX can be seen in relation to the discrepancy in the form factor ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' With additional crystals at back angles the 4 GeV runs would also be possible in a reasonable time frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The TPEX experiment at DESY would also measure the charge-averaged cross section just like the recent result from OLYMPUS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' As mentioned above this cross section is insensitive to charge-odd radiative corrections including “hard” two-photon exchange terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Thus, it provides a more robust measure of the proton form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 9 0 1 2 3 4 5 6 7 8 9 ] 2 [(GeV/c) 2 Q 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 2 p M / G p E G p µ TPEX Range 2 GeV 3 GeV 4 GeV Unpolarized Measurements Janssens 66 Berger 71 Litt 70 Bartel 73 Andivahis 94 Walker 94 Christy 04 Qattan 05 Polarization Measurements Jones 00 Pospischil 01 Gayou 02 Punjabi 05 Crawford 07 Puckett 10 Ron 11 Puckett 12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 8: Proton form factor ratio as before but also showing the Q2 range accessible with the proposed TPEX configuration at 2 and 3 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The 4 GeV range would be possible with additional crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The expected charge-averaged cross section uncertainties (assuming dipole cross section) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 9 for TPEX assuming 6 days of running at 2 GeV and 6 weeks of running at 3 GeV with only 50% data collection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The recent OLYMPUS results are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' LIQUID HYDROGEN TARGET AND SCATTERING CHAMBER The OLYMPUS experiment that previously ran on the DORIS storage ring at DESY used an internal gas target with typical areal density of 3 × 1015 atoms·cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The lepton current averaged around 60 mA, yielding an instantaneous luminosity about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 × 10−6 fb−1· s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' For this new experiment we propose to build a liquid hydrogen target that will yield a luminosity about a factor of 200 times higher than that of the OLYMPUS experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This higher luminosity will greatly shorten the run time needed at 2 GeV and help to make up for the lower cross section at higher beam energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' TABLE III: Target system requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Parameter Performance Requirements Liquid hydrogen T≈20 K and P≥1 atm Cool down time < 3 hrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Exit windows scattering into 25° – 120° and 7° – 9° allowed Target Cell end cap wall thickness tc ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 mm, inner diameter 10 mm < i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' < 20 mm, wall thickness tw ≤ 1 mm, 20 cm in length In order to satisfy the science needs for TPEX, and the safety requirements that always have to be taken into consideration for liquid hydrogen targets, we propose to build a liquid hydrogen target system that is tailored for this new experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The experimental requirements for the target system, detailed in Table III, include a single, 20-cm long liquid hydrogen target with an areal density of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='46×1023 atoms·cm−2 that can accommodate lepton currents up 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 5 ] 2 [(GeV/c) 2 Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 dipole σ / averaged σ Charge Averaged Cross Section / Dipole Measurements Olympus TPEX 2 GeV TPEX 3 GeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 9: Charge-averaged cross section divided by the dipole cross section from OLYMPUS and expected uncertainties and coverage from TPEX at 2 and 3 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' to 60 nA (30 nA for positrons and 60 nA for electrons);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' and long, thin scattering chamber windows to allow particles to be accepted over the large solid angle subtended by the detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Thus, the TPEX cryotarget system requires appropriate engineering and safety considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The Michigan group plans to work closely with the MIT-Bates engineers and an external company, Creare, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=', to design and fabricate the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Figure 10 presents the conceptual design of the target system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Panel (a) shows a schematic overview of the target system, which consists of the scattering chamber, the cryo-cooler system, and the 20 cm long, and 2 cm wide single target cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Details of the target system are shown in panels (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' In order to maximize rigidity and withstand the enormous force from atmospheric pressure, as well as to avoid welded and bolted joints, we propose to machine the scattering chamber from a single piece of aluminum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The dimensions of the scattering chamber windows shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 10a are determined from the solid angle subtended by the calorimeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The two side exit windows cover the polar angles for the PbWO4 crystal calorimeters in the range of 25° < θ < 120°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' At the end of the 3 m long beampipes leading to the luminosity monitors are two tiny exit windows cover a range of 7° < θ < 9°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The vertical dimensions of the two side exit windows cover an azimuthal angle of φ = 0° ± 10°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A schematic drawing of the 20 cm long target cell is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' At the time of this proposal, it had not yet been decided if the target cell diameter should be 10 mm or 20 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This will in part depend on the lepton beam properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The general aim is to minimize the target cell diameter to restrict the amount of hydrogen present in the target system, while minimizing heat load in the cell walls by the beam halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The cell walls are made of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='25 mm thick, drawn aluminum tubes, similar to those used for cigar tube travel cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' It is expected that the entire target system contains not more than 60 gas liters (or 75 ml LH2) of hydrogen gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The maximum beam heat load for the 3 GeV electron/positron beam impinging on the 20 cm long cell at 60 nA is H = 60 nA · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='070 g/cm3 · 20 cm · 30 MeV/(g/cm2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The thermal, or radiation heat load on the 20 cm long and 20 mm diameter target cell is about 730 mW/(n + 1), where n is the number of superinsulation layers wrapped loosely around the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' So, for the expected 10 layers of superinsulation, the thermal heat load will be approximately 70 mW, which is much smaller than the beam heat load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' We expect some bubble formation in the liquid H2 due to the heat load from the lepton beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The long, rectangular slot in the target cell, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 11, allows the bubbles to escape into the top aluminum block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This helps to minimize density fluctuations and target thickness variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The entrance and exit cups of the target cell will be thinned by chemical etching to reduce the amount of material in the beam, and thus the background caused by e± scattering off the target cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 11 (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 10: Conceptual design of the TPEX target chamber: (a) shows the full chamber view with the lepton beam entering from the left;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' (b) is a sectional drawing of the cryocooler system (1 – CH110-LT cryocooler, 2 – hydrogen supply and exhaust lines, 3 – condenser with a cooling loop, 4 – target cell), and (c) is the top view of the target chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 11: Design overview of the 20 cm long target cell: 1 – top block with liquid hydrogen level sensor, 2 – target cell, 3 – bottom block with temperature sensor and heater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 0 O 012 Liquid hydrogen will be filled through a single fill tube that serves as the return tube for the boiled off hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The fill/return tube connects the condenser, which is bolted to a cryo-cooler, with the top aluminum block also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This aluminum block also houses two liquid hydrogen level sensors (with one serving as a backup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Each sensor is a 100 Ω Allen Bradley carbon resistor driven at 20 mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' One Lakeshore Cernox® thin film resistance cryogenic temperature sensor and one (50 Ω, 50 W) cartridge heater are inside the bottom copper block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The temperature sensor, the level sensors, and the heater are all monitored and controlled by a slow control system similar to that used in the MUSE experiment [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The cryo-cooler/condenser combination will closely follow the successful MUSE design [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' We will therefore use the CH110-LT single-stage cryo-cooler from Sumitomo Heavy Industries Ltd [34] for refrigeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This cryo-cooler, in combination with the Sumitomo F-70 compressor [35], was chosen for MUSE [36] over Cryomech partly because Sumitomo has a service center in Darmstadt, Germany, while Cryomech does not have a service center in Europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 43, the cryo-cooler has a cooling power of 25 W at 20 K, which is more than sufficient to cool down and fill the 70 ml LH2 target cell in approximately 2 hours [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Geant4 Monte Carlo simulations will be performed for this conceptual design to verify that the experimental requirements can be met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' These simulations should tell us whether the basic cell design is acceptable, or whether modifications to the scattering chamber exit windows are needed to reduce background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Towards a functional LH2 Target for TPEX The Michigan group plans to work closely with the MIT-Bates engineers and an external company, Creare, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=', to design and fabricate the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The U-M group will start with the current conceptual design, and improvements informed by Geant4 simulations as well as the many lessons learned from building the cryogenic targets for the MUSE and SeaQuest experiments, to complete the engineering design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This will be done in close collaboration with the MIT- Bates engineers, and the external company, Creare1, who has performed the engineering design and the construction of the MUSE target system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' It is anticipated that this process will take about 6 months to allow sufficient time to include periodic reviews by the DESY for compliance and safety issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Safety precautions and the lack of a fully developed slow control system at Creare will not allow a full-blown cool-down test with LH2 to be performed before shipment to DESY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Instead a cool-down test with neon, which has a similar boiling point as hydrogen but is not explosive, has to be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Once general cool-down performance and target operation in vacuum, near 20 K, has been demonstrated, the target system will be shipped to DESY where the neon test will be repeated in a staging area to ensure that all components are still functioning properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A cool-down test with about 5 cm3 of hydrogen, an amount small enough to be safe even if it exploded in the cryostat vessel, will then be performed to start testing the slow control system and the safety procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Finally, a complete integration test in the Hall 2 testbeam area to fully test all components, including slow controls and safety procedures will be performed before starting the production run in the 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' LEAD TUNGSTATE CALORIMETERS For the proposed experiment we are leveraging the R&D experience [37, 38] from the CMS experiment and sub- sequent applications by the Bonn and Mainz groups at CEBAF [39] and for PANDA [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' We would start with ten 5 × 5 arrays of lead tungstate (PbWO4) crystals for a total of 250 crystals, some of which may be available from Mainz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Other configurations are possible and will be investigated with more detailed Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Some properties for lead tungstate are provided in appendix K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' We plan to use crystals 2×2×20 cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The density is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='3 g·cm−3 so each crystal weighs around 664 g, or 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 kg for a 5 × 5 array of crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Lead tungstate has a radiation length X0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8904 cm, so these crystals are approximately 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 X0 for good longitudinal electromagnetic shower confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The Moli`ere radius is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='959 cm, so using just the central 3 × 3 array of crystals for acceptance, the outer ring of crystals contains the transverse shower adequately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The nuclear interaction length for lead tungstate is λI = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='28 cm, so the crystals are roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='986 λI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Again, other configurations are possible and further studies and simulations are in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The energy resolution obtained with lead tungstate for the lepton energy range of interest is approximately 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The Mainz Panda readout design uses Avalanche Photo-Diodes (APD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' We are also considering SiPM and PMT readout schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 1 Creare is a relatively large Small Business of approximately 150 people, including 60 engineers, 50 technicians, machinists and technical specialists, and an in-house machine shop that is accustomed high-demanding high tolerance work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 13 Lead tungstate resolution varies with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' To achieve the best energy resolution, the crystal arrays should be maintained at a constant temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The best energy resolution has been obtained at −25° C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This requires refrigeration and complicates what would otherwise be very simple and compact calorimeter modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Results from the test runs at the DESY test beam facility on prototype lead tungstate calorimeters will be used to determine whether or not such cooling is required or if adequate resolution can be obtained with more modest cooling to have a stable temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' An alternative to lead tungstate is being investigated by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Horn at Catholic University of America.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' She is developing high-density, ceramic glass crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' These would be approximately 15% less dense than lead tungstate, so a larger crystal might be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' But the ceramic glass is much easier to produce and would be 5–10 times cheaper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' In addition, the ceramic glass is not as sensitive to temperature, which would simplify the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' We will be testing both lead tungstate and ceramic glass in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Clearly, the lead tungstate crystals will be a crucial part of the TPEX experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' It will therefore be very important to test and maintain the quality of the crystals whether they are produced by the firm Crytur in the Czech Republic or obtained from existing supplies in Europe or America.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The collaboration has colleagues from Charles University in Prague, Czech Republic who have volunteered to take responsibility for testing the crystals, verifying the quality and maintaining a database for tracking the crystals from delivery to the final calorimeter modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' GEM DETECTORS It is proposed to stack two Gas Electron Multiplier (GEM) detector layers with two-dimensional readout in front of each calorimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Thin absorbers will be placed between the target and the GEMs to stop low-energy Møller or Bhabha leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The GEMs provide spatial information of the traversing charged particle at the 100 micrometer precision level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The hits on two GEM elements are used to form a track segment providing directional information between the impact point on the calorimeter and the event origin in the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This serves to suppress charged- particle backgrounds from regions other than the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Also, the GEMs are insensitive to neutral particles, hence they provide a veto against photons and neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Using the calorimeter hit as a third tracking point will allow a measure of the efficiency of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' An active area of slightly more than 20x20 cm2 is required to fully cover the area of the calorimeter entrance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A total of 20 elements is required to instrument ten calorimeter arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The standard readout strip pitch of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 mm results in 500 channels per axis, or 1,000 channels per GEM element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The full experiment would have 20,000 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Since the occupancy will be at the few percent level at most, zero suppression will reduce the amount of recorded data substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The Hampton group has developed GEM detectors for OLYMPUS, MUSE and DarkLight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Recently, the group has established the novel scheme (NS2) of assembling GEM detectors without gluing, while stretching GEM foils mechanically within a double frame structure, for the first time for nuclear physics applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The scheme makes the assembly fast and low risk, such that even a larger number of GEM elements can be produced fairly easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' LUMINOSITY AND BEAM ALIGNMENT MONITOR The relative luminosity between the electron and positron running modes is the crucial normalization for the proposed measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The luminosity could be monitored by a pair of small-angle detectors positioned downstream on either side of the beamline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This approach was also used in the OLYMPUS experiment [41], and based on the lessons learned from that experiment, could be improved substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Given the running conditions of the proposed measurement, we favor a pair of quartz Cherenkov counters positioned 8° from the beamline to monitor the rates of Møller and Bhabha scattering from atomic electrons in the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' In OLYMPUS, the most accurate determination of the relative luminosity was obtained from the rates of multi- interaction events—in which a Møller or Bhabha event occurred in the same bunch as a forward elastic e±p event [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This method had an overall uncertainty of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='36% and looked promising for future measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Unfortunately it is not feasible for the proposed measurement because of the higher rate per bunch crossing, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The multi-interaction event method requires that the event per bunch rate to be much less than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' However, the monitors for the proposed measurement will see approximately 104 Møller or Bhabha events per bunch crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Instead, the proposed monitor can work by integrating the signal from all particles produced during each bunch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A monitor placed at 8° has a number of advantages relative to the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='3° placement of the OLYMPUS luminosity monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' First, at 8°, the Møller and Bhabha cross sections are only a few percent different, whereas for the OLYMPUS monitors, which covered the symmetric angle (90° in the center-of-mass frame), the two cross sections differed by over 50%, with significant angular dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Second, the Møller/Bhabha rate completely dwarfs the e±p elastic scattering 14 10−8 10−7 10−6 10−5 10−4 10−3 10−2 2◦ 4◦ 6◦ 8◦ 10◦ 12◦ 14◦ 100 101 102 103 104 105 2 GeV 3 GeV TPEX monitors OLYMPUS monitors OLYMPUS TPEX Scattering angle Counts per bunch ep elastic Møller Bhabha (total) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 12: Whereas the forward monitors in OLYMPUS had an event per bunch rate well below 1, the TPEX monitors will see 104 Møller or Bhabha events per bunch crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' PMT PMT Target 3 m Collimator Quartz Cherenkov Radiator 1 cm radius aperture (Not to scale) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 13: Schematic for the proposed luminosity monitor, consisting of two quartz Cherenkov detectors with an acceptance defined by 1 cm radius apertures in high-Z collimators rate, meaning that it is really only sensitive to QED processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' No form factors or any other hadronic corrections2 are needed to calculate the Møller and Bhabha cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Third, the sensitivity to alignment scales as 1/ sin θ, meaning the monitor will be much more robust to small misalignments, which were a significant problem for the OLYMPUS luminosity monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' To test the feasibility of the proposed luminosity monitor, we have developed a preliminary design, and run a Monte Carlo simulation to test the sensitivity to misalignments and beam position shifts, which were the dominant systematic errors for the OLYMPUS luminosity determination [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A schematic of the design is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The monitor consists of two quartz Cherenkov detectors, which act as independent monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Cherenkov detectors were chosen because they are widely used for monitoring in high-rate applications, such as in parity-violating electron scattering [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The two detectors operate independently and can cross-check each other, helping to reduce systematic errors from beam alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' In this design, the monitors are placed 3 m away from the center of the target, along the 8° scattering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The acceptance is defined by a collimator with a circular aperture with a radius of 1 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 other than the radiative correction from vacuum polarization 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6% −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4% −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2% 0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6% −800 −600 −400 −200 0 200 400 600 800 Bias: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='01 %/mm Bias in Le+/Le− Asymmetric Beam Offset [µm] Single Monitor (a) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6% −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4% −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2% 0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6% −800 −600 −400 −200 0 200 400 600 800 Bias: 0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='009 %/mm Bias in Le+/Le− Asymmetric Beam Offset [µm] With Both Monitors (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 14: The potential bias from a charge-asymmetric beam misalignment is a mere 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2%/mm in a single monitor (a), and this is completely eliminated by using a pair of monitors (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8% −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6% −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4% −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2% 0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8% −20 −10 0 10 20 Bias: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0235 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0004 %/mm Bias in Le+/Le− Error in Collimator Position [mm] Single Monitor FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 15: Errors in the positioning of the collimator aperture also have a minimal impact on the relative luminosity determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 14 shows the effect on the luminosity determination from beam misalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The most pernicious misalign- ment would be one that is asymmetric between electron and positron modes, and so this was the focus of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' When using a single monitor, an asymmetric misalignment would cause a mere 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1 %/mm bias in the determi- nation of the relative luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' When using the combination of both a left and right monitor, this bias is completely eliminated to the uncertainty of this simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' For comparison, the OLYMPUS luminosity monitor was sensitive to asymmetric misalignments at the level of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='7 %/mm, and it was estimated that the beam position monitors could control the asymmetric misalignment to within 20 µm [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Such control will probably not be possible in the proposed measurement due to the much smaller beam current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' However the simulation demonstrates that such control is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This is largely due to the flatness of the Møller and Bhabha cross sections at 8° (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' One downside of this robustness is that the proposed monitors are not very sensitive as beam alignment monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 15 shows the simulation results for the effect on the luminosity determination if one of the collimator apertures were to be positioned in a different place than expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The maximum effect occurs when the collimator is shifted to a larger or smaller scattering angle, and so this was the focus of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' For a single monitor, the effect is a mere 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='02 %/mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' In practice, both monitors can have positioning errors, and these effects could end up adding or partially canceling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Regardless, because of the positioning at 8◦, the bias is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' For comparison, the effect on OLYMPUS 16 was approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='13 %/mm, with a survey accuracy of approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Based on the experience gained from OLYMPUS, we can make improvements in the collimator positioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' One obvious improvement is to include integral survey marks on the collimator itself, since the aperture defines the monitor acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The simulation shows that two of the major systematic limitations of the OLYMPUS luminosity monitor will be minimal for the proposed design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The third major systematic, stemming from the residual magnetic field along the beamline, will be irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The proposed design does have other systematic limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The biggest concern will be the amplitude stability of the photomultiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' With about 104 particles passing through the aperture every bunch crossing, there is no way to calibrate the light yield from the data itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' To guard against gain drifts, an external calibration source will be vital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A pulsed light source, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' a UV laser, coupled to a fiber-optic distribution system can be used to monitor the gain of both photomultipliers throughout the experiment, while an independent photodiode can be used to cross check that the laser intensity itself does not drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Several of us have experience with laser calibration systems used in previous experiments [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Sub-percent level accuracy should be achievable, though this will almost certainly be the limiting systematic effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' BEAMDUMP / FARADAY CUP A new extracted beam facility from DESY II will need a beamdump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Schmitz (DESY) and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Tschalar (MIT) have looked into the requirements and proposed very similar configurations with an aluminum core and a copper shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Schmitz’s design was an aluminum cylinder 10 cm in diameter and 50 cm long embedded in a copper shell 22 cm in diameter and 65 cm long and had water cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Tschalar’s design was larger with 20 cm diameter and 50 cm long aluminum in a 32 cm diameter and 75 cm long copper shell but was air cooled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Both recommended that the beamdump be surrounded by neutron absorbing material like cement blocks or borated polyethylene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Assuming a maximum current of 100 nA and a beam energy of 7 GeV the maximum power to be handled is 700 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' To contain the showering you want order of 5 Moli´ere radii laterally and order of 25 radiation lengths longitudinally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' To be conservative we have selected C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Tschalar’s numbers as a starting point Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' To augment the luminosity Beam Dump and Faraday cup Copper Aluminium Insulator Negative Voltage secondary emission Window Vacuum Water cooling FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 16: Schematic of a possible beamdump / Faraday cup for TPEX measurement proposed above we thought it might be useful to modify the beamdump to function as a Faraday cup as 17 well to integrate the charge that passes through the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Then, assuming the length of the target cell and density of liquid hydrogen are known we can get a quick measure of the luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' As shown in the figure an insulated ring held at negative voltage of a few hundred volts is needed to suppress secondary emission from back scattering out of the Faraday cup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The beamdump / Faraday cup is under vacuum but this need only be roughing vacuum pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' ELECTRONICS AND READOUT SYSTEM The requirements for data acquisition are comparatively modest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Less than 300 channels have to be read out, including the luminosity monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' With a readout at a low and fixed frequency, no busy logic is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Trigger The beam has a very low and fixed bunch frequency of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 Hz, allowing us to trigger on the bunch clock instead of a trigger detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The only complication is that for proper gate alignment, the beam bunch signal has to be shifted by up to 80 ms, with stability on the 10 ns level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This can be achieved with an FPGA, which can also generate the required gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A V2495 module from CAEN is available in the collaboration and would be adequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Front end electronics For the calorimeter and luminosity monitors, all proposed readout devices require the acquisition of a time-integrated current pulse with a QDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' However, they differ in the required front end electronics: PMTs require only a base for the high-voltage distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Active bases can minimize power losses and improve stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Such bases are available commercially, or can be manufactured by the collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Circuit designs are readily available and can be adapted to fit the required form factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' No further signal conditioning (except attenuation in the case of the luminosity monitors) is required for interfacing with standard QDCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' APDs and SiPMs require driver and preamplifier circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' In the case of APDs, we would copy the tested design for the PANDA detector [47] from Mainz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' For SiPMs, the MUSE collaboration produced an amplifier design which could be adapted [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Baseline DAQ hardware and software In addition to the V2495 for trigger generation, the baseline configuration for 250+2 detectors would require eight 32-channel QDC modules like CAEN’s V792 or Mesytec’s MQDC-32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This can be housed in a single VME crate and read out via a single board computer (SBC) as the VME controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The data rates are low, with about 6 kB/s for the readout of 252 channels at 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This makes it possible to store all experiment data to a single server outside of the experimental area via standard network file systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This rate requires less than 4 GB per week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The SBU group already developed the DAQ software for the test beams, which will be basis for the DAQ solution of the actual experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' For the GEMs, multiple readout solutions are possible: APV based readout, based on the MPD-4 readout boards already used for SBS@JLAB and MUSE [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' While APVs are out of production, these collaborations have a significant number of APV readout cards and MPDs available, and experience in operating these components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' SAMPA based readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This is currently in development at JLab for TDIS and other projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Compared to GEMs, the signal quality is better and the wave form can be sampled as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' There has been a lot of progress on the testing of the chips, which will be in production for the foreseeable future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' However, a switch would require the procurement of new hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' VMM based readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The VMM chip is considered to be the successor to the APV chip in the Scalable Readout System (SRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This new development is cost-effective and scalable, and has been adopted and recommended by RD-51 in the framework of the SRS readout scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The Mainz MAGIX collaboration recently decided 18 to start using VMM for their GEM readout at MESA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The VMM offers readout with time and pulse shape digitization directly on the front-end card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Two VMM chips are housed on one front-end card to process 128 readout channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The data rate from the GEMs is considerably higher, but still manageable, particularly with zero suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' With 1,000 channels per detector and 20 detectors, the estimated rate is between 500 kB/s (1 sample per event and channel) to 5MB/s (10 samples per event and channel), resulting in about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 TB per week of beam time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' With zero suppression this could be reduced to a level of 300 GB per week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Possible improvements We are evaluating multiple improvements over this baseline design: Higher trigger rate: Instead of triggering just on the beam clock, the FPGA can generate additional gates before and after each trigger, spaced so that all conversion and data transmission can happen before the real gate opens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This triples the trigger and thus data rate—easily handled by the proposed system—but would allow for baseline and background monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Instead of QDCs, which only give information about the integrated charge, the signal wave forms could be digitized with high speed ADCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This would allow even better baseline control, but would increase the bandwidth considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' For example, sampling the signals for 1 µs with 250 MHz at 14 bit would result in about 6 kB/s per channel, less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 MB/s in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' These data rates are still readily managed by the system outlined above, and about 1 TB of storage per week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Commercial solutions for these digitizers exist, but are about factor four more costly than QDCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Cost-effective alternatives are the 12-ch WaveBoard 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 designed by INFN Roma/Genova or commercial boards using the DRS4 chip [49], like CAEN’s V1742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The DRS4 chip realizes an analog buffer to allow for the cost effective and high-speed (multi-GSamples/s) sampling of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The trade-off is considerable dead-time for the conversion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' however this is completely hidden in the proposed experiment by the comparably low trigger rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The digitization of the waveform would provide additional insights into the detected particle and it’s timing, allowing us to improve background rejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The decision on these improvements will be based on our experience with these options in test beams planned for the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' UPGRADES / IMPROVEMENTS TO THE PROPOSED EXPERIMENT While the configuration proposed so far is possible and would allow the two-photon exchange contribution to be investigated in a region where the observed form factor discrepancy is clear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' a number of upgrades are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The current configuration assumes that 250 lead tungstate crystals can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Clearly if less or more crystals are possible the configuration would change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Adding more crystals to the back angle calorimeters would increase the acceptance in a region of low count rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' With an additional 5 × 5 array above and below the current modules the solid angle would be increased from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 msr to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 msr an increase of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' If the showering in the 5 × 5 arrays of PbWO4 is well understood;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' it may be possible to accept a larger area of the calorimeter, say an effective area of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 msr rather than the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 msr using just the central 3 × 3 array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This would increase the acceptance by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Placing a tracking detector, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' GEM, immediately before the calorimeter would help to define the acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This needs to be investigated with test beam studies with a 5 × 5 calorimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Move the back angle calorimeters closer to the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Going to a radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 m would increase the count rate by a factor of four, though increasing the angular range subtended and thus reduce the Q2 resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Addition of GEM tracking may help this but needs further Monte Carlo simulation and study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' These options could increase the count rate significantly, making even higher beam energies accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Table IV shows the kinematic reach and differential cross section for just the back angle, 110°, for various lepton beam en- ergies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A measurement at lepton beam energy of 4 GeV would extend the two-photon exchange measurements to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='39 (GeV/c)2, and only requires an improvement by a factor of five to be comparable to the proposed measurement rate at 3 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 19 Ebeam θ Q2 ϵ dσ/dΩ GeV (GeV/c)2 fb 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 110° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='120 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='22 × 104 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 110° 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='096 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 × 103 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 110° 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='080 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='23 × 102 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 110° 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='068 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='92 × 101 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 110° 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='060 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='00 × 101 TABLE IV: Kinematics and cross section for measurements at 110° for lepton beam energies possible with DESY-II 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' BACKGROUND CONSIDERATIONS 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Protons from e±p elastic scattering As proposed, the experiment does not measure the scattered lepton and proton in coincidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' While this would have some benefits it would also require detectors at far forward angles where the event rates from elastic lepton scattering, Møller and Bhabha scattering, and pion production would be problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Nevertheless, protons from elastic lepton- proton scattering will strike the proposed detectors and will be a source of background for the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 20 30 40 50 60 70 80 90 100 110 120 [deg] lepton θ 0 10 20 30 40 50 60 70 [deg] proton θ ep Elastic Kinematics beam E 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV (a) 20 30 40 50 60 70 80 90 100 110 120 [deg] θ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 3 P [GeV/c] ep Elastic Kinematics Lepton momentum (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV) Lepton momentum (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV) Proton momentum (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV) Proton momentum (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 17: Kinematics for ep elastic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' (a) Proton scattering angle as a function of the lepton scattering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' (b) Lepton and proton momenta as a function of their respective scatting angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 17a shows the relation between the proton polar scattering angle and the lepton scattering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 17b shows the momentum for the lepton and proton as a function of their scattering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The proton momentum is greater than that of the lepton at forward angles but drops more rapidly as its scattering angle increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Protons will also be detected in the calorimeters and will have to be identified and corrected for on an event by event basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The calorimeter modules at over 22 X0 will adequately contain the electromagnetic showers and detect most of the lepton energy, but the proton will not deposit its full energy as the calorimeter is only around one nuclear interaction length in depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Monte Carlo studies (presented below) indicate that the proton will deposit at most 300 to 400 MeV in the calorimeters at 30° and 50° and significantly less at larger angles, particularly if an absorber shield is placed in front of the calorimeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This will allow the lepton signal to be clearly resolved from the proton signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' At 30° the proton rate will be an occasional nuisance as it is significantly less than the lepton rate as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 18a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' However, at 50° the proton rate will be 10–100 times that for the lepton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This is still not a problem, as the rate is manageable (approximately once every three beam spills) plus the deposited proton energy will be more than 700 MeV lower than the lepton’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' But, at 70° the rate for protons is 104 − 105 greater than that for leptons resulting in multiple protons from every beam spill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' However, as discussed in the Monte Carlo section, with a suitable absorber the protons can be stopped before the calorimeter without significantly affecting the lepton signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' It may also be possible to eliminate this if the calorimeter timing and readout is sufficiently fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The β value for the proton is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 18b and is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 at 70°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' That means the protons will arrive around 3 ns after the lepton possibly allowing a timing window to exclude them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' At 90° the proton rate is even higher but the energy is much lower and 20 20 30 40 50 60 70 80 90 100 110 120 [deg] θ 4 10 5 10 6 10 7 10 8 10 9 10 10 10 [fb] σ ep Elastic Kinematics Lepton cross section (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV) Lepton cross section (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV) Proton cross section (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV) Proton cross section (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV) (a) 20 30 40 50 60 70 80 90 100 110 120 [deg] proton θ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='9 1 proton β ep Elastic Kinematics (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV) β Proton (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV) β Proton (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 18: Kinematics for ep elastic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' (a) Cross sections for lepton and proton as a function of their polar angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' (b) Beta for the proton as a function of its polar angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' can be handled by an absorber and/or timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Møller and Bhabha scattering The cross sections for Møller and Bhabha scattering into the detector angles being considered in this proposal are given in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' For an average lepton current of 40 nA incident on a 20 cm liquid hydrogen target the luminosity is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='11 × 10−4 fb−1· s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' If we consider the face of the entire 5 × 5 array of each calorimeter module this corresponds to 10 msr so the luminosity factor becomes 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='11 × 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Multiplying this factor through the cross sections in Table V yields rates ranging from around 4 to 2 × 109 events per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' θ Møller Bhabha e+ Bhabha e− fb fb fb 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV 30° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='223 × 1014 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='863 × 108 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='219 × 1014 50° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='991 × 1014 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='866 × 107 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='989 × 1014 70° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='986 × 1015 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='089 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='985 × 1015 90° diverges 0 diverges 110° 0 0 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV 30° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='223 × 1014 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='274 × 108 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='220 × 1014 50° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='991 × 1014 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='719 × 107 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='989 × 1014 70° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='985 × 1015 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='041 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='985 × 1015 90° diverges 0 diverges 110° 0 0 0 TABLE V: Cross section for Møller and Bhabha scattering as a function of the polar scattering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The energies of these Møller and Bhabha scattered leptons are low, less than 4 MeV at 30° and even lower at the larger angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' For the most part they are still relativistic so a timing cut is not possible except at 90° and possibly at 70° if the calorimeter electronics are fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' To sweep these leptons away would require a magnetic field around 400 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' However, Monte Carlo studies show that a simple 10 mm aluminum absorber before the calorimeter modules will stop these particles from producing any signal in the calorimeter without degrading the response to the higher energy leptons of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Since a 10 mm aluminum plate over the front face of the calorimeter array would work well as part of the cooling system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' the high rate of Møller and Bhabha scattered leptons is not a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 21 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Pion Production Another source of background comes from pion production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' There are four reactions to consider: e− + p → e− + p + π0 (2) e− + p → e− + n + π+ (3) e+ + p → e+ + p + π0 (4) e+ + p → e+ + n + π+ (5) A Monte Carlo pion event generator was used to simulate these four reactions at 2 and 3 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The calorimeter modules in the proposed configuration will be struck by the leptons (electrons or positrons), baryons (protons or neutrons), and pions (π0 or π+) from the various pion production reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' In the case of π0 production the most likely decay to two photons must also be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The event rate per day and the momentum distribution of the leptons, baryons, and pions incident on the 5 × 5 calorimeter array at 30° for the reactions e− + p → e− + p + π0 and → e− + n + π+ for an incident electron beam energy of 2 and 3 GeV are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' No accounting for π0 decay or the energy actually deposited in the calorimeters has been made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A more complete Monte Carlo simulation is in progress and further plots for pion production are provided in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=') The total event rate for electrons from pion production at 2 GeV striking the 5 × 5 face of the calorimeter array at 30° is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='07 × 106 per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This is comparable to the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='92 × 105 events per day expected in the central 3 × 3 array for the elastic scattering events we wish to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' However, the elastic events are peaked around a momentum of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='56 GeV/c while the lepton momentum from pion production has a small peak around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='35 GeV/c and a long tail to much lower momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' With PbWO4’s excellent energy resolution this difference should be easily resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The rates at this angle are such that we can expect one of these pion events every beam spill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This background must be detected and corrected on an event by event basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The deposited energies of the baryons and pions are significantly lower but will also contribute to the background and will need to be handled in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' θ e− + p + π0 e− + n + π+ e+ + p + π0 e+ + n + π+ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV 30° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='08 × 106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='06 × 106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='06 × 106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='08 × 106 50° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='64 × 105 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='54 × 105 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 × 105 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='56 × 105 70° 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='22 × 104 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 × 104 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='28 × 104 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 × 104 90° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='86 × 104 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 × 104 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='84 × 104 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 × 104 110° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 × 104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='45 × 104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='41 × 104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='38 × 104 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV 30° 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='02 × 105 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='08 × 105 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='08 × 105 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='06 × 105 50° 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 × 104 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 × 104 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='64 × 104 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='62 × 104 70° 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='58 × 103 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='54 × 103 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='64 × 103 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='72 × 103 90° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='24 × 103 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='26 × 103 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 × 103 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='26 × 103 110° 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='22 × 102 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 × 102 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 × 102 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='32 × 102 TABLE VI: Event rates per day for leptons from pion production striking the 5 × 5 calorimeter detector arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 19 shows the number of particles detected per day at the calorimeter positioned at 30° for each particle produced in pion production from e−p at 2 and 3 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The plots for e+p are similar and plots for all detector angles are given in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' At higher angles the lepton rates fall quickly and are broadly distributed in momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The rates for protons and neutrons also fall quickly plus they deposit little energy in the calorimeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Pion rates remain fairly uniform with angle but are at a low momentum and as Monte Carlo studies indicate the deposited energy is even less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The π0 decay to two photons however will be a rather uniform, low energy background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Further Monte Carlo studies are needed to verify that the background events can be cleanly resolved from the lepton signals that need to be measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Table VI gives the daily event rates for the leptons from pion production striking the 5×5 array of each calorimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' These rates should be compared with those in Table I and Table II that give the rates for the events of interest striking the central 3 × 3 array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The lepton rate from pion production is generally higher than the elastic scattered events of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' However, the lepton events of interest, arising from elastic ep scattering, are peaked at significantly higher energies while the lepton energies from pion production are lower in energy and distributed over a broad range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 22 0 200 400 600 800 1000 1200 1400 1600 1800 Momentum [MeV/c] 0 50 100 150 200 250 300 350 400 3 10 × p and 30 degrees production at 2 GeV e 0 π 30 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e- e p 0 π p and 30 degrees production at 2 GeV e 0 π (a) 0 200 400 600 800 1000 1200 1400 1600 1800 Momentum [MeV/c] 0 50 100 150 200 250 3 10 × p and 30 degrees production at 2 GeV e + π 30 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e- e n + π p and 30 degrees production at 2 GeV e + π (b) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 Momentum [MeV/c] 0 20 40 60 80 100 120 140 160 180 200 3 10 × p and 30 degrees production at 3 GeV e 0 π 30 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e- e p 0 π p and 30 degrees production at 3 GeV e 0 π (c) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 Momentum [MeV/c] 0 20 40 60 80 100 120 140 160 3 10 × p and 30 degrees production at 3 GeV e + π 30 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e- e n + π p and 30 degrees production at 3 GeV e + π (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 19: Number of particles directed towards the 5 × 5 calorimeter array situated at 30° from the reactions (a) e− + p → e− + p + π0 and (b) e− + p → e− + p + π+ at 2 GeV and (c) e− + p → e− + p + π0 and (d) e− + p → e− + p + π+ at 3 GeV during one day of running at the nominal luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' θ e− + p + π0 e− + n + π+ e+ + p + π0 e+ + n + π+ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV 30° 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='96 × 106 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='04 × 106 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='08 × 106 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='64 × 106 50° 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 × 106 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 × 106 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 × 106 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='06 × 106 70° 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 × 105 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='74 × 105 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 × 105 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='86 × 105 90° 0 0 0 0 110° 0 0 0 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV 30° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='42 × 106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='46 × 106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='62 × 106 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='26 × 106 50° 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='76 × 106 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='56 × 106 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='38 × 106 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='86 × 106 70° 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='02 × 104 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='56 × 104 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='26 × 104 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 × 104 90° 0 0 0 0 110° 0 0 0 0 TABLE VII: Event rates per day for baryons from pion production striking the 5 × 5 calorimeter detector arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 23 θ e− + p + π0 e− + n + π+ e+ + p + π0 e+ + n + π+ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV 30° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='78 × 106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 × 106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='02 × 106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='54 × 106 50° 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 × 106 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='04 × 106 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='36 × 106 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 × 106 70° 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='36 × 106 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='44 × 106 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 × 106 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='94 × 106 90° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 × 106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='52 × 106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 × 106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='86 × 106 110° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='68 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='81 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='21 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='95 × 106 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 GeV 30° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='04 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='34 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='35 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='48 × 106 50° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='96 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='46 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 × 106 70° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='53 × 105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='39 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='81 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='34 × 106 90° 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 × 105 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='58 × 105 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 × 105 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='68 × 105 110° 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 × 105 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 × 105 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='22 × 105 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 × 105 TABLE VIII: Event rates per day for pions from pion production striking the 5 × 5 calorimeter detector arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Table VII and Table VIII give the corresponding daily rates for the baryons and pions striking the 5 × 5 array of each calorimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' While these rates by themselves are comparable to the elastic ep events of interest the energy actually deposited in the calorimeter will be significantly less and should be readily distinguished from the elastic lepton signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Of course more detailed Monte Carlo simulations are necessary and these are in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' MONTE CARLO SIMULATIONS In order to study the energy deposited in the 5×5 calorimeter arrays proposed in this document a simple GEANT4 Monte Carlo [50] simulation was developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Beams of electrons, protons, and pions (π+ and π0) were directed through 1 m of air at the center of a 5×5 calorimeter array at normal incidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Initial momenta of 100 MeV/c to 2500 MeV/c in 100 MeV/c steps were studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Four combinations of absorbers (none, 10 mm Al, 10 mm Al + 10 mm Pb, and 10 mm Al + 20 mm Pb) were placed at the front face of the calorimeter array to study the effect this would have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The 10 mm aluminum plate would naturally form part of the cooling system needed to obtain a stable energy resolution from the PbWO4 crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The various thicknesses of lead were introduced to study how this could be used to reduce background from protons and pions and the effect this would have on the lepton signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 20 illustrates the Monte Carlo studies performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' With the simulations, details of the longitudinal and trans- verse energy distributions can be studied though in an actual experiment only the energy deposited in the individual crystals are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' However, these studies show that the electron shower is effectively contained longitudinally and the transverse distribution is narrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Further studies will investigate the reconstruction of position and angle from the energy deposited in the crystals alone and also unfolding events with multiple incident particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Results for various incident particles and absorbers are presented as a function of the particle momentum and absorber thicknesses in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 24 E_vs_Z Entries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='182026e+07 Mean 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='34 − Std Dev 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='69 100 − 80 − 60 − 40 − 20 − 0 20 40 60 80 100 0 2 4 6 8 10 12 14 E_vs_Z Entries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='182026e+07 Mean 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='34 − Std Dev 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='69 Z Energy Distribution (a) 5 − 4 − 3 − 2 − 1 − 0 1 2 3 4 5 5 − 4 − 3 − 2 − 1 − 0 1 2 3 4 50 100 200 300 400 500 600 700 800 E_Crystal Entries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='182026e+07 Mean x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='001336 Mean y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='001096 − Std Dev x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='9165 Std Dev y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='9169 E_Crystal Entries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='182026e+07 Mean x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='001336 Mean y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='001096 − Std Dev x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='9165 Std Dev y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='9169 Crystal Energy (b) 50 − 40 − 30 − 20 − 10 − 0 10 20 30 40 50 50 − 40 − 30 − 20 − 10 −0 10 20 30 40 500 10 20 30 40 50 E_vs_XY Entries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='182026e+07 Mean x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='01195 Mean y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='005381 Std Dev x 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='874 Std Dev y 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='879 E_vs_XY Entries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='182026e+07 Mean x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='01195 Mean y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='005381 Std Dev x 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='874 Std Dev y 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='879 XY Energy Distribution (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 20: Monte Carlo studies of electron showering in a 5 × 5 PbWO4 calorimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Incident electron momentum was 1000 MeV and a 10 mm aluminum absorber was placed before the crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' (a) Longitudinal energy distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' (b) Total energy detected by each crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' (c) Transverse energy distribution in the calorimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 25 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Electrons and Positrons As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 21 a lepton incident on the central crystal of the calorimeter array deposits almost all its energy in the calorimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Most of that energy (∼ 80%) is in the central crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The shower width is also quite narrow (∼ 10 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Note that the 10 mm aluminum absorber has almost no effect on the lepton shower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The lead absorber increases the transverse width of the shower significantly at low momenta and to a lesser degree at higher momenta resulting in some losses in total energy and the percentage deposited in the central crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Not surprisingly positrons have a virtually identical behavior and are therefore not plotted separately here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 21: Electron showering in a 5 × 5 PbWO4 calorimeter array as a function of incident electron momentum with different absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' (a) Sum of energies in all 25 crystals, (b) Percentage of energy in the central crystal, and (c) RMS width of transverse shower development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 26 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Protons Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 22 shows the results for proton incident on the calorimeter array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The total energy deposited in the calorimeter is significantly less than the incident energies and for the most part is a third for momenta below 1000 MeV/c and between 300 and 400 MeV/c for higher momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This is consistent with the calorimeter being only one nuclear interaction length in depth so the proton has a tendency to pass straight through depositing only a fraction of its energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Most of the energy deposited (∼ 70%) is in the struck crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The absorbers have little effect except at low incident momenta where the proton can be completely absorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This may be useful in stopping the large number of lower energy protons produced at backward angles as well as low energy protons from pion production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Monte Carlo 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Energy Deposition in Peak ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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+page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 22: Proton showering in a 5 × 5 PbWO4 calorimeter array as a function of incident proton momentum and different absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' (a) Sum of energies deposited in all 25 crystals, (b) Percentage of energy in the central crystal, and (c) RMS width of transverse shower development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 27 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Neutrons Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 23 shows the results for neutrons incident on the calorimeter array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The total energy deposited in the calorimeter just 5%–15% of the incident energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' About 50% of the energy deposited is in the struck crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The absorbers have little effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Energy Deposition in Peak 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Monte Carlo 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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+page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Monte Carlo Shower Width ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 23: Neutron showering in a 5 × 5 PbWO4 calorimeter array as a function of incident neutron momentum and different absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' (a) Sum of energies deposited in all 25 crystals, (b) Percentage of energy in the central crystal, and (c) RMS width of transverse shower development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 28 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' π+ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 24 shows the calorimeter response to incident π+ mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The total energy deposited in the calorimeter array varies from around 50% of the incident momenta below 500 MeV to 25% at higher momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The various absorber thicknesses have a small effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 50%–60% is deposited in the central crystal and the RMS width for the transverse shower development is fairly constant around 14 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This reduced signal from π+ will aid in distinguishing them from the leptons of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Response with π− mesons is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='300 ' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='350 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Energy Deposition in Peak ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm 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+page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi+ Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi+ Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Energy Deposition in Peak ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Energy Deposition in Peak ' 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+page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi+ Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi+ Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi+ Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Monte Carlo Shower Width ' 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+page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Energy Deposition in Peak ' 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+page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' 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+page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi+ Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi+ Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi+ Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi+ Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi+ Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi+ Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 24: π+ showering in a 5 × 5 PbWO4 calorimeter array as a function of incident pion momentum with different absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' (a) Sum of energies deposited in all 25 crystals, (b) Percentage of energy in the central crystal, and (c) RMS width of transverse shower development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 29 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' π0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 25 shows the calorimeter response to π0 mesons originating at the target 1 m away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The π0s primarily decay isotropically to two photons that may or may not strike the calorimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' For low energies the probability is small and very little energy is deposited in the calorimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' At higher energies the two photons are boosted in the direction of the calorimeter and deposit a more significant fraction of their original energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The energy deposited is, in any case, spread over a large area and not restricted to the central crystal as can be seen from the plots of the percentage in the central crystal and the RMS width of the shower development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Energy Deposition in Peak ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi+ Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi+ Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi+ Monte Carlo Shower Width ' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi0 Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Monte Carlo Shower Width ' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Energy Deposition in Peak ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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+page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='350 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi+ Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi+ Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi+ Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi0 Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi0 Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi0 Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Electron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Positron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Proton Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Energy Deposition in Peak ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Neutron Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi+ Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi+ Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Percentage Deposited in Peak [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi+ Energy Deposition in Peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Energy Deposited [MeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi0 Energy Deposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi0 Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi0 Momenta [MeV/c] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='MC RMS Shower Width [mm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Pi0 Monte Carlo Shower Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='Absorber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 10 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='10 mm Al + 20 mm Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 25: π0 showering in a 5 × 5 PbWO4 calorimeter array as a function of incident pion momentum with different absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' (a) Sum of energies deposited in all 25 crystals, (b) Percentage of energy in the central crystal, and (c) RMS width of transverse shower development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 30 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' TEST BEAM AT DESY The Monte Carlo studies discussed in the previous section are encouraging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The proposed experiment can make a significant and direct measurement of the two-photon contribution in a region of Q2 and ϵ where the discrepancy is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' However, the Monte Carlo studies must be verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' It is therefore important that the performance of the calorimeter modules be studied in a test beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' An initial prototype calorimeter was tested at the DESY test beam facility in the fall, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The results are reported here in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' These initial tests with a small 3× 3 prototype design are encouraging with good agreement with the Monte Carlo but further tests are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' We propose to perform these measurements at the DESY test beam facility [32] using a 5×5 calorimeter array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The purpose of the test beam activity will be to measure the performance of the PbWO4 calorimeter array and to verify the Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' We would use various energies, with and without the absorber plates, and incident at various positions and angles across the calorimeter array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Monte Carlo simulations can suggest and be used to train reconstruction algorithms but these need to be verified with actual measurements therefore the proposed test beam studies are very important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' If the ceramic glass crystals being developed by Tanja Horn (CUA) are available we would also test these in the prototype calorimeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This clearly has links with efforts underway in Europe and the United States for future detectors for the proposed Electron-Ion Collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' CONCLUSION The observed discrepancy in the proton form factor ratio is a fundamental problem in nuclear physics and possibly in quantum electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Why are the leading order QED radiative corrections insufficient to resolve the discrepancy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Are higher order corrections necessary or are more detailed models for the intermediate hadronic state needed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Or is some other process responsible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' An extracted positron and electron beam facility at DESY would provide a unique opportunity to measure the two-photon exchange contribution to elastic lepton-proton scattering over a kinematic range where the observed discrepancy is clearly evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The above proposal outlines an initial plan for an experimental configuration that could help resolve this issue and provide insight to the radiative corrections needed to understand the proton form factors at higher momentum transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 31 Appendix A: Test Beam Results Over two weeks in September-October, 2019, a calorimeter consisting of a 3×3 array of PbWO4 crystals was studied at the DESY test beam facility [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Tests were made scanning the electron beam across the face of the calorimeter and with different thicknesses of absorber plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' We also compared, in parallel, a traditional, triggered readout with a streaming readout scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Further test beam studies were made in the fall of 2021 and spring of 2022 using a more realistic 5 × 5 array of lead tungstate crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This calorimeter was also cooled and measured at 25°, 10°, -10°, and 25°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A small number of high-density ceramic glass crystals were also tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The analysis of the 2022 test run is ongoing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Calorimeter Setup and Tests The calorimeter used nine 2×2×20 cm3 lead tungstate crystals read out using Hamamatsu R1166 PMTs attached to one end of each crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The crystals were wrapped with one layer of white Tyvek (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 mm thick) and an outer layer of opaque aluminum foil (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='09 mm thick).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The crystal-PMT assemblies were placed inside a black anodized aluminum housing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Copper tubes for water cooling were installed on the outside of the aluminum box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The calorimeter assembly was mounted on an XY translation table but was electrically isolated from the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A collimator and a set of four thin scintillators upstream of the calorimeter were using in the triggered readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 26: Photo of 3x3 lead tungstate calorimeter prototype and trigger detectors used in initial test run at DESY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' High voltage for the PMTs was provided by LeCroy 1461N modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Signals from the PMTs were divided by a 50 Ω splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' One side of each splitter output was connected through a 100 ns delay cable to CAEN V792 QDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The signals from the four thin scintillators were combined in a coincidence unit requiring a triple coincidence that was used to trigger the QDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The other splitter output was connected to a CAEN V1725 digitizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Since the digitizer had only 8 channels a decision was made to read out crystals 1 to 7, and use channel 0 to record the trigger signal in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The gain from each crystal and PMT was matched using a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 GeV beam incident on the center of the crystal and the HV adjusted to give a common value close to the end of the QDC range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Data were collected at beam energies of 2, 3, 4, and 5 GeV and with 2 × 2 mm2 and 8 × 8 mm2 collimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' For all conditions scans were made over the face of the calorimeter and using 0, 1, and 2 cm thick lead absorber plates before the calorimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Typical energy spectra for all four beam energies can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The left figure shows the QDC spectra (with pedestal subtraction) and the right figure the digitizer spectra for events which are in coincidence with the trigger signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 28 shows the sum of all 25 signals with a 5 GeV beam centered on the central crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The root analysis tool functions “gaus” and “crystalball” were used to fit the spectra to determine peak position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Linearity of the peak position with incident energy is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Streaming and triggered readout In the triggered readout scheme all channels of the QDC are read out together after receiving a trigger signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This takes some time during which the QDC is unable to record new events (deadtime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' On the other hand the streaming readout scheme using the digitizer continuously records events in all channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Thus, the streaming system records Trigger 3x3 detector detectors assembly32 0 500 1000 1500 2000 2500 3000 3500 4000 QDC [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='] 0 50 100 150 200 250 Counts = 5 GeV 0 E = 4 GeV 0 E = 3 GeV 0 E = 2 GeV 0 E Crystal 4 0 2000 4000 6000 8000 10000 DIGI [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='] 50 100 150 200 250 300 Counts = 5 GeV 0 E = 4 GeV 0 E = 3 GeV 0 E = 2 GeV 0 E Crystal 4 coincident with channel 0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 27: Deposited energy in central crystal recorded by QDC (left) and digitizer (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The digitizer spectra also required a coincidence with a trigger signal in digitizer channel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 28: Sum of energies deposited in all crystals recorded by QDC (left) and digitizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The digitizer spectrum also required a coincidence with the trigger signal in channel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 29: Energy dependence of the peak position in QDC (left) and digitizer (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' more events in individual channels though many signals may be uncorrelated from cosmic rays, noise, or background events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' To make sense of the large amount of data collected by the digitizer it is necessary to determine the relative timing of all channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Then signals at a common time can be reasonably assumed to arise from the same event, like corresponding to showering in the calorimeter (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Similarly, the relative timing to the trigger signal used for the QDC (connected to channel 0 of the digitizer) can be determined and used to compare the same event collected by the QDC with that recorded by the digitizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' QDC SUM 220 Entries 8627 200 Mean 9227 Std Dev 1679 180 160 140 ounts data 120 gaus fit C 100 crystalball fit 80 60 40 20 0 0 2000 4000 6000 8000 10000 12000 14000 16000 QDC [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' ]DIGLSUM7 DET Entries 35119 250 Mean 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='605e+04 Std Dev 3280 200 Counts data 150 gaus fit crystalball fit 100 50 0 0 5000 10000 15000 20000 25000 30000 35000 40000 DIGI[a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' ]10,000 Mean Channel Sum 5,000 Data LinearFit A·X + B A=1438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='3± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 B = 4247 ± 2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 Beam Energy [GeV]30,000 Data Ax + B A=3322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='7 Sum B = 11087 ± 2 LinearFit1 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='000 Mean Channel 20,000 15,000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 BeamEnergy[GeV]33 DIGI CH4 Entries 21452 Mean 7182 Std Dev 454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 0 2000 4000 6000 8000 10000 12000 DIGI [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='] 0 100 200 300 400 500 Counts DIGI CH4 Entries 21452 Mean 7182 Std Dev 454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5 DIGI CH4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 30: Deposited energy in central detector (channel 4) with coincidence signals in at least 6 neighboring crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' ● ● ●● ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='●● ● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●● ' metadata={'source': 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+page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='7000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='DIGI [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='] QDC [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 31: Left figure shows the energy deposited in the digitizer versus that for the QDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The right figure is the 3D histogram plot of the same data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Monte Carlo Simulation of Test Beam A Monte Carlo simulation of the test beam was developed in Geant4 [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' We use the FTFP BERT physics list provided by Geant to simulate the showers and energy loss processes in the crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' We reproduced the TB24/1 area from the available drawings and technical details provided by DESY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This included the calorimeter, absorber plates, trigger scintillators, collimator, and the origin of the beam source at the DESY II ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' This last item was found to be very important to account for the significant energy straggling observed in the measured spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The Geant4 visualization of the front face of the 3 × 3 calorimeter array is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 33 shows a comparison between the measured QDC data and the simulation for the central crystal for a 2 GeV incident beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 34 shows a similar comparison between simulation and the digitizer data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The discrepancies between simulation and data can be ascribed to an incomplete model of the experimental hall, specifically any material causing energy loss upstream of the hall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' We believe with improved modeling the agreement could be better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 600 $400 200 0 3000 2000 2000 4000 QDC [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='] 6000 1000 DIGI [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='] 800034 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 32: The Geant4 simulation view of the front face of the calorimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 0 500 1000 1500 Energy Deposited (AU) 1 10 2 10 3 10 Counts Geant4 Simulation QDC Data 0 500 1000 1500 Energy Deposited (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 3 10 × Counts Geant4 Simulation QDC Data Geant4 Simulation QDC Data FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 33: Comparison between Geant4 simulation and data from the QDCs using logarithmic (left) and linear (right) scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The simulation is scaled to the height of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 35 0 500 1000 1500 Energy Deposited (AU) 1 10 2 10 3 10 Counts Geant4 Simulation Digitizer Data 0 500 1000 1500 Energy Deposited (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 3 10 × Counts Geant4 Simulation Digitizer Data Geant4 Simulation Digitizer Data FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 34: Comparison between Geant4 simulation and data from the digitizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The simulation is scaled to the height of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 36 Appendix B: Monte Carlo Simulation for e− + p → e− + p + π0 at 2 GeV 0 200 400 600 800 1000 1200 1400 1600 1800 Momentum [MeV/c] 0 50 100 150 200 250 300 350 400 3 10 × p and 30 degrees production at 2 GeV e 0 π 30 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e- e p 0 π p and 30 degrees production at 2 GeV e 0 π (a) 0 200 400 600 800 1000 1200 Momentum [MeV/c] 0 50 100 150 200 250 300 350 400 450 3 10 × p and 50 degrees production at 2 GeV e 0 π 50 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e- e p 0 π p and 50 degrees production at 2 GeV e 0 π (b) 0 100 200 300 400 500 600 700 800 Momentum [MeV/c] 0 20 40 60 80 100 120 140 160 180 3 10 × p and 70 degrees production at 2 GeV e 0 π 70 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e- e p 0 π p and 70 degrees production at 2 GeV e 0 π (c) 0 100 200 300 400 500 600 Momentum [MeV/c] 0 20 40 60 80 100 120 140 160 180 200 220 3 10 × p and 90 degrees production at 2 GeV e 0 π 90 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e- e p 0 π p and 90 degrees production at 2 GeV e 0 π (d) 0 50 100 150 200 250 300 350 400 450 500 Momentum [MeV/c] 0 20 40 60 80 100 120 140 160 180 3 10 × p and 110 degrees production at 2 GeV e 0 π 110 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e- e p 0 π p and 110 degrees production at 2 GeV e 0 π (e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 35: Number of electrons, protons, and π0 directed towards the 5 × 5 calorimeter arrays at 30°, 50°, 70°, 90°, and 110° during one day of running at the nominal luminosity for the reaction e− + p → e− + p + π0 at 2 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 37 Appendix C: Monte Carlo Simulation for e− + p → e− + n + π+ at 2 GeV 0 200 400 600 800 1000 1200 1400 1600 1800 Momentum [MeV/c] 0 50 100 150 200 250 3 10 × p and 30 degrees production at 2 GeV e + π 30 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e- e n + π p and 30 degrees production at 2 GeV e + π (a) 0 200 400 600 800 1000 1200 Momentum [MeV/c] 0 50 100 150 200 250 300 350 3 10 × p and 50 degrees production at 2 GeV e + π 50 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e- e n + π p and 50 degrees production at 2 GeV e + π (b) 0 100 200 300 400 500 600 700 800 Momentum [MeV/c] 0 50 100 150 200 250 3 10 × p and 70 degrees production at 2 GeV e + π 70 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e- e n + π p and 70 degrees production at 2 GeV e + π (c) 0 100 200 300 400 500 600 Momentum [MeV/c] 0 20 40 60 80 100 120 140 160 180 200 220 3 10 × p and 90 degrees production at 2 GeV e + π 90 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e- e n + π p and 90 degrees production at 2 GeV e + π (d) 0 50 100 150 200 250 300 350 400 450 500 Momentum [MeV/c] 0 20 40 60 80 100 120 140 3 10 × p and 110 degrees production at 2 GeV e + π 110 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e- e n + π p and 110 degrees production at 2 GeV e + π (e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 36: Number of electrons, neutrons, and π+ directed towards the 5 × 5 calorimeter arrays at 30°, 50°, 70°, 90°, and 110° during one day of running at the nominal luminosity for the reaction e− + p → e− + n + π+ at 2 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 38 Appendix D: Monte Carlo Simulation for e+ + p → e+ + p + π0 at 2 GeV 0 200 400 600 800 1000 1200 1400 1600 1800 Momentum [MeV/c] 0 20 40 60 80 100 120 140 160 3 10 × p and 30 degrees + production at 2 GeV e 0 π 30 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e+ + e p 0 π p and 30 degrees + production at 2 GeV e 0 π (a) 0 200 400 600 800 1000 1200 Momentum [MeV/c] 0 50 100 150 200 250 300 350 400 450 3 10 × p and 50 degrees production at 2 GeV e 0 π 50 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e- e p 0 π p and 50 degrees production at 2 GeV e 0 π (b) 0 100 200 300 400 500 600 700 800 Momentum [MeV/c] 0 50 100 150 200 250 3 10 × p and 70 degrees + production at 2 GeV e 0 π 70 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e+ + e p 0 π p and 70 degrees + production at 2 GeV e 0 π (c) 0 100 200 300 400 500 600 Momentum [MeV/c] 0 20 40 60 80 100 120 140 3 10 × p and 90 degrees + production at 2 GeV e 0 π 90 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e+ + e p 0 π p and 90 degrees + production at 2 GeV e 0 π (d) 0 50 100 150 200 250 300 350 400 450 500 Momentum [MeV/c] 0 20 40 60 80 100 120 3 10 × p and 110 degrees + production at 2 GeV e 0 π 110 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e+ + e p 0 π p and 110 degrees + production at 2 GeV e 0 π (e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 37: Number of positrons, protons, and π0 directed towards the 5 × 5 calorimeter arrays at 30°, 50°, 70°, 90°, and 110° during one day of running at the nominal luminosity for the reaction e+ + p → e+ + p + π0 at 2 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 39 Appendix E: Monte Carlo Simulation for e+ + p → e+ + n + π+ at 2 GeV 0 200 400 600 800 1000 1200 1400 1600 1800 Momentum [MeV/c] 0 50 100 150 200 250 3 10 × p and 30 degrees + production at 2 GeV e + π 30 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e+ + e n + π p and 30 degrees + production at 2 GeV e + π (a) 0 200 400 600 800 1000 1200 Momentum [MeV/c] 0 50 100 150 200 250 300 3 10 × p and 50 degrees + production at 2 GeV e + π 50 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e+ + e n + π p and 50 degrees + production at 2 GeV e + π (b) 0 100 200 300 400 500 600 700 800 Momentum [MeV/c] 0 50 100 150 200 250 3 10 × p and 70 degrees + production at 2 GeV e + π 70 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e+ + e n + π p and 70 degrees + production at 2 GeV e + π (c) 0 100 200 300 400 500 600 Momentum [MeV/c] 0 50 100 150 200 250 3 10 × p and 90 degrees + production at 2 GeV e + π 90 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e+ + e n + π p and 90 degrees + production at 2 GeV e + π (d) 0 50 100 150 200 250 300 350 400 450 500 Momentum [MeV/c] 0 20 40 60 80 100 120 140 160 180 3 10 × p and 110 degrees + production at 2 GeV e + π 110 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 2 GeV e+ + e n + π p and 110 degrees + production at 2 GeV e + π (e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 38: Number of positrons, neutrons, and π+ directed towards the 5 × 5 calorimeter arrays at 30°, 50°, 70°, 90°, and 110° during one day of running at the nominal luminosity for the reaction e+ + p → e+ + n + π+ at 2 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 40 Appendix F: Monte Carlo Simulation for e− + p → e− + p + π0 at 3 GeV 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 Momentum [MeV/c] 0 20 40 60 80 100 120 140 160 180 200 3 10 × p and 30 degrees production at 3 GeV e 0 π 30 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e- e p 0 π p and 30 degrees production at 3 GeV e 0 π (a) 0 200 400 600 800 1000 1200 1400 Momentum [MeV/c] 0 50 100 150 200 250 3 10 × p and 50 degrees production at 3 GeV e 0 π 50 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e- e p 0 π p and 50 degrees production at 3 GeV e 0 π (b) 0 100 200 300 400 500 600 700 800 900 Momentum [MeV/c] 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 p and 70 degrees production at 3 GeV e 0 π 70 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e- e p 0 π p and 70 degrees production at 3 GeV e 0 π (c) 0 100 200 300 400 500 600 700 Momentum [MeV/c] 0 10000 20000 30000 40000 50000 60000 p and 90 degrees production at 3 GeV e 0 π 90 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e- e p 0 π p and 90 degrees production at 3 GeV e 0 π (d) 0 50 100 150 200 250 300 350 400 450 500 Momentum [MeV/c] 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 p and 110 degrees production at 3 GeV e 0 π 110 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e- e p 0 π p and 110 degrees production at 3 GeV e 0 π (e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 39: Number of electrons, protons, and π0 directed towards the 5 × 5 calorimeter arrays at 30°, 50°, 70°, 90°, and 110° during one day of running at the nominal luminosity for the reaction e− + p → e− + p + π0 at 3 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 41 Appendix G: Monte Carlo Simulation for e− + p → e− + n + π+ at 3 GeV 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 Momentum [MeV/c] 0 20 40 60 80 100 120 140 160 3 10 × p and 30 degrees production at 3 GeV e + π 30 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e- e n + π p and 30 degrees production at 3 GeV e + π (a) 0 200 400 600 800 1000 1200 1400 Momentum [MeV/c] 0 20 40 60 80 100 120 140 160 180 3 10 × p and 50 degrees production at 3 GeV e + π 50 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e- e n + π p and 50 degrees production at 3 GeV e + π (b) 0 100 200 300 400 500 600 700 800 900 Momentum [MeV/c] 0 10000 20000 30000 40000 50000 60000 70000 p and 70 degrees production at 3 GeV e + π 70 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e- e n + π p and 70 degrees production at 3 GeV e + π (c) 0 100 200 300 400 500 600 700 Momentum [MeV/c] 0 10000 20000 30000 40000 50000 p and 90 degrees production at 3 GeV e + π 90 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e- e n + π p and 90 degrees production at 3 GeV e + π (d) 0 50 100 150 200 250 300 350 400 450 500 Momentum [MeV/c] 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 p and 110 degrees production at 3 GeV e + π 110 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e- e n + π p and 110 degrees production at 3 GeV e + π (e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 40: Number of electrons, neutrons, and π+ directed towards the 5 × 5 calorimeter arrays at 30°, 50°, 70°, 90°, and 110° during one day of running at the nominal luminosity for the reaction e− + p → e− + n + π+ at 3 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 42 Appendix H: Monte Carlo Simulation for e+ + p → e+ + p + π0 at 3 GeV 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 Momentum [MeV/c] 0 20 40 60 80 100 120 3 10 × p and 30 degrees + production at 3 GeV e 0 π 30 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e+ + e p 0 π p and 30 degrees + production at 3 GeV e 0 π (a) 0 200 400 600 800 1000 1200 1400 Momentum [MeV/c] 0 50 100 150 200 250 3 10 × p and 50 degrees production at 3 GeV e 0 π 50 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e- e p 0 π p and 50 degrees production at 3 GeV e 0 π (b) 0 100 200 300 400 500 600 700 800 900 Momentum [MeV/c] 0 20 40 60 80 100 120 3 10 × p and 70 degrees + production at 3 GeV e 0 π 70 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e+ + e p 0 π p and 70 degrees + production at 3 GeV e 0 π (c) 0 100 200 300 400 500 600 700 Momentum [MeV/c] 0 5000 10000 15000 20000 25000 30000 p and 90 degrees + production at 3 GeV e 0 π 90 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e+ + e p 0 π p and 90 degrees + production at 3 GeV e 0 π (d) 0 50 100 150 200 250 300 350 400 450 500 Momentum [MeV/c] 0 5000 10000 15000 20000 25000 p and 110 degrees + production at 3 GeV e 0 π 110 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e+ + e p 0 π p and 110 degrees + production at 3 GeV e 0 π (e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 41: Number of positrons, protons, and π0 directed towards the 5 × 5 calorimeter arrays at 30°, 50°, 70°, 90°, and 110° during one day of running at the nominal luminosity for the reaction e+ + p → e+ + p + π0 at 3 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 43 Appendix I: Monte Carlo Simulation for e+ + p → e+ + n + π+ at 3 GeV 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 Momentum [MeV/c] 0 20 40 60 80 100 120 140 160 180 3 10 × p and 30 degrees + production at 3 GeV e + π 30 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e+ + e n + π p and 30 degrees + production at 3 GeV e + π (a) 0 200 400 600 800 1000 1200 1400 Momentum [MeV/c] 0 20 40 60 80 100 120 140 160 3 10 × p and 50 degrees + production at 3 GeV e + π 50 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e+ + e n + π p and 50 degrees + production at 3 GeV e + π (b) 0 100 200 300 400 500 600 700 800 900 Momentum [MeV/c] 0 10000 20000 30000 40000 50000 60000 p and 70 degrees + production at 3 GeV e + π 70 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e+ + e n + π p and 70 degrees + production at 3 GeV e + π (c) 0 100 200 300 400 500 600 700 Momentum [MeV/c] 0 5000 10000 15000 20000 25000 30000 35000 40000 p and 90 degrees + production at 3 GeV e + π 90 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e+ + e n + π p and 90 degrees + production at 3 GeV e + π (d) 0 50 100 150 200 250 300 350 400 450 500 Momentum [MeV/c] 0 10000 20000 30000 40000 50000 110 deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 3 GeV e+ + e n + π p and 110 degrees + production at 3 GeV e + π (e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 42: Number of positrons, neutrons, and π+ directed towards the 5 × 5 calorimeter arrays at 30°, 50°, 70°, 90°, and 110° during one day of running at the nominal luminosity for the reaction e+ + p → e+ + n + π+ at 3 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 44 Appendix J: Hydrogen Properties Hydrogen normally exists as a diatomic molecule, H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The molecule occurs in two forms or allotropes: orthohy- drogen, where the nuclear spins of the two atoms are parallel (J = 0, 2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' and parahydrogen, where the nuclear spins are anti-parallel (J = 1, 3, 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' The concentrations of the two allotropes vary with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' At 80 K the concentration of each is roughly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' At room temperature and above it is generally 75% orthohydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' At 19 K it is 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='75% parahydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Various parameters are given in Table IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Para-Equilibrium Normal Critical point Temperature 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='976 K 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='19 K Pressure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2928 MPa (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='759 atm) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='315 MPa (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='98 atm) Density 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='43 kg/m3 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='59 mol/L) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='12 kg/m3 (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='94 mol/L) Normal boiling point Temperature 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='268 K 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='39 K Pressure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='101325 MPa (1 atm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='101325 MPa (1 atm) Density (liquid) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='78 kg/m3 (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='11 mol/L) 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 kg/m3 (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 mol/L) Density (vapor) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='338 kg/m3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 6636 mol/L) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='331 kg/m3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6604 mol/L) Triple point Temperature 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='803 K 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='957 K Pressure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='00704 MPa (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0695 atm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='00720 MPa (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0711 atm) Density (solid) 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='50 kg/m3 (42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='91 mol/L) 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='71 kg/m3 (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='01 mol/L) Density (liquid) 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='03 kg/m3 (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='21 mol/L) 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 kg/m3 (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='3 mol/L) Density (vapor) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='126 kg/m3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0623 mol/L) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='130 kg/m3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0644 mol/L) Molecular Weight 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='01588 TABLE IX: Parameters for hydrogen FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 43: Cooling power of the cold head being considered for the TPEX liquid hydrogen target system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' CH-110 System SHI Cryogenics Group Performance Low Temperature Version CH11oLT Cold Head Capacity Map Using F-70 Compressor at 60 Hz 300 275 250 225 200 175 [W] 150 HeatLoad 125 100 75 50 25 0 0 20 40 60 80 100 120 140 160 180 Temperature [K]45 Appendix K: Lead Tungstate, PbWO4, Properties From the 2020 Particle Data Group Atomic and Nuclear Properties of Materials: Density PbWO4 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='300 g·cm−3 2 × 2 × 20 cm3 crystal = 664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 g Moli`ere radius = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='959 cm Nuclear Interaction Length λI = 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='3 g·cm−2 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='28 cm Radiation Length X0 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='39 g·cm−2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='8904 cm Energy Loss dE/dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='229 MeV·g−1·cm2 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2 MeV·cm−1 Appendix L: Numbers Used for Calculations in this Proposal Density LH2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='07078 g·cm−3 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='2289×1022 atoms·cm−3 20 cm LH2 target = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='4578×1023 atoms·cm−2 40 nA on 20 cm LH2 target = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1116×1035 cm−2·s−1·sr−1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='1116×10−4 fb−1·s−1·sr−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='0 × 107 fb = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6018 events·s−1 into 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6 msr [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Janssens, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Hofstadter, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Hughes, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Yearian, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 142, 922 (1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Berger, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Burkert, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Knop, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Langenbeck, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Rith, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 35B, 87 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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+page_content=' Busser, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Dix, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Felst, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Harms, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Krehbiel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Andivahis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' D50, 5491 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' D49, 5671 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Christy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' (E94110), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' C70, 015206 (2004), nucl-ex/0401030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' [8] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Qattan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 94, 142301 (2005), nucl-ex/0410010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' (Jefferson Lab Hall A), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 84, 1398 (2000), nucl-ex/9910005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' [10] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Pospischil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' (A1), Eur.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 98, 052301 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Puckett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 104, 242301 (2010), 1005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='3419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' [15] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Ron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' (Jefferson Lab Hall A), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' (OLYMPUS), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 118, 092501 (2017), 1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='04685.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' [18] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Tomalak and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Vanderhaeghen, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' A51, 24 (2015), 1408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='5330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='6908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' 46 [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' Bernauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' (2020), 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content='05349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE3T4oBgHgl3EQfxAvE/content/2301.04708v1.pdf'} +page_content=' [24] J.' 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b/OtFRT4oBgHgl3EQf5Dhx/content/tmp_files/2301.13671v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..37ed08061a8ac35186ffeacff32a798135a4d74d --- /dev/null +++ b/OtFRT4oBgHgl3EQf5Dhx/content/tmp_files/2301.13671v1.pdf.txt @@ -0,0 +1,696 @@ +arXiv:2301.13671v1 [cs.LG] 31 Jan 2023 +Enhancing Hyper-To-Real Space Projections +Through Euclidean Norm Meta-Heuristic +Optimization⋆ +Luiz C. F. Ribeiro[0000−0003−1265−0273], Mateus Roder[0000−0002−3112−5290], +Gustavo H. de Rosa[0000−0002−6442−8343], Leandro A. +Passos[0000−0003−3529−3109], and Jo˜ao P. Papa[0000−0002−6494−7514] +Department of Computing, S˜ao Paulo State University, Bauru, Brazil +{luiz.felix, mateus.roder, gustavo.rosa, leandro.passos, +joao.papa}@unesp.br +Abstract. The continuous computational power growth in the last +decades has made solving several optimization problems significant to +humankind a tractable task; however, tackling some of them remains +a challenge due to the overwhelming amount of candidate solutions to +be evaluated, even by using sophisticated algorithms. In such a context, +a set of nature-inspired stochastic methods, called meta-heuristic opti- +mization, can provide robust approximate solutions to different kinds of +problems with a small computational burden, such as derivative-free real +function optimization. Nevertheless, these methods may converge to in- +adequate solutions if the function landscape is too harsh, e.g., enclosing +too many local optima. Previous works addressed this issue by employ- +ing a hypercomplex representation of the search space, like quaternions, +where the landscape becomes smoother and supposedly easier to opti- +mize. Under this approach, meta-heuristic computations happen in the +hypercomplex space, whereas variables are mapped back to the real do- +main before function evaluation. Despite this latter operation being per- +formed by the Euclidean norm, we have found that after the optimiza- +tion procedure has finished, it is usually possible to obtain even better +solutions by employing the Minkowski p-norm instead and fine-tuning p +through an auxiliary sub-problem with neglecting additional cost and no +hyperparameters. Such behavior was observed in eight well-established +benchmarking functions, thus fostering a new research direction for hy- +percomplex meta-heuristic optimization. +Keywords: Hypercomplex Space, Real-Valued Projection, Euclidean +Norm, Meta-Heuristic Optimization, Benchmarking Functions +⋆ The authors would like to thank S˜ao Paulo Research Foundation (FAPESP) grants +#2013/07375-0, #2014/12236-1, #2017/25908-6, #2019/02205-5, #2019/07825-1, +and #2019/07665-4, and National Council for Scientific and Technological Develop- +ment (CNPq) grants #307066/2017-7 and #427968/2018-6. + +2 +Luiz C. F. Ribeiro et al. +1 +Introduction +Humanity sharpened their mathematical skills over several years of evolution +by researching and studying formal and elegant tools to model world events’ +behavior. In such a context, when dealing with non-trivial problems, it is common +to apply mathematical programming to overcome the before-mentioned tasks or +even to streamline the process. Furthermore, once any prior knowledge might not +be available, mathematical programming, commonly known as optimization [19], +provides an attractive approach to tackle the burden of empirical setups. +In the past decades, a new optimization paradigm called meta-heuristic +has been used to solve several optimization problems [21]. Essentially, a meta- +heuristic is a high-level abstraction of a procedure that generates and selects a +heuristic that aims to provide a feasible solution to the problem. It combines +concepts of exploration and exploitation, i.e., globally searching throughout the +space and enhancing a promising solution based on its neighbors, respectively, +constituted of complex learning procedures and simple searches usually inspired +by biological behaviors. Additionally, they do not require specific domain knowl- +edge and provide mechanisms to avoid susceptibility to local optima convergence. +Although meta-heuristic techniques seem to be an exciting proposal, they still +might perform poorly on challenging objective functions, being trapped in local +optima and not achieving the most suitable solutions. Some attempts as hybrid +variants [12], aging mechanisms [1], and fitness landscape analysis [3] try to deal +with this issue. Relying on more robust search spaces, such as representing each +decision variable as a hypercomplex number, is an alternative approach that is +not fully explored in the literature. +One can perceive that handling hypercomplex spaces is based on the like- +lihood of having more natural fitness landscapes, although mathematically not +proved yet. The most common representations are the quaternions [7] and oc- +tonions [6], which have compelling traits to describe the object’s orientation +in n-dimensional search spaces, being extremely useful in performing rotations +in such spaces [8]. These representations have been successfully used in differ- +ent areas, as in deep learning [14], feature selection [17], special relativity [11] +and quantum mechanics [4]. Regarding meta-heuristic optimization, interesting +results have been achieved in global optimization [5,13,16], although not yet +mathematically guaranteed. +Notwithstanding, hypercomplex optimization also has its particular prob- +lems, i.e., before attempting to feed quaternions or octonions to a real-valued +objective function, one needs to project their values onto a real-valued space, usu- +ally accomplished by the Euclidean norm function. However, to the best of our +knowledge, there is no work in the literature regarding how using the standard +Euclidean norm function might affect the loss of information when projecting +one space onto another. Thus, we are incredibly interested in exploring the pos- +sibility of employing the p-norm function and finding the most suitable p value +that minimizes the loss of information throughout the projection. +In this work, we investigate how employing the p-norm to refine the solution +found by a standard hypercomplex meta-heuristic can affect the obtained re- + +Enhancing Space Projections Through Meta-Heuristic Optimization +3 +sult. In short, we optimize a real function using the standard quaternion-based +variant of the Particle Swarm Algorithm (Q-PSO) [15], i.e., meta-heuristic oper- +ations are performed in the hypercomplex space. In contrast, decision variables +are mapped to the real domain through the Euclidean Norm for function eval- +uation. Notwithstanding, the best solution is refined by finding a more suitable +projection between domains using the p-norm. The rationale for this decision +lies in the fact that this operation is a Euclidean norm generalization. Hence we +resort to fine-tuning new, yet not explored, hyperparameter in the optimization +procedure, thus allowing more robust solutions to be found. Regardless, such a +procedure can be applied to any hypercomplex-based meta-heuristic. Therefore, +this work’s main contributions are twofold: (i) to introduce a generic and in- +expensive procedure to refine solutions found by hypercomplex meta-heuristics; +and (ii) to foster research regarding how to map better hypercomplex to real +values in the context of meta-heuristic optimization. +The remainder of this paper is organized as follows. Sections 2 and 3 present +the theoretical background related to hypercomplex-based spaces (quaternions +and Minkowski p-norm) and meta-heuristic optimization, respectively. Section 4 +discusses the methodology adopted in this work, while Section 5 presents the +experimental results. Finally, Section 6 states conclusions and future works1. +2 +Hypercomplex Representation +A quaternion q is a hypercomplex number, composed of real and complex parts, +being q = a + bi + cj + dk, where a, b, c, d ∈ R and i, j, k are fundamental +quaternions units. The basis equation that defines what a quaternion looks like +is described as follows: +i2 = j2 = k2 = ijk = −1. +(1) +Essentially, a quaternion q is a four-dimensional space representation over +the real numbers, i.e., R4. Given two arbitrary quaternions q1 = a + bi + cj + dk +and q2 = α+βi+γj+δk and a scalar κ ∈ R, we define the quaternion algebra [2] +used throughout this work: +q1 + q2 = (a + bi + cj + dk) + (α + βi + γj + δk) += (a + α) + (b + β)i + (c + γ)j + (d + δ)k, +(2) +q1 − q2 = (a + bi + cj + dk) − (α + βi + γj + δk) += (a − α) + (b − β)i + (c − γ)j + (d − δ)k, +(3) +κq1 = κ(a + bi + cj + dk) += κa + (κb)i + (κc)j + (κd)k. +(4) +1 The source code is available online at https://github.com/lzfelix/lio. + +4 +Luiz C. F. Ribeiro et al. +2.1 +Minkowski p-norm +Another crucial operator that needs to be defined is the p-norm, which is respon- +sible for mapping hypercomplex values to real numbers. Let q be a hypercomplex +number with real coefficients {zd}D−1 +d=0 , one can compute the Minkowski p-norm +as follows: +∥q∥p = +�D−1 +� +d=0 +|zd|p +�1/p +, +(5) +where D is the number of dimensions of the space (2 for complex numbers, and +4 for quaternions, for instance) and p ≥ 1. Common values for the latter variable +are 1 or 2 for the Taxicab and Euclidean norms, respectively. Hence, one can see +the p-norm as a generalization of such norm operators. +3 +Meta-Heuristic Optimization +Optimization is the task of selecting a solution that best fits a function among a +set of possible solutions. Several methods have been applied in this context, such +as grid-search and gradient-based methods. Nevertheless, these methods carry +a massive amount of computation, leading to burdened states in more complex +problems, e.g., exponential and NP-complete problems. +An attempt to overcome such behaviors is to employ a meta-heuristic-based +approach. Meta-heuristic techniques are nature-inspired stochastic algorithms +that mimic an intelligence behavior, often observed in groups of animals, humans, +or nature. Such approaches combine exploration and exploitation mechanisms +in order to achieve sub-optimal solutions with low effort. +In this work, we employed the quaternion variant of the state-of-the-art Par- +ticle Swarm Optimization (PSO) [10] algorithm for function optimization. On +the other hand, since fine-tuning the p hyperparameter is a single-variable opti- +mization task with a small search interval, we resort to the hyperparameter-less +Black Hole (BH) [9] algorithm. +4 +Methodology +This section discusses how the presented meta-heuristics can be combined with +quaternions to perform the so-called “hypercomplex-based meta-heuristic op- +timization”. The proposed approach designated “Last Iteration Optimization” +(LIO) is presented along with the considered benchmarking functions to evaluate +it and the experimental setup. +4.1 +Hypercomplex Optimization +In their original formulation, meta-heuristic algorithms were conceived to opti- +mize real-valued target functions with multiple real parameters. However, one +may decide to represent each decision variable as quaternions. + +Enhancing Space Projections Through Meta-Heuristic Optimization +5 +In this case, each decision variable is represented by a quaternion with its +real coefficients randomly initialized from a uniform distribution in the interval +[0, 1]. Furthermore, the mapping from quaternions to real numbers for function +evaluation becomes a paramount operation, which is usually carried out through +the Euclidean norm. Still, care must be taken to ensure that this transformation +does not yield numbers outside the feasibility region. Hence, hypercomplex co- +efficients are clipped individually to the real interval [0, 1] and the mapping for +each decision variable is performed by the following mapping function: +ˆqj = M(qj, p) += lj + (uj − lj) ∥qj∥p +D1/p , +(6) +such that j = {1, 2, . . ., n}, D is the number of hypercomplex dimensions (4 +for quaternions), lj and uj are the lower and upper bounds for each decision +variable, respectively, and p = 2 in this particular case. +4.2 +Last Iteration Optimization +The main goal of this work consists of refining the solution found by a +hypercomplex-based meta-heuristic using a low-cost procedure. To such an ex- +tent, given a fitness function f : Rn → R, we first optimize it through the Q-PSO +algorithm, which consists in representing each decision variable as a quaternion +with the relations defined in Equations 2, 3, and 4. Once this step is finished, we +have the best candidate solution q⋆ with a real representation ˆq⋆ ∈ Rn, which +is obtained through Equation 6 with p = 2. Shortly, one can compute the best +solution fitness µ as follows: +µ = f +� +M(q⋆ +1, 2), M(q⋆ +2, 2), . . . , M(q⋆ +n, 2) +� +. +(7) +where M(·) is computed according to Equation 6. +We propose a second phase to the optimization pipeline, where the best +solution found is q⋆ is kept fixed, while the hyper-parameter p is unfrozen. Such +an approach allows obtaining a better real representation of q⋆, which translates +to an even smaller fitness value µ⋆. Namely, we aim at solving the following +auxiliary optimization problem: +p⋆ = arg min +p +f +� +M(q⋆ +1, p), M(q⋆ +2, p), . . . , M(q⋆ +n, p) +� +, +st. 1 ≤ p ≤ pmax, +(8) +where pmax denotes the maximum possible value for parameter p. If pmax = 2, +for instance, the problem consists in finding a suitable norm between the Taxicab +and Euclidean ones. +Since the new search interval is usually small, as it is going to be discussed in +Section 4.4, we resort to the traditional BH algorithm since it does not contain + +6 +Luiz C. F. Ribeiro et al. +hyperparameters to be tuned, thus making the process even simpler. As this +procedure is performed for a single decision variable in a small search space, the +time spent in this phase is negligible compared to the Q-PSO step. Furthermore, +since this new step is performed as the new last iteration of the optimization +pipeline, we name it Last Iteration Optimization (LIO). +4.3 +Benchmarking Functions +Table 1 introduces the eight benchmarking functions used to evaluate the pro- +posed approach. +Table 1. Benchmarking functions. +Function +Equation +Bounds +f(x∗) +Sphere +f1(x) = +n +� +i=1 +x2 +i +−10 ≤ xi ≤ 10 +0 +Csendes +f2(x) = �n +i=1 x6 +i +� +2 + sin 1 +xi +� +−1 ≤ xi ≤ 1 +0 +Salomon +f3(x) = 1 − cos(2π +��n +i=1 x2 +i ) + 0.1 +��n +i=1 x2 +i +−100 ≤ xi ≤ 100 +0 +Ackley #1 f4(x) = −20e−0.02√ +n−1 �n +i=1 x2 +i − en−1 �n +i=1 cos(2πxi) + 20 + e +−35 ≤ xi ≤ 35 +0 +Alpine #1 +f5(x) = �n +i=1 |xisin(xi) + 0.1xi| +−10 ≤ xi ≤ 10 +0 +Rastrigin +f6(x) = 10n + �n +i=1 +� +x2 +i − 10cos(2πxi) +� +−5.12 ≤ xi ≤ 5.12 +0 +Schwefel +f7(x) = +� n +� +i=1 +x2 +i +�√π +−100 ≤ xi ≤ 100 +0 +Brown +f8(x) = �n−1 +i=1 +� +(x2 +i )(x2 +i+1+1) + (x2 +i+1)(x2 +i +1)� +−1 ≤ xi ≤ 4 +0 +4.4 +Experimental Setup +The proposed approach divides function optimization into two parts: global and +fine-tuning phases, which correspond to finding µ using Q-PSO and µ⋆ by solving +Equation 8 through the BH algorithm. +Regarding the first phase, we use the same experimental setup from [16]. +Namely, each benchmark function is optimized with n ∈ {10, 25, 50, 100} de- +cision variables, for (2000 · n) iterations using 100 agents. As the amount of +iterations grows considerably fast, we adapt to the early stopping mechanism. +Such a strategy allows detecting if the optimization is stuck for too long in a + +Enhancing Space Projections Through Meta-Heuristic Optimization +7 +local optimum and unlikely to find a better solution, saving computational time. +If the difference of fitness between two consecutive iterations is smaller than +δ = 10−5 for 50 iterations or more, the optimization is halted, and the best fit- +ness found so far is deemed the solution. Despite these values being determined +empirically, they often present the same results as those obtained using all avail- +able iterations, despite using, at most 4% of all iterations for the extreme case +when n = 100. For the Q-PSO hyperparameters we use w = 0.7, c1 = c2 = 1.7, +as well established in the literature. +In the second phase, optimization is performed with pmax = 5, using 20 agents +for 50 iterations, which were determined on preliminary experiments. Further, +we do not rely on early stopping for this phase since it is performed much faster +than the previous one. Finally, we compare the results obtained by Q-PSO and +Q-PSO with LIO. Each experiment is executed 15 times, and the best results +with significance smaller than 0.05, according to the Wilcoxon signed-rank [20], +are highlighted in bold. Regarding the implementation, we used Opytimizer [18] +library. +5 +Experimental Results +Experimental results are presented in Table 2, where the average fitness values +obtained by Q-PSO are compared against their refined versions, computed with +LIO. More specifically, we ran Q-PSO, stored the results, and continued the LIO +(denoted as Q-PSO + LIO). +5.1 +Overall Discussion +Experimental results provided in Table 2 confirm the robustness of the proposed +approach since the Q-PSO + LIO outperformed the standard Q-PSO in the mas- +sive majority of benchmarking functions and configurations. One can highlight, +for instance, that LIO obtained the best results overall, considering all dimen- +sional configurations, in half of the functions, i.e., Sphere, Csendes, Schwefel, +and Brown. Besides, Alpine1 and Rastrigin can also be deliberated, although +Q-PSO obtained similar statistical results. Further, LIO also obtained the best +results considering all functions over three-out-of-four configurations, i.e., 25, 50, +and 100 dimensions, being Q-PSO statistical similar in only two of them. +On the other hand, Q-PSO obtained the best results over two functions, i.e., +Salomon and Rastrigin, considering a 10-dimensional configuration. Such be- +havior is very interesting since Q-PSO performed better over two functions who +share similar characteristics: both are continuous, differentiable, non-separable, +scalable, and multimodal, contemplating the same dimensionality, which may +denote some specific constraint to the model. +Finally, as an overview, the proposed approach can significantly improve Q- +PSO, with an almost insignificant computational burden, and whose growth is +barely insignificant compared to the increase in the number of dimensions, as +discussed in the next section. + +8 +Luiz C. F. Ribeiro et al. +Table 2. Best fitness found by varying the number of decision variables for each +benchmark function. +Functions Dimensions +Q-PSO +Q-PSO + LIO +p +Q-PSO time (s) LIO time (s) +Sphere +10 +1.3447 · 10−7 ± 1.8964 · 10−7 +1.2169 · 10−7 ± 1.6903 · 10−7 +1.99 ± 5.78 +5.66 ± 0.35 +0.29 ± 0.01 +25 +3.1993 · 10−1 ± 1.9855 · 10−1 +3.0657 · 10−1 ± 1.9018 · 10−1 +1.98 ± 0.06 +33.08 ± 3.93 +0.30 ± 0.01 +50 +5.3962 · 100 ± 2.0896 · 100 +5.3205 · 100 ± 2.0266 · 100 +2.01 ± 0.19 +77.55 ± 14.45 +0.37 ± 0.01 +100 +2.3679 · 101 ± 3.3931 · 100 +2.3433 · 101 ± 3.4593 · 100 +1.84 ± 0.38 +158.04 ± 24.79 +0.40 ± 0.01 +Csendes +10 +4.1345 · 10−13 ± 1.2771 · 10−12 +2.3502 · 10−13 ± 6.5480 · 10−13 1.99 ± 0.00 +2.43 ± 0.12 +0.31 ± 0.01 +25 +6.2160 · 10−7 ± 6.2587 · 10−7 +5.8670 · 10−7 ± 5.6680 · 10−7 +1.97 ± 0.11 +6.16 ± 0.35 +0.34 ± 0.02 +50 +8.5587 · 10−6 ± 6.5273 · 10−6 +8.3406 · 10−6 ± 6.2563 · 10−6 +2.01 ± 0.05 +13.31 ± 0.71 +0.39 ± 0.01 +100 +4.9290 · 10−5 ± 2.3214 · 10−5 +4.8460 · 10−5 ± 2.3491 · 10−5 +1.94 ± 0.15 +31.88 ± 2.95 +0.48 ± 0.02 +Salomon +10 +3.4654 · 10−1 ± 1.0242 · 10−1 3.4655 · 10−1 ± 1.0243 · 10−1 +2.04 ± 0.08 +4.97 ± 0.58 +0.30 ± 0.02 +25 +2.0332 · 100 ± 4.3919 · 10−1 +2.0199 · 100 ± 4.4000 · 10−1 +2.07 ± 0.29 +11.73 ± 1.78 +0.31 ± 0.01 +50 +4.1332 · 100 ± 5.6529 · 10−1 +4.1000 · 100 ± 5.4903 · 10−1 +2.22 ± 0.52 +24.11 ± 3.30 +0.36 ± 0.01 +100 +6.3799 · 100 ± 7.2682 · 10−1 +6.3665 · 100 ± 7.0016 · 10−1 +2.07 ± 0.46 +58.78 ± 9.67 +0.44 ± 0.02 +Ackley #1 +10 +8.7839 · 10−1 ± 2.9911 · 10−1 8.7843 · 10−1 ± 2.9914 · 10−1 +2.00 ± 0.00 +7.61 ± 1.23 +0.35 ± 0.02 +25 +1.2330 · 100 ± 2.5165 · 10−1 +1.2293 · 100 ± 2.5283 · 10−1 +1.99 ± 0.00 +35.09 ± 5.11 +0.38 ± 0.02 +50 +1.8135 · 100 ± 1.8909 · 10−1 +1.8062 · 100 ± 1.9024 · 10−1 +2.00 ± 0.01 +61.51 ± 9.02 +0.40 ± 0.02 +100 +2.2038 · 100 ± 1.0338 · 10−1 +2.2006 · 100 ± 1.0280 · 10−1 +2.00 ± 0.01 +125.12 ± 14.76 +0.49 ± 0.03 +Alpine #1 +10 +7.9560 · 10−2 ± 1.2551 · 10−1 7.9515 · 10−2 ± 1.2565 · 10−1 +2.00 ± 0.00 +9.95 ± 2.36 +0.29 ± 0.02 +25 +2.2345 · 100 ± 1.3898 · 100 +2.2240 · 100 ± 1.3957 · 100 +2.01 ± 0.05 +27.84 ± 3.47 +0.31 ± 0.02 +50 +1.2124 · 101 ± 4.1844 · 100 +1.2088 · 101 ± 4.1675 · 100 +2.00 ± 0.05 +63.12 ± 11.53 +0.35 ± 0.01 +100 +2.7375 · 101 ± 7.7322 · 100 +2.7322 · 101 ± 7.7351 · 100 +1.95 ± 0.20 +124.72 ± 20.72 +0.42 ± 0.02 +Rastrigin +10 +1.1608 · 101 ± 5.1079 · 100 +1.1608 · 101 ± 5.1079 · 100 +2.00 ± 0.00 +7.21 ± 0.60 +0.33 ± 0.02 +25 +3.7845 · 101 ± 1.3993 · 101 +3.7673 · 101 ± 1.4040 · 101 +1.99 ± 0.01 +39.67 ± 5.66 +0.33 ± 0.02 +50 +1.4677 · 102 ± 2.3428 · 101 +1.4506 · 102 ± 2.3524 · 101 +2.00 ± 0.04 +94.89 ± 20.97 +0.37 ± 0.02 +100 +4.8812 · 102 ± 4.7400 · 101 +4.8437 · 102 ± 4.7186 · 101 +1.99 ± 0.05 +208.48 ± 39.45 +0.44 ± 0.02 +Schwefel +10 +4.9247 · 10−9 ± 9.7025 · 10−9 +3.5510 · 10−9 ± 7.0349 · 10−9 +1.99 ± 7.36 +6.20 ± 0.43 +0.31 ± 0.01 +25 +1.5213 · 103 ± 2.5509 · 103 +1.1048 · 103 ± 1.2858 · 103 +1.97 ± 0.07 +64.56 ± 12.69 +0.31 ± 0.01 +50 +9.4460 · 104 ± 5.9571 · 104 +8.8915 · 104 ± 5.3935 · 104 +1.88 ± 0.23 +163.12 ± 52.00 +0.35 ± 0.01 +100 +9.4876 · 105 ± 3.8055 · 105 +9.0762 · 105 ± 4.0064 · 105 +2.29 ± 0.56 +349.40 ± 112.34 +0.40 ± 0.02 +Brown +10 +3.3230 · 100 ± 4.3978 · 100 +1.0051 · 100 ± 9.4007 · 10−1 +1.58 ± 0.37 +8.21 ± 3.22 +0.33 ± 0.01 +25 +1.4622 · 101 ± 7.9266 · 100 +5.6664 · 100 ± 2.9834 · 100 +1.13 ± 0.11 +56.16 ± 13.62 +0.34 ± 0.02 +50 +2.8192 · 102 ± 1.0655 · 102 +1.3212 · 102 ± 5.4533 · 101 +1.01 ± 0.02 +120.97 ± 36.77 +0.41 ± 0.03 +100 +1.8173 · 103 ± 3.1725 · 102 +1.2501 · 103 ± 3.0196 · 102 +1.01 ± 0.01 +245.45 ± 82.48 +0.49 ± 0.02 +5.2 +Computational Burden +Germane to this aspect, the results in Table 2 show that LIO takes significantly +less time than the main meta-heuristic to be evaluated. This phenomenon is +expected since the latter involves solving an optimization problem with a single +real variable in a small search interval. Nonetheless, despite this simplicity, our +results show promising results by performing such a task. In the worst-case +scenario, i.e., Csendes function with 10 variables, LIO takes only 12.6% of the + +Enhancing Space Projections Through Meta-Heuristic Optimization +9 +time consumed by Q-PSO, which amounts to 0.31 seconds, while decreasing the +fitness value by a factor of 1.7. +5.3 +How does p Influence Projections? +From the results in Table 2, one can highlight the variation in p-norm value. As +expected, such a variable is highly correlated to the optimization performance, +since small changes in its value resulted in better functions minima. On the other +hand, one can notice that expressive changes in p may also support performance +improvement, as in Brown function. Besides that, as p is changed, the mapping +process, i.e., the projection, from the hypercomplex space to the real one becomes +“less aggressive” to the latter, since the proposed approach gives margin to a +smooth fit for the values obtained in the former space. +Therefore, examining the performance on the optimization functions, one +can observe that employing LIO’s projection, different optimization landscapes +are achieved, and such a process provides better value’s representation from the +hypercomplex search space. It is worth observing that for Rastrigin, Alpine #1, +and Ackley #1 functions, LIO found optimal p values with mean 2 and minimal +standard deviations, thus showing this parameter’s sensitiveness for some bench- +marking functions. Moreover, only LIO optimization for the Schwefel function +with 10 dimensions showed a large standard deviation for this hyperparameter. +In contrast, in the remaining cases, there was no norm larger than 3, suggesting +that in further experiments, and even smaller search intervals (with pmax = 3, +for instance) could be employed. +6 +Conclusion +In this work, we introduced the Last Iteration Optimization (LIO) procedure, +which consists of refining the solution found by a hypercomplex-based meta- +heuristic optimization algorithm by solving a low-cost hyperparameter-less aux- +iliary problem after the primary heuristic has found the best candidate solution. +Such a procedure provided robust results in various benchmarking functions, +showing statistically significant gains in 24 out of 32 experiments, over functions +with diverse characteristics. Since LIO has a low computational burden and is +easy to implement, it can be readily incorporated into other works. +In future studies, we intend to investigate how changing the p parameter +during the global optimization procedure can affect the obtained results. Fur- +thermore, LIO can be extended to find a different p for each decision variable, +making it more flexible, and even other functions can be employed (or learned) +to perform the hypercomplex-to-real mapping process. Ultimately, fine-tuning +the p hyper-parameter of the Minkowski norm opens new research directions for +hypercomplex-based meta-heuristic function optimization methods. + +10 +Luiz C. F. Ribeiro et al. +References +1. Deng, L., Sun, H., Li, C.: Jdf-de: A differential evolution with jrand number de- +creasing mechanism and feedback guide technique for global numerical optimiza- +tion. Applied Intelligence 51(1), 359–376 (2021) +2. Eberly, D.: Quaternion algebra and calculus. 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Papa, J.P., Rosa, G.H., Pereira, D.R., Yang, X.S.: Quaternion-based deep belief +networks fine-tuning. Applied Soft Computing 60, 328–335 (2017) +15. Papa, J.P., de Rosa, G.H., Yang, X.S.: On the hypercomplex-based search spaces +for optimization purposes. In: Nature-Inspired Algorithms and Applied Optimiza- +tion, pp. 119–147. Springer (2018) +16. Passos, L.A., Rodrigues, D., Papa, J.P.: Quaternion-based backtracking search +optimization algorithm. In: 2019 IEEE Congress on Evolutionary Computation +(CEC). pp. 3014–3021. IEEE (2019) +17. de Rosa, G.H., Papa, J.P., Yang, X.S.: A nature-inspired feature selection approach +based on hypercomplex information. Applied Soft Computing 94, 106453 (2020) +18. de Rosa, G.H., Papa, J.P.: Opytimizer: A nature-inspired python optimizer (2019) +19. T¨orn, A., ˇZilinskas, A.: Global optimization, vol. 350. Springer (1989) +20. Wilcoxon, F.: Individual comparisons by ranking methods. Biometrics Bulletin +1(6), 80–83 (1945) +21. Yang, X.S.: Review of meta-heuristics and generalised evolutionary walk algorithm. +International Journal of Bio-Inspired Computation 3(2), 77–84 (2011) + diff --git a/OtFRT4oBgHgl3EQf5Dhx/content/tmp_files/load_file.txt b/OtFRT4oBgHgl3EQf5Dhx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8010ef9d5cbb70bebfe8db583497f2e59af501a5 --- /dev/null +++ b/OtFRT4oBgHgl3EQf5Dhx/content/tmp_files/load_file.txt @@ -0,0 +1,677 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf,len=676 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='13671v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='LG] 31 Jan 2023 Enhancing Hyper-To-Real Space Projections Through Euclidean Norm Meta-Heuristic Optimization⋆ Luiz C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Ribeiro[0000−0003−1265−0273], Mateus Roder[0000−0002−3112−5290], Gustavo H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' de Rosa[0000−0002−6442−8343], Leandro A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Passos[0000−0003−3529−3109], and Jo˜ao P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Papa[0000−0002−6494−7514] Department of Computing, S˜ao Paulo State University, Bauru, Brazil {luiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='felix, mateus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='roder, gustavo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='rosa, leandro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='passos, joao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='papa}@unesp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='br Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' The continuous computational power growth in the last decades has made solving several optimization problems significant to humankind a tractable task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' however, tackling some of them remains a challenge due to the overwhelming amount of candidate solutions to be evaluated, even by using sophisticated algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' In such a context, a set of nature-inspired stochastic methods, called meta-heuristic opti- mization, can provide robust approximate solutions to different kinds of problems with a small computational burden, such as derivative-free real function optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Nevertheless, these methods may converge to in- adequate solutions if the function landscape is too harsh, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=', enclosing too many local optima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Previous works addressed this issue by employ- ing a hypercomplex representation of the search space, like quaternions, where the landscape becomes smoother and supposedly easier to opti- mize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Under this approach, meta-heuristic computations happen in the hypercomplex space, whereas variables are mapped back to the real do- main before function evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Despite this latter operation being per- formed by the Euclidean norm, we have found that after the optimiza- tion procedure has finished, it is usually possible to obtain even better solutions by employing the Minkowski p-norm instead and fine-tuning p through an auxiliary sub-problem with neglecting additional cost and no hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Such behavior was observed in eight well-established benchmarking functions, thus fostering a new research direction for hy- percomplex meta-heuristic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Keywords: Hypercomplex Space, Real-Valued Projection, Euclidean Norm, Meta-Heuristic Optimization, Benchmarking Functions ⋆ The authors would like to thank S˜ao Paulo Research Foundation (FAPESP) grants #2013/07375-0, #2014/12236-1, #2017/25908-6, #2019/02205-5, #2019/07825-1, and #2019/07665-4, and National Council for Scientific and Technological Develop- ment (CNPq) grants #307066/2017-7 and #427968/2018-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' 2 Luiz C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Ribeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' 1 Introduction Humanity sharpened their mathematical skills over several years of evolution by researching and studying formal and elegant tools to model world events’ behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' In such a context, when dealing with non-trivial problems, it is common to apply mathematical programming to overcome the before-mentioned tasks or even to streamline the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Furthermore, once any prior knowledge might not be available, mathematical programming, commonly known as optimization [19], provides an attractive approach to tackle the burden of empirical setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' In the past decades, a new optimization paradigm called meta-heuristic has been used to solve several optimization problems [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Essentially, a meta- heuristic is a high-level abstraction of a procedure that generates and selects a heuristic that aims to provide a feasible solution to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' It combines concepts of exploration and exploitation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=', globally searching throughout the space and enhancing a promising solution based on its neighbors, respectively, constituted of complex learning procedures and simple searches usually inspired by biological behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Additionally, they do not require specific domain knowl- edge and provide mechanisms to avoid susceptibility to local optima convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Although meta-heuristic techniques seem to be an exciting proposal, they still might perform poorly on challenging objective functions, being trapped in local optima and not achieving the most suitable solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Some attempts as hybrid variants [12], aging mechanisms [1], and fitness landscape analysis [3] try to deal with this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Relying on more robust search spaces, such as representing each decision variable as a hypercomplex number, is an alternative approach that is not fully explored in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' One can perceive that handling hypercomplex spaces is based on the like- lihood of having more natural fitness landscapes, although mathematically not proved yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' The most common representations are the quaternions [7] and oc- tonions [6], which have compelling traits to describe the object’s orientation in n-dimensional search spaces, being extremely useful in performing rotations in such spaces [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' These representations have been successfully used in differ- ent areas, as in deep learning [14], feature selection [17], special relativity [11] and quantum mechanics [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Regarding meta-heuristic optimization, interesting results have been achieved in global optimization [5,13,16], although not yet mathematically guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Notwithstanding, hypercomplex optimization also has its particular prob- lems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=', before attempting to feed quaternions or octonions to a real-valued objective function, one needs to project their values onto a real-valued space, usu- ally accomplished by the Euclidean norm function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' However, to the best of our knowledge, there is no work in the literature regarding how using the standard Euclidean norm function might affect the loss of information when projecting one space onto another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Thus, we are incredibly interested in exploring the pos- sibility of employing the p-norm function and finding the most suitable p value that minimizes the loss of information throughout the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' In this work, we investigate how employing the p-norm to refine the solution found by a standard hypercomplex meta-heuristic can affect the obtained re- Enhancing Space Projections Through Meta-Heuristic Optimization 3 sult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' In short, we optimize a real function using the standard quaternion-based variant of the Particle Swarm Algorithm (Q-PSO) [15], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=', meta-heuristic oper- ations are performed in the hypercomplex space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' In contrast, decision variables are mapped to the real domain through the Euclidean Norm for function eval- uation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Notwithstanding, the best solution is refined by finding a more suitable projection between domains using the p-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' The rationale for this decision lies in the fact that this operation is a Euclidean norm generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Hence we resort to fine-tuning new, yet not explored, hyperparameter in the optimization procedure, thus allowing more robust solutions to be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Regardless, such a procedure can be applied to any hypercomplex-based meta-heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Therefore, this work’s main contributions are twofold: (i) to introduce a generic and in- expensive procedure to refine solutions found by hypercomplex meta-heuristics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' and (ii) to foster research regarding how to map better hypercomplex to real values in the context of meta-heuristic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Sections 2 and 3 present the theoretical background related to hypercomplex-based spaces (quaternions and Minkowski p-norm) and meta-heuristic optimization, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Section 4 discusses the methodology adopted in this work, while Section 5 presents the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Finally, Section 6 states conclusions and future works1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' 2 Hypercomplex Representation A quaternion q is a hypercomplex number, composed of real and complex parts, being q = a + bi + cj + dk, where a, b, c, d ∈ R and i, j, k are fundamental quaternions units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' The basis equation that defines what a quaternion looks like is described as follows: i2 = j2 = k2 = ijk = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' (1) Essentially, a quaternion q is a four-dimensional space representation over the real numbers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=', R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Given two arbitrary quaternions q1 = a + bi + cj + dk and q2 = α+βi+γj+δk and a scalar κ ∈ R, we define the quaternion algebra [2] used throughout this work: q1 + q2 = (a + bi + cj + dk) + (α + βi + γj + δk) = (a + α) + (b + β)i + (c + γ)j + (d + δ)k, (2) q1 − q2 = (a + bi + cj + dk) − (α + βi + γj + δk) = (a − α) + (b − β)i + (c − γ)j + (d − δ)k, (3) κq1 = κ(a + bi + cj + dk) = κa + (κb)i + (κc)j + (κd)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' (4) 1 The source code is available online at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='com/lzfelix/lio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' 4 Luiz C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Ribeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='1 Minkowski p-norm Another crucial operator that needs to be defined is the p-norm, which is respon- sible for mapping hypercomplex values to real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Let q be a hypercomplex number with real coefficients {zd}D−1 d=0 , one can compute the Minkowski p-norm as follows: ∥q∥p = �D−1 � d=0 |zd|p �1/p , (5) where D is the number of dimensions of the space (2 for complex numbers, and 4 for quaternions, for instance) and p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Common values for the latter variable are 1 or 2 for the Taxicab and Euclidean norms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Hence, one can see the p-norm as a generalization of such norm operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' 3 Meta-Heuristic Optimization Optimization is the task of selecting a solution that best fits a function among a set of possible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Several methods have been applied in this context, such as grid-search and gradient-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Nevertheless, these methods carry a massive amount of computation, leading to burdened states in more complex problems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=', exponential and NP-complete problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' An attempt to overcome such behaviors is to employ a meta-heuristic-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Meta-heuristic techniques are nature-inspired stochastic algorithms that mimic an intelligence behavior, often observed in groups of animals, humans, or nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Such approaches combine exploration and exploitation mechanisms in order to achieve sub-optimal solutions with low effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' In this work, we employed the quaternion variant of the state-of-the-art Par- ticle Swarm Optimization (PSO) [10] algorithm for function optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' On the other hand, since fine-tuning the p hyperparameter is a single-variable opti- mization task with a small search interval, we resort to the hyperparameter-less Black Hole (BH) [9] algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' 4 Methodology This section discusses how the presented meta-heuristics can be combined with quaternions to perform the so-called “hypercomplex-based meta-heuristic op- timization”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' The proposed approach designated “Last Iteration Optimization” (LIO) is presented along with the considered benchmarking functions to evaluate it and the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='1 Hypercomplex Optimization In their original formulation, meta-heuristic algorithms were conceived to opti- mize real-valued target functions with multiple real parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' However, one may decide to represent each decision variable as quaternions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Enhancing Space Projections Through Meta-Heuristic Optimization 5 In this case, each decision variable is represented by a quaternion with its real coefficients randomly initialized from a uniform distribution in the interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Furthermore, the mapping from quaternions to real numbers for function evaluation becomes a paramount operation, which is usually carried out through the Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Still, care must be taken to ensure that this transformation does not yield numbers outside the feasibility region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Hence, hypercomplex co- efficients are clipped individually to the real interval [0, 1] and the mapping for each decision variable is performed by the following mapping function: ˆqj = M(qj, p) = lj + (uj − lj) ∥qj∥p D1/p , (6) such that j = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=', n}, D is the number of hypercomplex dimensions (4 for quaternions), lj and uj are the lower and upper bounds for each decision variable, respectively, and p = 2 in this particular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='2 Last Iteration Optimization The main goal of this work consists of refining the solution found by a hypercomplex-based meta-heuristic using a low-cost procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' To such an ex- tent, given a fitness function f : Rn → R, we first optimize it through the Q-PSO algorithm, which consists in representing each decision variable as a quaternion with the relations defined in Equations 2, 3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Once this step is finished, we have the best candidate solution q⋆ with a real representation ˆq⋆ ∈ Rn, which is obtained through Equation 6 with p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Shortly, one can compute the best solution fitness µ as follows: µ = f � M(q⋆ 1, 2), M(q⋆ 2, 2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' , M(q⋆ n, 2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' (7) where M(·) is computed according to Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' We propose a second phase to the optimization pipeline, where the best solution found is q⋆ is kept fixed, while the hyper-parameter p is unfrozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Such an approach allows obtaining a better real representation of q⋆, which translates to an even smaller fitness value µ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Namely, we aim at solving the following auxiliary optimization problem: p⋆ = arg min p f � M(q⋆ 1, p), M(q⋆ 2, p), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' , M(q⋆ n, p) � , st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' 1 ≤ p ≤ pmax, (8) where pmax denotes the maximum possible value for parameter p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' If pmax = 2, for instance, the problem consists in finding a suitable norm between the Taxicab and Euclidean ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Since the new search interval is usually small, as it is going to be discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='4, we resort to the traditional BH algorithm since it does not contain 6 Luiz C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Ribeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' hyperparameters to be tuned, thus making the process even simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' As this procedure is performed for a single decision variable in a small search space, the time spent in this phase is negligible compared to the Q-PSO step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Furthermore, since this new step is performed as the new last iteration of the optimization pipeline, we name it Last Iteration Optimization (LIO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='3 Benchmarking Functions Table 1 introduces the eight benchmarking functions used to evaluate the pro- posed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Benchmarking functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Function Equation Bounds f(x∗) Sphere f1(x) = n � i=1 x2 i −10 ≤ xi ≤ 10 0 Csendes f2(x) = �n i=1 x6 i � 2 + sin 1 xi � −1 ≤ xi ≤ 1 0 Salomon f3(x) = 1 − cos(2π ��n i=1 x2 i ) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='1 ��n i=1 x2 i −100 ≤ xi ≤ 100 0 Ackley #1 f4(x) = −20e−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='02√ n−1 �n i=1 x2 i − en−1 �n i=1 cos(2πxi) + 20 + e −35 ≤ xi ≤ 35 0 Alpine #1 f5(x) = �n i=1 |xisin(xi) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='1xi| −10 ≤ xi ≤ 10 0 Rastrigin f6(x) = 10n + �n i=1 � x2 i − 10cos(2πxi) � −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='12 ≤ xi ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='12 0 Schwefel f7(x) = � n � i=1 x2 i �√π −100 ≤ xi ≤ 100 0 Brown f8(x) = �n−1 i=1 � (x2 i )(x2 i+1+1) + (x2 i+1)(x2 i +1)� −1 ≤ xi ≤ 4 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='4 Experimental Setup The proposed approach divides function optimization into two parts: global and fine-tuning phases, which correspond to finding µ using Q-PSO and µ⋆ by solving Equation 8 through the BH algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Regarding the first phase, we use the same experimental setup from [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Namely, each benchmark function is optimized with n ∈ {10, 25, 50, 100} de- cision variables, for (2000 · n) iterations using 100 agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' As the amount of iterations grows considerably fast, we adapt to the early stopping mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Such a strategy allows detecting if the optimization is stuck for too long in a Enhancing Space Projections Through Meta-Heuristic Optimization 7 local optimum and unlikely to find a better solution, saving computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' If the difference of fitness between two consecutive iterations is smaller than δ = 10−5 for 50 iterations or more, the optimization is halted, and the best fit- ness found so far is deemed the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Despite these values being determined empirically, they often present the same results as those obtained using all avail- able iterations, despite using, at most 4% of all iterations for the extreme case when n = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' For the Q-PSO hyperparameters we use w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='7, c1 = c2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='7, as well established in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' In the second phase, optimization is performed with pmax = 5, using 20 agents for 50 iterations, which were determined on preliminary experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Further, we do not rely on early stopping for this phase since it is performed much faster than the previous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Finally, we compare the results obtained by Q-PSO and Q-PSO with LIO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Each experiment is executed 15 times, and the best results with significance smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='05, according to the Wilcoxon signed-rank [20], are highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Regarding the implementation, we used Opytimizer [18] library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' 5 Experimental Results Experimental results are presented in Table 2, where the average fitness values obtained by Q-PSO are compared against their refined versions, computed with LIO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' More specifically, we ran Q-PSO, stored the results, and continued the LIO (denoted as Q-PSO + LIO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='1 Overall Discussion Experimental results provided in Table 2 confirm the robustness of the proposed approach since the Q-PSO + LIO outperformed the standard Q-PSO in the mas- sive majority of benchmarking functions and configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' One can highlight, for instance, that LIO obtained the best results overall, considering all dimen- sional configurations, in half of the functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=', Sphere, Csendes, Schwefel, and Brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Besides, Alpine1 and Rastrigin can also be deliberated, although Q-PSO obtained similar statistical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Further, LIO also obtained the best results considering all functions over three-out-of-four configurations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=', 25, 50, and 100 dimensions, being Q-PSO statistical similar in only two of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' On the other hand, Q-PSO obtained the best results over two functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=', Salomon and Rastrigin, considering a 10-dimensional configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Such be- havior is very interesting since Q-PSO performed better over two functions who share similar characteristics: both are continuous, differentiable, non-separable, scalable, and multimodal, contemplating the same dimensionality, which may denote some specific constraint to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Finally, as an overview, the proposed approach can significantly improve Q- PSO, with an almost insignificant computational burden, and whose growth is barely insignificant compared to the increase in the number of dimensions, as discussed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' 8 Luiz C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Ribeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Best fitness found by varying the number of decision variables for each benchmark function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Functions Dimensions Q-PSO Q-PSO + LIO p Q-PSO time (s) LIO time (s) Sphere 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='3447 · 10−7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='8964 · 10−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='2169 · 10−7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='6903 · 10−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='99 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='78 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='01 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='1993 · 10−1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='9855 · 10−1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='0657 · 10−1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='9018 · 10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='06 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='08 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='01 50 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='3962 · 100 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='0896 · 100 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='3205 · 100 ± 2.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='05 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='12 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='01 100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='7375 · 101 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='7322 · 100 2.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='02 Rastrigin 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='1608 · 101 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='1079 · 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='1608 · 101 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='1079 · 100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='00 ± 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='89 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='02 100 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='8812 · 102 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='7400 · 101 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='8437 · 102 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='7186 · 101 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='05 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='48 ± 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='02 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='2858 · 103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='07 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='56 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='01 50 9.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='01 25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='4622 · 101 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='9266 · 100 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='6664 · 100 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} 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+page_content='8192 · 102 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='0655 · 102 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='3212 · 102 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='4533 · 101 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='02 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='97 ± 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='03 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='8173 · 103 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='1725 · 102 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='2501 · 103 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='0196 · 102 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='01 245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='45 ± 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='02 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='2 Computational Burden Germane to this aspect, the results in Table 2 show that LIO takes significantly less time than the main meta-heuristic to be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' This phenomenon is expected since the latter involves solving an optimization problem with a single real variable in a small search interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Nonetheless, despite this simplicity, our results show promising results by performing such a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' In the worst-case scenario, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=', Csendes function with 10 variables, LIO takes only 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='6% of the Enhancing Space Projections Through Meta-Heuristic Optimization 9 time consumed by Q-PSO, which amounts to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='31 seconds, while decreasing the fitness value by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='3 How does p Influence Projections?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' From the results in Table 2, one can highlight the variation in p-norm value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' As expected, such a variable is highly correlated to the optimization performance, since small changes in its value resulted in better functions minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' On the other hand, one can notice that expressive changes in p may also support performance improvement, as in Brown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Besides that, as p is changed, the mapping process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=', the projection, from the hypercomplex space to the real one becomes “less aggressive” to the latter, since the proposed approach gives margin to a smooth fit for the values obtained in the former space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Therefore, examining the performance on the optimization functions, one can observe that employing LIO’s projection, different optimization landscapes are achieved, and such a process provides better value’s representation from the hypercomplex search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' It is worth observing that for Rastrigin, Alpine #1, and Ackley #1 functions, LIO found optimal p values with mean 2 and minimal standard deviations, thus showing this parameter’s sensitiveness for some bench- marking functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Moreover, only LIO optimization for the Schwefel function with 10 dimensions showed a large standard deviation for this hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' In contrast, in the remaining cases, there was no norm larger than 3, suggesting that in further experiments, and even smaller search intervals (with pmax = 3, for instance) could be employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' 6 Conclusion In this work, we introduced the Last Iteration Optimization (LIO) procedure, which consists of refining the solution found by a hypercomplex-based meta- heuristic optimization algorithm by solving a low-cost hyperparameter-less aux- iliary problem after the primary heuristic has found the best candidate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Such a procedure provided robust results in various benchmarking functions, showing statistically significant gains in 24 out of 32 experiments, over functions with diverse characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Since LIO has a low computational burden and is easy to implement, it can be readily incorporated into other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' In future studies, we intend to investigate how changing the p parameter during the global optimization procedure can affect the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Fur- thermore, LIO can be extended to find a different p for each decision variable, making it more flexible, and even other functions can be employed (or learned) to perform the hypercomplex-to-real mapping process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Ultimately, fine-tuning the p hyper-parameter of the Minkowski norm opens new research directions for hypercomplex-based meta-heuristic function optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' 10 Luiz C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Ribeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Deng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=', Sun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=', Li, C.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=', Papa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=', Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' : A nature-inspired feature selection approach based on hypercomplex information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' Applied Soft Computing 94, 106453 (2020) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' de Rosa, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=', Papa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=': Opytimizer: A nature-inspired python optimizer (2019) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} +page_content=' T¨orn, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQf5Dhx/content/2301.13671v1.pdf'} 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0000000000000000000000000000000000000000..5cb408b819921c29eb764ce91dd98f129f8fa2e9 --- /dev/null +++ b/P9E4T4oBgHgl3EQf-g5_/content/tmp_files/2301.05364v1.pdf.txt @@ -0,0 +1,2391 @@ +Revealing the nature of hidden charm pentaquarks with machine learning +Zhenyu Zhang,1, 2 Jiahao Liu,1, 2 Jifeng Hu,1, 2, ∗ Qian Wang,1, 2, † and Ulf-G. Meißner3, 4, 5, ‡ +1Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, +South China Normal University, Guangzhou 510006, China +2Guangdong-Hong Kong Joint Laboratory of Quantum Matter, +Southern Nuclear Science Computing Center, South China Normal University, Guangzhou 510006, China +3Helmholtz-Institut f¨ur Strahlen- und Kernphysik and Bethe Center +for Theoretical Physics, Universit¨at Bonn, D-53115 Bonn, Germany +4Institute for Advanced Simulation, Institut f¨ur Kernphysik and J¨ulich Center for Hadron Physics, +Forschungszentrum J¨ulich, D-52425 J¨ılich, Germany +5Tbilisi State University, 0186 Tbilisi, Georgia +(Dated: January 16, 2023) +We study the nature of the hidden charm pentaquarks, i.e. the Pc(4312), Pc(4440) and Pc(4457), +with a neural network approach in pionless effective field theory. In this framework, the normal χ2 +fitting approach cannot distinguish the quantum numbers of the Pc(4440) and Pc(4457). In contrast +to that, the neural network-based approach can discriminate them. In addition, we also illustrate +the role of each experimental data bin of the invariant J/ψp mass distribution on the underlying +physics in both neural network and fitting methods. Their similarities and differences demonstrate +that neural network methods can use data information more effectively and directly. This study +provides more insights about how the neural network-based approach predicts the nature of exotic +states from the mass spectrum. +I. +INTRODUCTION +In last two decades we have witnessed the emergence of +tens of candidates for the so-called exotic hadrons which +are beyond the conventional meson and baryon configu- +rations, such as the hidden charm pentaquarks [1–3], the +fully charmed tetraquark X(6900) [4–6], or the doubly +charmed tetraquark T + +cc [7, 8]. The first hidden charm +pentaquarks were reported by the LHCb Collaboration +in 2015 in Ref. [1] by observing the J/ψp invariant +mass via the process Λb → J/ψpK−. +Four years lat- +er, the Pc(4312), Pc(4440) and Pc(4457), were reported +with an order of magnitude larger luminosity in Ref. [3]. +Since then, many interpretations were proposed for these +states, e.g. +compact multi-quark, hadronic molecule, +hybrid, and triangle singularity [9–19]. +Traditionally, +judging whether a model works or not is to compare +it with the experimental data. +However, such a top- +down approach makes the conclusion model-dependent. +Physicists are thus trying to find bottom-up approach- +es [20] to obtain a definitive conclusion. One direction +is to use machine learning to explore hadron properties, +which benefits from the evolution of computing capa- +bilities during the last decades. +For instance, genetic +algorithms [21–23] and neural networks [24–27] can be +utilized to explore the nature of hadrons or the proper- +ties of nuclei. Machine learning has been widely used in +nuclear physics [28–34], high energy nuclear physics [35– +37], experimental data analysis [38, 39] and theoretical +physics [40, 41], even to discover the physical principles +∗ hujf@m.scnu.edu.cn +† qianwang@m.scnu.edu.cn +‡ meissner@hiskp.uni-bonn.de +underneath the experimental data [20, 42, 43]. However, +its application in hadron physics is in its early stage. A +preliminary attempt was made in Refs. [20, 44–46] for the +molecular picture [12], which is one of natural explana- +tions of near-threshold peaks. Refs. [46, 47] demonstrate +that deep learning can be applied to classify poles and +regress the model parameters in the one-channel coupling +case. +Here, extending and improving the works [44, 45], we +consider the multi-channel coupling case. We take the +hidden charm pentaquark states, i.e. Pc(4312), Pc(4440), +Pc(4457) [3] as examples, to achieve the following goals: +• In the Σ(∗) +c +¯D(∗) hadronic molecular picture [48– +50], both the Pc(4440) and the Pc(4457) are relat- +ed to the Σc ¯D∗ threshold. Their spin-parity (JP ) +could be either 1 +2 +−, 3 +2 +− (solution A) or 3 +2 +−, +1 +2 +− (so- +lution B) [48–50], from the viewpoint of pionless +Effective Field Theory. +In solution A, the lower +mass hidden charm pentaquark has lower spin, and +vice versa in solution B. Although the χ2 fit can- +not distinguish the two solutions, we try to find out +which solution is preferred. +• In the traditional fitting approach, the physics is +embedded in the model parameters. One has to ex- +tract the model parameters from the experimental +data and further extract the physical quantities of +interest. We use a neural network (NN) based ap- +proach to extract the pertinent physical quantities +directly and illustrate the role of each experimental +data point (here the bins in the mass distributions) +on the physics. +We demonstrate that the NN-based approach consis- +tently favors solution A over B compared to the fitting +arXiv:2301.05364v1 [hep-ph] 13 Jan 2023 + +2 +approach, and clearly gives the properties of the poles in +the multi-channel case. Our paper is organized as follows: +Sec. II describes how we describe the various pentaquark +states and label them according to their properties. The +model to analyze the LHCb data on the invariant J/ψp +mass distribution is discussed in Sec. III. To train the +NNs, a large number of pseudodata are produced by +Monte Carlo simulation, see Sec. IV. The training and +validation of the NNs are described in Sec. V. Then, we +analyse the data with these NNs, see Sec. VI. We sum- +marize our findings and conclude in Sec. VII. +II. +STATES AND LABELS +The hidden charm pentaquark states are described as +generated from the scattering of the (Σc, Σ∗ +c) and ( ¯D, ¯D∗) +doublets. +As the thresholds of the inelastic channels +J/ψp and ηcp are far away from those of elastic chan- +nels, their effects on the classification of the poles rele- +vant to the physical observables are marginal. The 1 +2 +− +and +3 +2 +− states are given by a three-channel case with +23 = 8 Riemann surfaces denoted as R±±±, see [19] for +a pictorial. Here, the “+” and “-” signs in the ith po- +sition mean the physical and unphysical sheet of the ith +channel, respectively. However, only poles on the phys- +ical sheet R+++ and those close-by ones, i.e. +R−++, +R−−+, R−−−, will contribute significantly to physical +observables. Thus we focus on those poles. As we aim +at extracting the most important information of those +poles, it is sufficient to work with leading order contact +interactions. In this case, the poles accounting for the +near threshold structures can be classified as in Fig. 1. +The labels are defined as follows: The case with three +poles on the R+++, R−++, R−−+ sheets below the first, +second and third thresholds, respectively, is defined as +“bound state” which is labeled as 0. The case with three +poles on the R−++, R−−+, R−−− sheets above the first, +second and third thresholds, respectively, is defined as +“resonance” which is labeled as 1. The case with three +poles on the R−++, R−−+, R−−− sheets below the first, +second and third thresholds, respectively, is defined as +“virtual state” which is labeled as 2. As the JP = 5 +2 +− +state corresponds to a one-channel case, i.e. the Σ∗ +c ¯D∗ +threshold, “bound state”, “resonance state”, and “vir- +tual state” are defined as usually, i.e. by a pole on the +physical sheet below threshold, a pole on the unphysical +sheet below threshold, and a pole on the unphysical sheet +above the threshold, in order. The nature of the poles for +the JP = 1 +2 +−, +3 +2 +−, +5 +2 +− channels are labeled in order as +“lll”, with l = 0, 1, 2. Another label “o” , which we put in +front, is used for the mass order. That is due to the inde- +terminacy of the quantum numbers of the Pc(4440) and +Pc(4457). As they are close to the Σc ¯D∗ threshold, the +S-wave interaction can give both JP = 1 +2 +− and 3 +2 +− states. +In the limit of heavy quark spin symmetry (HQSS), there +are two solutions, i.e. Solution A and Solution B defined + ："Bound state" (0) +："Resonance" (1) +："Virtual state" (2) +FIG. 1. +The definition of the poles for the three coupled +channel case, i.e. +the JP = +1 +2 +− and +3 +2 +− states. +For more +details, see the text. +in Refs. [48–50], as discussed before. In Solution A, the +higher spin state has higher mass, i.e. JPc(4440) = 1 +2 and +JPc(4457) = 3 +2. In Solution B, the situation is reversed, +i.e. +JPc(4457) = +1 +2 and JPc(4440) = +3 +2. +To distinguish +these two scenarios, another label o = 1 and o = 0 for +Solution A and Solution B, respectively, is required. In +total, we have a four-digit label “olll” to denote the sit- +uation of the poles. Taking the “0000” case for example, +it means that the poles for all the quantum numbers are +“bound state” and JPc(4440) = 3 +2, JPc(4457) = 1 +2. Note +that we can also have cases with the poles different from +the three situations discussed above, e.g. the poles are +far away from the thresholds. However their probabilities +are almost zero, and are thus neglected. +III. +MODEL DESCRIPTION +The Pc(4312) and Pc(4440)/Pc(4457) states are close +to the Σc ¯D and Σc ¯D∗ thresholds, respectively, making +them prime candidates for hadronic molecules. +In the +heavy quark limit, these hidden charm pentaquarks are +related to the scattering between the (Σc, Σ∗ +c) and the +( ¯D, ¯D∗) heavy quark spin doublets. +By constructing +the interactions between these two doublets in the ef- +fective field theory (EFT) respecting the heavy quark +spin symmetry (HQSS), Refs. [49, 50] extracted the pole +positions of the seven hidden charm pentaquarks by fit- +ting the J/ψp invariant mass distribution for the pro- +cess Λb → J/ψpK−. +Following the same procedure, +we consider the Σ(∗) +c +¯D(∗) and J/ψp, ηcp channels as +elastic and inelastic channels, respectively. The poten- +tial quantum numbers of hidden charm pentaquarks are +1 +2 +−, +3 +2 +−, +5 +2 +− which correspond to the scattering of the +Σc ¯D−Σc ¯D∗−Σ∗ +c ¯D∗, Σ∗ +c ¯D−Σc ¯D∗−Σ∗ +c ¯D∗, Σ∗ +c ¯D∗, respec- +tively. In the HQSS limit, these scatterings are described +by two parameters C 1 +2 and C 3 +2 where the subscripts refer + +3 +to the light degrees of freedom [50]. The coupling for the +S-wave and D-wave inelastic channels, i.e. the J/ψp and +ηcp channels, are described by two parameters gS and +gD, respectively. +The bare production amplitudes per +JP channel is parameterized by FJ +i with the subscript i +indicating the ith channel. Putting pieces together, the +inelastic amplitude of the J/ψp for the decay process +Λ0 +b → J/ψpK− can be expressed as [49, 50] +U J +i (E, k) = − +� +β +� +d3q +(2π)3 VJ +iβ(k)Gβ(E, q)U J +β (E, q). (1) +Here, U J +i is the ith J/ψp inelastic channel amplitude with +the quantum number J. VJ +iβ is the transition vertex bet- +ween the βth elastic and the ith inelastic channel, and +described by the coupling constants gS for the S-wave +and gD for the D-wave. Gβ is the two-body propagator +of the βth elastic channel. The physical production am- +plitude of the βth elastic channel U J +β for total spin J is +obtained from the Lippmann-Schwinger equation (LSE) +U J +α(E, p) += P J +α − +� +β +� +d3q +(2π)3 V J +αβ(E, p, q)Gβ(E, q)U J +β (E, q), (2) +with P J +α , constructed by seven parameters FJ +n , the pro- +duction amplitude for the αth elastic channel. Note that +above integral equations can be reduced to algebric equa- +tions in the framework of a pionless EFT. For a general +introduction to this type of EFT, see [51]. V J +αβ is the +effective potential, which contains both the scattering +between the elastic channels described by two parame- +ters C1/2, C3/2 and that between the elastic and inelas- +tic channels described by the two parameters gS, gD. In +total, eleven parameters are combined in a vector +P = +� +gS, gD, C 1 +2 , C 3 +2 , +F +1 +2 +1 , F +1 +2 +2 , F +1 +2 +3 , F +3 +2 +1 , F +3 +2 +2 , F +3 +2 +3 , F +5 +2 +1 +� +. +(3) +To describe the invariant mass distribution M(J/ψp), +a composite model is constructed via a probability dis- +tribution function (PDF) as: +PDF(E; P) = α +� +J +� +|U J|2p.s.(E)G(E′ − E)dE′ ++ +(1 − α)Chebyshev6(E) , +(4) +with U J the Λb → J/ψpK− amplitude (2) for total spin +J = 1 +2, +3 +2, +5 +2 and p.s.(E) is the phase space of the corre- +sponding process. A Gaussian function G(E′ −E) which +represents the experimental detector resolution is con- +voluted with the physical invariant mass distribution. +Here, we take take the energy resolution as a constant +of σ ∼ 2.3 MeV [3] without considering its energy depen- +dence. Further, Chebyshev6 is the sixth-order Chebyshev +polynomial which represents the background distribu- +tion, and 1 − α its fraction, α ∈ [0, 1]. The background +fraction in the data is determined to be (96.0±0.8%) +as discussed in Ref. [52]. The coefficients (c0, ..., c6) for +the background component are obtained by fitting the +Chebyshev polynomials to data [53], as listed in Ref. [52]. +Note that the area enclosed by the signal (background) +distribution, i.e. the remnant of the first term after re- +moving the factor α (the second term after removing the +factor 1 − α ) in Eq. (4) are normalized to one. +IV. +MONTE CARLO SIMULATION +Based on the PDF given in Eq. (4), 240184 samples +corresponding to physical states, i.e. with poles not too +far away from the real axis in the considered energy +range, are selected among 1.85 million samples uniform- +ly generated in the space of P, distributed in the mass +window from 4.25 GeV to 4.55 GeV. These samples are +represented as a set: {Hj, Lj} where the j index refers +to a given sample. A histogram H with 150 bins (see +Fig. 2) denotes the invariant mass spectrum of the Pc +states (background included) and the label L indicates +the state label defined in Sec. II. The values of P are +4.25 +4.30 +4.35 +4.40 +4.45 +4.50 +4.55 +mJ/ p[Gev] +200 +400 +600 +800 +1000 +1200 +Monte Carlo Data +Monte Carlo Data +FIG. 2. An example of the simulated invariant mass distribu- +tion M(J/ψp). +uniformly sampled in the following ranges, +gS ∈ [0, 10] GeV−2, +F +1 +2 +3 ∈ [−3600, −3300], +gD ∈ [0.5, 1.5] × gS, +F +3 +2 +1 ∈ [−3900, −3600], +C1/2 ∈ [−20, 0]GeV−2, +F +3 +2 +2 ∈ [−1900, −1600], +(5) +C3/2 ∈ [0.5, 1.5] × C1/2, +F +3 +2 +3 ∈ [−4800, −4500], +F +1 +2 +1 ∈ [0, 300], +F +5 +2 +1 ∈ [600, 900]. +F +1 +2 +2 ∈ [700, 1000], +The ranges are empirically taken to produce a mass spec- +trum similar to the experimental data. Note that gS and +gD as well as C1/2 and C3/2 are strongly correlated, so +the second of these coefficients is related by a factor in + +4 +the range [0.5, 1.5] to the corresponding first one. Note +that these strong correlations were found in [49, 50], and +is also found if one uses much larger set of samples to +train the NNs. The MC production is performed with +the open source software ROOT [53] and GSL [54], as +categorized in Tab. I. Considering different background +fractions, the set of MC samples produced at the back- +ground fraction 1 − α = 90% is denoted as {S90}, so are +the other sets. +TABLE I. +Distribution of parameters. +The state label of +the second column is defined as in Fig. 1. The first label “0” +means JPc(4440) = 3 +2 and JPc(4457) = 1 +2. The first label “1” +means JPc(4440) = +1 +2 and +JPc(4457) = +3 +2. The last column +represents the number of samples for this label. +Mass Relation Label State Label Number of Samples +0 +000 +46951 +1 +000 +4283 +1 +001 +1260 +1 +002 +4360 +0 +100 +3740 +0 +110 +4320 +0 +111 +7520 +1 +111 +360 +0 +200 +9590 +1 +200 +280 +1 +210 +3980 +1 +211 +2690 +1 +220 +50240 +1 +221 +50512 +1 +222 +50098 +V. +TRAINING AND VERIFICATION +The neural network (NN) is implemented with an in- +frastructure of ResNet [55] as discussed in Ref. [52], and +solved with the Adam [56] optimizer parametrized by the +cross-entropy loss function. A reasonable solution can be +obtained after training 500 epochs. Note that 70% of MC +simulation samples are used for training and the rest 30% +for a benchmark. The training details are summarized +in Ref. [52]. In short, the NNs successfully retrieve the +state label with an accuracy of 77.29%, 72.94%, 67.89%, +55.64% for MC simulation samples {S90}, {S92}, {S94}, +{S96}. The prediction accuracy decreases as the back- +ground increases, as expected. +Turning to the mass relation, we verified 100 {S90} +samples corresponding to the label “1xxx”, and another +100 samples corresponding to the label “0xxx”. Fig. 3 +(top)/(bottom) shows the predicted probability distribu- +tion for the label 1xxx/0xxx samples, in which all labels +are successfully retrieved. In other words, the NN can ac- +curately predict the mass relationship. For comparison, +we also perform an analysis with the χ2-fitting method. +Fig. 4 (top)/(bottom) shows the normalized χ2/dof dis- +tributions corresponding to the label 1xxx/0xxx. A 3% +misidentification is observed for the label 1xxx samples. +0.00 +0.01 +0.02 +0.03 +0 +5 +10 +15 +20 +0.97 +0.98 +0.99 +1.00 +Predicted Probability +Number of MC data samples +Predicted label:1000 +Predicted label:0000 +0.00 0.01 0.02 0.03 0.04 +0 +5 +10 +15 +20 +25 +30 +0.95 0.96 0.97 0.98 0.99 1.00 +Predicted Probability +Number of MC data samples +Predicted label:1000 +Predicted label:0000 +FIG. 3. The normalized χ2/dof distributions of the predicted +probability. (top) samples corresponding to the “1000” label. +(bottom) samples corresponding to the “0000” label. +VI. +APPLICATION TO DATA +We now use the well-trained NNs to analyse the exper- +imental data. The results are listed in Tab. II. To reduce +the systematic uncertainties arising from the NNs, we +trained five NN models which have an identical struc- +ture under different initialization, for each group of sam- +ples with different background fractions. The probabili- +ty of the sum of all the other labels is smaller than 1%. +The labels with top three probabilities are 1000, 1001, +1002, which means that the NNs favors Solution A, i.e. +JPc(4440) = 1 +2 and JPc(4457) = 3 +2, as the first label is 1. +The second label 0 and the third label 0 mean that the +poles for the JP = 1 +2 +− and JP = 3 +2 +− channels behave as +“bound states”, which is different from the virtual state +conclusion for the Pc(4312) in Ref. [20] also using ma- +chine learning. The pole situation for the JP = 5 +2 +− is +undetermined, i.e. all the three labels 0, 1, 2 appearing +for the forth label, as the structure around the Σ∗ +c ¯D∗ is +not significant. To resolve this issue, precise data in this +energy region would be needed. + +5 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +²/d.o.f. +0 +2 +4 +6 +8 +10 +12 +14 +16 +Number of MC data samples +Success +Failure +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +²/d.o.f. +0 +2 +4 +6 +8 +10 +12 +14 +16 +Number of MC data samples +Success +Failure +FIG. 4. The normalized χ2/dof distributions for (top) sam- +ples corresponding to the “1000” label, (bottom) samples cor- +responding to the “0000” label. +Of importance is to understand how the NN learns in- +formation from the input mass spectrum H. To achieve +this goal, the neuron weights between the input and a +NN are shown in Fig. 5, where the neuron weight a1,i +projects the first bin of H onto the ith neuron of the +NN’s hidden layer. The neuron weight is usually inter- +preted as an impact power (I) of the input on a neuron, +or a response power of a neuron on the input. A positive +larger (negative smaller) weight implies that an input is +more important than the others in activating (deactivat- +ing) the neuron, or transferring more decision informa- +tion from the input to the neuron. If we treat a NN as +a unity, the sum Ij ≡ �N +i=1 |aj,i| (here, N denotes the +number of neurons in the first layer of the NN) means an +overall impact power of the jth bin of H on the NN. Half +of the bins correspond to the counts and half of the bins +to the statistical uncertainties. Figure 6 shows the I dis- +tributions before and after training the NN with {S90} +samples. After training, the four bumps clearly seen in +the I distribution are just around the peaks in the mass +spectrum. This can also be explained by the theoreti- +cal model which contains four thresholds. It is worth to +TABLE II. Predicted probability for the 15 states correspond- +ing to the different labels L. The bold numbers in each line +are the maximum probability. The probability of the sum of +other labels is listed in the last column. +Network +Output +Label +0000 +1000 +1001 +1002 +others +prediction of NN trained with S90 samples. +Network 1 +0.69% 89.13% +1.42% +8.75% +0.01% +Network 2 +0.03% +5.83% +38.47% 55.30% 0.37% +Network 3 +2.40% 94.45% +0.15% +2.99% +0.01% +Network 4 +0.05% +2.76% +31.64% 65.36% 0.17% +Network 5 +0.11% +4.16% +35.17% 60.23% 0.32% +Average +0.67% 39.27% 21.37% +38.53% 0.18% +prediction of NN trained with S92 samples. +Network 1 +0.00% +0.15% +5.37% +94.47% 0.00% +Network 2 +0.30% 38.32% 25.87% +35.03% 0.38% +Network 3 +0.00% +0.52% +14.76% 84.72% 0.00% +Network 4 +0.00% +0.07% +4.11% +95.81% 0.00% +Network 5 +0.00% +0.78% +13.57% 85.61% 0.03% +Average +0.06% +7.97% +12.74% 79.13% 0.08% +prediction of NN trained with S94 samples. +Network 1 +0.02% +2.23% +11.51% 86.24% 0.00% +Network 2 +0.80% 97.38% +1.43% +0.37% +0.03% +Network 3 +0.00% +2.94% +21.80% 75.25% 0.01% +Network 4 +0.15% 24.05% 48.79% 26.89% 0.13% +Network 5 +0.02% +4.12% +72.61% 23.23% 0.02% +Average +0.05% 26.14% +31.23% 42.36% 0.04% +prediction of NN trained with S96 samples. +Network 1 +0.00% +1.29% +19.79% 78.92% 0.00% +Network 2 +0.00% +3.38% +25.64% 70.97% 0.01% +Network 3 +0.00% +6.57% +32.20% 61.23% 0.00% +Network 4 +3.89% 87.35% +3.84% +4.91% +0.01% +Network 5 +0.00% +5.16% +21.02% 73.82% 0.00% +Average +0.78% 20.75% +20.50% 57.97% 0.00% +point out that a peak around the right-most bump cor- +responding to the 5/2− state threshold does not appear +in the experimental data. Thus the NN makes an inac- +curate prediction for the 5/2− state as mentioned before. +A comparison is performed by checking the traditional +χ2-fitting approach, which minimizes the +χ2 = +N +� +k +�N(k; P) − N d(k) +σN d(k) +�2 +(6) +between the theoretical prediction N(k; P) and experi- +mental data N d(k) by summing over all bins of H, where +P represents the model parameters to be determined and +k the bin index. We check how the jth bin of H impacts +on the parameters P and the pole positions. +For this +purpose, we define the uncertainty ∆Ui ≡ | Pi(on)−Pi(off) +Pi(on) +| +for the ith parameter of P, which is determined with the +central values obtained by switching on/off each bin of +H in the fitting. The bigger uncertainty observed by ex- +cluding a bin means a larger impact power of the bin on +a parameter. Fig. 7 illustrates the ∆U distributions for +two of the eleven parameters. The distributions for other +parameters can be found in App. C. The bins near the + +6 +Input Layer +Next Layer +Hidden Layer +and +Output Layer +FIG. 5. A schematic representation between the input data +to the first layer of the NN, where a1,i denotes a weight of +impacting the first input on the first neutron. +Blue (red) +circles represent binned values (errors) of H. +peaks in the mass spectrum do not show stronger con- +straints on the uncertainties of the parameters C1/2 and +F1/2 +1 +than the others. That is because that the model +parameters are correlated to each other and do not re- +flect the underlying physics directly. +In addition, to +check the impact of the jth experimental point on the +pole positions of the JP channel, the uncertainty +∆U JP +i +≡ +���� +Re[Polei](on) − Re[Polei](off) +Re[Polei](on) +���� +(7) +for the real part of the ith pole position is defined by +switching on/off each bin of H in the fitting. +Figs. 8, +9, +10 illustrate the ∆U JP +i +distributions for the JP = +1 +2 +−, 3 +2 +−, 5 +2 +− channels, respectively, without considering +the correlation among the parameters. Although those +values are of the order 10−5, one can still see that the ex- +perimental data around the Σc ¯D, Σc ¯D∗ and Σ∗ +c ¯D thresh- +olds are more important than the others. The reason why +the data around the Σ∗ +c ¯D∗ threshold is not important is +due to the small production amplitude of the Σ∗ +c ¯D∗ chan- +nel and the experimental data have little constraint about +the corresponding poles [49, 50]. One can also obtain the +same conclusion from Fig. 10, where the experimental +data around the JP = +5 +2 +− dynamic channel Σ∗ +c ¯D∗ do +not show any significance. Due to the coupled-channel +effect, the experimental data around the coupled chan- +nels still have strong constraints on the physics, e.g. the +seven poles, around the other coupled channels. Taking +the first pole of the JP = +1 +2 +− channel as an example, +e.g. Fig. 8(a), the data around the Σc ¯D∗ threshold are +as significant as those around the Σc ¯D threshold. As the +sample data of Figs. 8, 9, 10 corresponds to Solution A, +the data around Pc(4440) (Pc(4457)) are more important +0 +20 +40 +60 +80 +100 +120 +140 +Neuron Number +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +1.4 +1.5 +Neuron Weight +Neuron Weight +0 +200 +400 +600 +800 +1000 +1200 +LHCb Data +LHCb Data +0 +20 +40 +60 +80 +100 +120 +140 +Neuron Number +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Neuron Weight +Neuron Weight +0 +200 +400 +600 +800 +1000 +1200 +LHCb Data +LHCb Data +FIG. 6. Upper pannel: Distribution of data weights in the +neurons. The points are the 150 experimental data, and the +bars are the normalized neuron weights after random initial- +ization of the neural network without training. Lower pannel: +Distribution of the data weights in the neurons. The points +are the 150 experimental data, and bars are the normalized +neuron weights after training. +than those around the Pc(4457) (Pc(4440)) in Fig. 8(b) +(Fig. 9(b)) as expected. +VII. +CONCLUSION +We have investigated the nature of the famous hid- +den charm pentaquarks with a NN-based approach in a +pionless EFT, which strongly favors JPc(4440) = +1 +2 and +JPc(4457) = 3 +2, i.e. solution A in Refs [48–50]. By com- +paring the training of 100 “1xxx” and 100 “0xxx” sam- +ples, we find that solution A is systematically preferred +over solution B. Furthermore, we also performed checks +on both the NN-based approach and the χ2-fitting ap- +proach. +Our conclusion is that both approaches work +well on the MC simulation samples. +In the NN-based +approach, the role of each data bin on the underlying +physics is well reflected by the impact power I. For the +χ2-fitting approach, such a direct relation does not exist. +This further explains why the two solutions can be bet- + +7 +0 +20 +40 +60 +80 +100 +120 +140 +Number +0.0000 +0.0005 +0.0010 +0.0015 +0.0020 +0.0025 +0.0030 +0.0035 +Weight +C1/2 +0 +200 +400 +600 +800 +1000 +1200 +Sample Data +Sample Data +0 +20 +40 +60 +80 +100 +120 +140 +Number +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +Weight +1/2 +1 +0 +200 +400 +600 +800 +1000 +1200 +Sample Data +Sample Data +FIG. 7. The weight distribution of the data points regarding parameter C1/2 and F 1/2 +1 +. +0 +20 +40 +60 +80 +100 +120 +140 +Number +0 +1 +2 +3 +4 +5 +Weight +1e +5 +Pole1/2 +1 +0 +200 +400 +600 +800 +1000 +1200 +Sample Data +(a) +Sample Data +0 +20 +40 +60 +80 +100 +120 +140 +Number +0 +1 +2 +3 +4 +5 +6 +Weight +1e +5 +Pole1/2 +2 +0 +200 +400 +600 +800 +1000 +1200 +Sample Data +(b) +Sample Data +0 +20 +40 +60 +80 +100 +120 +140 +Number +0 +1 +2 +3 +4 +5 +6 +7 +Weight +1e +5 +Pole1/2 +3 +0 +200 +400 +600 +800 +1000 +1200 +Sample Data +(c) +Sample Data +FIG. 8. The ∆U JP +i +distributions for the three pole positions of the JP = 1 +2 +− channel. (a), (b), (c) are for the poles from lower +to higher energy. The data is a simulated sample for Solution A. The distributions do not consider the correlation among the +parameters. +ter distinguished in the NN-based approach than in the +χ2-fitting approach. At the same time, the effect of the +background fraction on the accuracy of the network is +accurately obtained, which provides a paradigm for neu- +ral networks to study exotic hadrons by first extracting +the background fraction from the data, and then analyz- +ing the physics of the possible states from the data with +the network trained from the corresponding background +fraction data. This study provides more insights about +how the NN-based approach predicts the nature of exotic +states from the mass spectrum. +Acknowledgements: +We are grateful to Meng- +Lin Du for the helpful discussion. This work is partly +supported by the National Natural Science Foundation +of +China +with +Grant +No. +12035007, +Guangdong +Provincial funding with Grant No. 2019QN01X172, +Guangdong +Major +Project +of +Basic +and +Applied +Basic Research No. 2020B0301030008. +Q.W. and +U.G.M. +are +also +supported +by +the +NSFC +and +the Deutsche Forschungsgemeinschaft (DFG, German +Research Foundation) through the funds provided to +the Sino-German Collaborative Research Center TRR110 +“Symmetries and the Emergence of Structure in QCD” +(NSFC Grant No. +12070131001, +DFG Project-ID +196253076-TRR 110). +The work of U.G.M. is further +supported by the Chinese Academy of Sciences (CAS) +President’s International Fellowship Initiative (PIFI) +(Grant No. +2018DM0034) and Volkswagen Stiftung +(Grant No. 93562). +[1] R. Aaij et al. (LHCb), Observation of J/ψp Resonances +Consistent with Pentaquark States in Λ0 +b → J/ψK−p +Decays, Phys. Rev. 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The template recursive functions (A1) for defining the Chebyshev +polynomials is, +T0(x) = 1, +T1(x) = x, +T2(x) = 2x2 − 1, +(A1) +... +TN(x) = 2xTN−1(x) − TN−2(x). +This allows for the implementation of the Chebyshev polynomials as +Chebyshev0(x) = c0, +Chebyshev1(x) = c0 + c1x, +Chebyshev2(x) = c2T2(x) + Chebyshev1(x), +(A2) +... +Chebyshev6(x) = c6T6(x) + Chebyshev5(x). +where the c0, c1, ... are coefficients, as given in Tab. III for the problem at hand. +TABLE III. The coefficients of the sixth-order Chebyshev polynomials. +Coefficients +value error +c0 +67.96 ±0.18 +c1 +4.13 ±0.18 +c2 +−16.60 ±0.15 +c3 +−2.55 ±0.15 +c4 +3.27 ±0.14 +c5 +2.88 ±0.14 +c6 +−1.66 ±0.14 +4.25 +4.30 +4.35 +4.40 +4.45 +4.50 +4.55 +mJ/ p[Gev] +200 +400 +600 +800 +1000 +Experimental Background +Experimental Background +Error of Background +FIG. 11. The experimental background distribution of the invariant mass M(J/ψp). + +12 +Appendix B: Determination of the Background Fraction in Data +We produce a group of samples for different background fractions from 0% to 90% every 10%, as well as (92%, 94%, +96%). We trained a ResNet-based NN which employs the MSELoss function to measure the Euclidean distance between +the predicted background fractions and the truth values. The NN successfully retrieves the background fraction, as +illustrated in Fig. 12 (left). The right plot shows the difference between the predicated values and the background +values. The bias and uncertainty values are -0.0583 and 0.71, respectively. We then use this NN to determine the +background fraction for experimental data. The background fraction for data is determined to be (96.02 ± 0.76)% in +case of training the NN with samples {S50...S90, S92, S94, S96}, and is determined to be (95.72 ± 0.72)% in case of +training the NN with samples {S0...S90, S92, S94, S96}. The background fraction is determined to be (95.79 ± 0.76)% +predicted with a different training. In short, the background fraction is taken as their average value (96.0 ± 0.8)%. +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +220 +3 +10 +× +0 +20 +40 +60 +80 +100 +Label +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Prediction +5 +− +0 +5 +R +∆ +0 +100 +200 +300 +400 +500 +600 +3 +10 +× +Entry +bkg +Entries + 2401840 +Mean +0.0583 +− + +Std Dev + 0.71 +FIG. 12. (left) Predicted background fractions versus the background values for samples: {S0...S90}, (right) the difference +distributions between predicted background fractions and ground-truth values. +Appendix C: The weights of neurons with respect to other parameters +We define the uncertainty +∆Ui ≡ +���� +Pi(on) − Pi(off) +Pi(on) +���� +(C1) +for the ith parameter of P, which is determined with the central values obtained by switching on/off each bin of H +in the fitting. We use ∆Ui to measure the influence of each data point on a parameter. Figure 13 illustrates ∆U +distributions w.r.t eleven parameters. The larger ∆U points correspond to data points deviating strongly from the +interpolation of its neighbour data points. + +13 +0 +20 +40 +60 +80 +100 +120 +140 +Number +0.0000 +0.0005 +0.0010 +0.0015 +0.0020 +0.0025 +0.0030 +0.0035 +Weight +C1/2 +0 +200 +400 +600 +800 +1000 +1200 +Sample Data +Sample Data +0 +20 +40 +60 +80 +100 +120 +140 +Number +0.000 +0.001 +0.002 +0.003 +0.004 +0.005 +0.006 +0.007 +0.008 +Weight +C3/2 +0 +200 +400 +600 +800 +1000 +1200 +Sample Data +Sample Data +0 +20 +40 +60 +80 +100 +120 +140 +Number +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +Weight +gS +0 +200 +400 +600 +800 +1000 +1200 +Sample Data +Sample Data +0 +20 +40 +60 +80 +100 +120 +140 +Number +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +Weight +g′D +0 +200 +400 +600 +800 +1000 +1200 +Sample Data +Sample Data +0 +20 +40 +60 +80 +100 +120 +140 +Number +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +Weight +1/2 +1 +0 +200 +400 +600 +800 +1000 +1200 +Sample Data +Sample Data +0 +20 +40 +60 +80 +100 +120 +140 +Number +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +Weight +1/2 +2 +0 +200 +400 +600 +800 +1000 +1200 +Sample Data +Sample Data +0 +20 +40 +60 +80 +100 +120 +140 +Number +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +0.035 +0.040 +Weight +1/2 +3 +0 +200 +400 +600 +800 +1000 +1200 +Sample Data +Sample Data +0 +20 +40 +60 +80 +100 +120 +140 +Number +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +0.035 +Weight +3/2 +1 +0 +200 +400 +600 +800 +1000 +1200 +Sample Data +Sample Data +0 +20 +40 +60 +80 +100 +120 +140 +Number +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +Weight +3/2 +2 +0 +200 +400 +600 +800 +1000 +1200 +Sample Data +Sample Data +0 +20 +40 +60 +80 +100 +120 +140 +Number +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +0.035 +0.040 +Weight +3/2 +3 +0 +200 +400 +600 +800 +1000 +1200 +Sample Data +Sample Data +0 +20 +40 +60 +80 +100 +120 +140 +Number +0.00 +0.05 +0.10 +0.15 +0.20 +Weight +5/2 +1 +0 +200 +400 +600 +800 +1000 +1200 +Sample Data +Sample Data +FIG. 13. The ∆U distributions for each parameter (from left to right and top to bottom): C1/2, C3/2, gS, g′ +D, F 1/2 +1 +, F 1/2 +2 +, +F 1/2 +3 +, F 3/2 +1 +, F 3/2 +2 +, F 3/2 +3 +, F 5/2 +1 +. + +14 +Appendix D: Detailed Predictions +To reduce the systematic uncertainties arising from the NNs, we trained several NN models with identical structure +but different initializations. In particular, for the background fractions {S90}, {S92}, {S94}, and {S96}, we train nine, +five, nine, five NNs, respectively. Their predicted results are presented in Tab. IV with maximum probabilities in +boldface. As shown in the table, the first labels of all the maximum probabilities are “1”, which indicates that all the +NNs favor solution A, i.e. JPc(4440) = 1 +2 and JPc(4457) = 3 +2. As these 9+5+9+5 = 28 NNs can distinguish solution A +from solution B, we do not train more NNs. +TABLE IV. Predicted probability of 15 state labels. The bold numbers in each line are the maximum probability. +Network +Output +Label +0000 +1000 +1001 +1002 +0100 +0110 +0111 +1111 +0200 +1200 +1210 +1211 +1220 +1221 +1222 +prediction of NN trained with samples at a background level 90%. +Network 1 +0.69% 89.13% +1.42% +8.75% +0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.01% 0.00% +Network 2 +0.03% +5.83% +38.47% 55.30% 0.00% 0.03% 0.08% 0.00% 0.07% 0.04% 0.02% 0.00% 0.00% 0.09% 0.04% +Network 3 +0.03% +5.39% +15.79% 78.41% 0.00% 0.02% 0.03% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.03% 0.03% +Network 4 +0.01% +1.90% +27.01% 70.95% 0.00% 0.01% 0.01% 0.00% 0.00% 0.01% 0.00% 0.00% 0.01% 0.03% 0.06% +Network 5 +2.40% 94.45% +0.15% +2.99% +0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.01% 0.00% +Network 6 +5.57% 73.04% +7.94% +13.26% 0.00% 0.04% 0.01% 0.00% 0.02% 0.00% 0.00% 0.00% 0.00% 0.08% 0.01% +Network 7 +0.05% +2.76% +31.64% 65.36% 0.00% 0.01% 0.03% 0.00% 0.00% 0.02% 0.00% 0.00% 0.01% 0.02% 0.08% +Network 8 +0.11% +4.16% +35.17% 60.23% 0.00% 0.04% 0.06% 0.00% 0.01% 0.04% 0.00% 0.00% 0.00% 0.04% 0.13% +Network 9 +0.18% +4.23% +16.10% 79.44% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.02% 0.02% +prediction of NN trained with samples at a background level 92%. +Network 1 +0.00% +0.15% +5.37% +94.47% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% +Network 2 +0.30% 38.32% 25.87% +35.03% 0.01% 0.03% 0.04% 0.01% 0.06% 0.01% 0.00% 0.00% 0.05% 0.16% 0.10% +Network 3 +0.00% +0.52% +14.76% 84.72% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% +Network 4 +0.00% +0.07% +4.11% +95.81% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% +Network 5 +0.00% +0.78% +13.57% 85.61% 0.00% 0.00% 0.01% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.02% +prediction of NN trained with samples at a background level 94%. +Network 1 +0.00% +1.20% +2.26% +96.53% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% +Network 2 +0.02% +2.23% +11.51% 86.24% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% +Network 3 +0.00% +0.26% +10.26% 89.47% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% +Network 4 +0.00% +1.22% +13.92% 84.86% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% +Network 5 +0.03% +1.31% +58.42% 40.21% 0.00% 0.00% 0.02% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% +Network 6 +0.80% 97.38% +1.43% +0.37% +0.00% 0.01% 0.01% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.01% 0.00% +Network 7 +0.00% +2.94% +21.80% 75.25% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.01% +Network 8 +0.15% 24.05% 48.79% 26.89% 0.00% 0.04% 0.02% 0.00% 0.01% 0.03% 0.00% 0.00% 0.00% 0.02% 0.01% +Network 9 +0.02% +4.12% +72.61% 23.23% 0.00% 0.00% 0.01% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.01% 0.00% +prediction of NN trained with samples at a background level 96%. +Network 1 +0.00% +1.29% +19.79% 78.92% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% +Network 2 +0.00% +3.38% +25.64% 70.97% 0.01% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% +Network 3 +0.00% +6.57% +32.20% 61.23% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% +Network 4 +3.89% 87.35% +3.84% +4.91% +0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.01% 0.00% 0.00% 0.00% +Network 5 +0.00% +5.16% +21.02% 73.82% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% + +15 +Appendix E: Neural Network Comparison +We have compared a MLP-based [58] NN to a ResNet-based [55] NN, which are implemented using PyTorch [57]. +Both NN models are trained with the {S90} samples. The Adam [56] optimizer is used to solve these models with a +combination of the momentum algorithm with the RMSProp algorithm [59]. A reasonable solution can be obtained +after 500 training epochs using an initial learning rate value of 0.001. At the beginning of training, the weights +of the neurons are randomly initialized with a normal distribution, and the biases of neurons are set to zero. The +threshold values of dropout layers are set to 0.2 for MLP-based NN and to 0.3 for ResNet-based NN. In summary, the +CrossEntropyLoss function converges as the training epochs increase, and the predicted accuracy increases as training +epochs increase. The ResNet-based NN achieves a better solution, as shown in Fig. 14. The ResNet-based NN is +structured as shown in Fig. 15. +0 +100 +200 +300 +400 +500 +Epochs +1.0 +1.5 +2.0 +2.5 +3.0 +Loss +ResNet(600-900-600_5) +ResNet(800-1500-800_5) +ResNet(800-1500-800_10) +ResNet(1000-1500-1000_15) +DNN(200-150-100-50) +DNN(600-900-600-300) +DNN(1000-1500-1000-600) +0 +100 +200 +300 +400 +500 +Epochs +1.0 +1.5 +2.0 +2.5 +3.0 +Loss +ResNet(600-900-600_5) +ResNet(800-1500-800_5) +ResNet(800-1500-800_10) +ResNet(1000-1500-1000_15) +DNN(200-150-100-50) +DNN(600-900-600-300) +DNN(1000-1500-1000-600) +0 +100 +200 +300 +400 +500 +Epochs +10 +20 +30 +40 +50 +60 +70 +Accuracy(%) +ResNet(600-900-600_5) +ResNet(800-1500-800_5) +ResNet(800-1500-800_10) +ResNet(1000-1500-1000_15) +DNN(200-150-100-50) +DNN(600-900-600-300) +DNN(1000-1500-1000-600) +FIG. 14. +(left) For training datasets, the loss functions used in different NNs converge as training epochs increase. Here, +“ResNet(600-900-600 5)” represents a ResNet-based NN consists of five ResBlock, and each block consists of a layer of 600 +neurons, a layer of 900 neurons, and a layer of 600 neurons. “DNN(200-150-100-50)” represents a MLP-based NN which consists +of four fully-connected layers, and each layer consists of 200, 150, 100, 50 neurons respectively (the input and output layers are +excluded). (middle) For testing datasets, the loss functions used in different NNs converge as training epochs increase. (right) +The predicted accuracy increases as training epochs increase. +Dropout +Dropout +Dropout +1000 +neurons +2000 +neurons +1000 +neurons +1000 +neurons +2000 +neurons +Dropout +Dropout +1000 +neurons + +Relu +Relu +Relu +Relu +Relu +Relu +Dropout +2000 +neurons +1000 +neurons +1000 +neurons +Auxiliary Classifier +Inputs: +300 neurons +Outputs: +15 neurons +Auxiliary Classifier +Relu +Relu +Relu + + + +FIG. 15. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' ‡ 1Guangdong Provincial Key Laboratory of Nuclear Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Institute of Quantum Matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' South China Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Guangzhou 510006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' China 2Guangdong-Hong Kong Joint Laboratory of Quantum Matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Southern Nuclear Science Computing Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' South China Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Guangzhou 510006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' China 3Helmholtz-Institut f¨ur Strahlen- und Kernphysik and Bethe Center for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Universit¨at Bonn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' D-53115 Bonn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Germany 4Institute for Advanced Simulation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Institut f¨ur Kernphysik and J¨ulich Center for Hadron Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Forschungszentrum J¨ulich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' D-52425 J¨ılich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Germany 5Tbilisi State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 0186 Tbilisi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Georgia (Dated: January 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 2023) We study the nature of the hidden charm pentaquarks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' the Pc(4312), Pc(4440) and Pc(4457), with a neural network approach in pionless effective field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' In this framework, the normal χ2 fitting approach cannot distinguish the quantum numbers of the Pc(4440) and Pc(4457).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' In contrast to that, the neural network-based approach can discriminate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' In addition, we also illustrate the role of each experimental data bin of the invariant J/ψp mass distribution on the underlying physics in both neural network and fitting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Their similarities and differences demonstrate that neural network methods can use data information more effectively and directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' This study provides more insights about how the neural network-based approach predicts the nature of exotic states from the mass spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' INTRODUCTION In last two decades we have witnessed the emergence of tens of candidates for the so-called exotic hadrons which are beyond the conventional meson and baryon configu- rations, such as the hidden charm pentaquarks [1–3], the fully charmed tetraquark X(6900) [4–6], or the doubly charmed tetraquark T + cc [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The first hidden charm pentaquarks were reported by the LHCb Collaboration in 2015 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' [1] by observing the J/ψp invariant mass via the process Λb → J/ψpK−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Four years lat- er, the Pc(4312), Pc(4440) and Pc(4457), were reported with an order of magnitude larger luminosity in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Since then, many interpretations were proposed for these states, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' compact multi-quark, hadronic molecule, hybrid, and triangle singularity [9–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Traditionally, judging whether a model works or not is to compare it with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' However, such a top- down approach makes the conclusion model-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Physicists are thus trying to find bottom-up approach- es [20] to obtain a definitive conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' One direction is to use machine learning to explore hadron properties, which benefits from the evolution of computing capa- bilities during the last decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' For instance, genetic algorithms [21–23] and neural networks [24–27] can be utilized to explore the nature of hadrons or the proper- ties of nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Machine learning has been widely used in nuclear physics [28–34], high energy nuclear physics [35– 37], experimental data analysis [38, 39] and theoretical physics [40, 41], even to discover the physical principles ∗ hujf@m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='scnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='cn † qianwang@m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='scnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='cn ‡ meissner@hiskp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='uni-bonn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='de underneath the experimental data [20, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' However, its application in hadron physics is in its early stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' A preliminary attempt was made in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' [20, 44–46] for the molecular picture [12], which is one of natural explana- tions of near-threshold peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' [46, 47] demonstrate that deep learning can be applied to classify poles and regress the model parameters in the one-channel coupling case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Here, extending and improving the works [44, 45], we consider the multi-channel coupling case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' We take the hidden charm pentaquark states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Pc(4312), Pc(4440), Pc(4457) [3] as examples, to achieve the following goals: In the Σ(∗) c ¯D(∗) hadronic molecular picture [48– 50], both the Pc(4440) and the Pc(4457) are relat- ed to the Σc ¯D∗ threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Their spin-parity (JP ) could be either 1 2 −, 3 2 − (solution A) or 3 2 −, 1 2 − (so- lution B) [48–50], from the viewpoint of pionless Effective Field Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' In solution A, the lower mass hidden charm pentaquark has lower spin, and vice versa in solution B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Although the χ2 fit can- not distinguish the two solutions, we try to find out which solution is preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' In the traditional fitting approach, the physics is embedded in the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' One has to ex- tract the model parameters from the experimental data and further extract the physical quantities of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' We use a neural network (NN) based ap- proach to extract the pertinent physical quantities directly and illustrate the role of each experimental data point (here the bins in the mass distributions) on the physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' We demonstrate that the NN-based approach consis- tently favors solution A over B compared to the fitting arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='05364v1 [hep-ph] 13 Jan 2023 2 approach, and clearly gives the properties of the poles in the multi-channel case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Our paper is organized as follows: Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' II describes how we describe the various pentaquark states and label them according to their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The model to analyze the LHCb data on the invariant J/ψp mass distribution is discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' To train the NNs, a large number of pseudodata are produced by Monte Carlo simulation, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The training and validation of the NNs are described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Then, we analyse the data with these NNs, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' We sum- marize our findings and conclude in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' STATES AND LABELS The hidden charm pentaquark states are described as generated from the scattering of the (Σc, Σ∗ c) and ( ¯D, ¯D∗) doublets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' As the thresholds of the inelastic channels J/ψp and ηcp are far away from those of elastic chan- nels, their effects on the classification of the poles rele- vant to the physical observables are marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The 1 2 − and 3 2 − states are given by a three-channel case with 23 = 8 Riemann surfaces denoted as R±±±, see [19] for a pictorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Here, the “+” and “-” signs in the ith po- sition mean the physical and unphysical sheet of the ith channel, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' However, only poles on the phys- ical sheet R+++ and those close-by ones, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' R−++, R−−+, R−−−, will contribute significantly to physical observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Thus we focus on those poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' As we aim at extracting the most important information of those poles, it is sufficient to work with leading order contact interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' In this case, the poles accounting for the near threshold structures can be classified as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The labels are defined as follows: The case with three poles on the R+++, R−++, R−−+ sheets below the first, second and third thresholds, respectively, is defined as “bound state” which is labeled as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The case with three poles on the R−++, R−−+, R−−− sheets above the first, second and third thresholds, respectively, is defined as “resonance” which is labeled as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The case with three poles on the R−++, R−−+, R−−− sheets below the first, second and third thresholds, respectively, is defined as “virtual state” which is labeled as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' As the JP = 5 2 − state corresponds to a one-channel case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' the Σ∗ c ¯D∗ threshold, “bound state”, “resonance state”, and “vir- tual state” are defined as usually, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' by a pole on the physical sheet below threshold, a pole on the unphysical sheet below threshold, and a pole on the unphysical sheet above the threshold, in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The nature of the poles for the JP = 1 2 −, 3 2 −, 5 2 − channels are labeled in order as “lll”, with l = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Another label “o” , which we put in front, is used for the mass order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' That is due to the inde- terminacy of the quantum numbers of the Pc(4440) and Pc(4457).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' As they are close to the Σc ¯D∗ threshold, the S-wave interaction can give both JP = 1 2 − and 3 2 − states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' In the limit of heavy quark spin symmetry (HQSS), there are two solutions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Solution A and Solution B defined ："Bound state" (0) ："Resonance" (1) ："Virtual state" (2) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The definition of the poles for the three coupled channel case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' the JP = 1 2 − and 3 2 − states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' For more details, see the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' [48–50], as discussed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' In Solution A, the higher spin state has higher mass, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' JPc(4440) = 1 2 and JPc(4457) = 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' In Solution B, the situation is reversed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' JPc(4457) = 1 2 and JPc(4440) = 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' To distinguish these two scenarios, another label o = 1 and o = 0 for Solution A and Solution B, respectively, is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' In total, we have a four-digit label “olll” to denote the sit- uation of the poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Taking the “0000” case for example, it means that the poles for all the quantum numbers are “bound state” and JPc(4440) = 3 2, JPc(4457) = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Note that we can also have cases with the poles different from the three situations discussed above, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' the poles are far away from the thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' However their probabilities are almost zero, and are thus neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' MODEL DESCRIPTION The Pc(4312) and Pc(4440)/Pc(4457) states are close to the Σc ¯D and Σc ¯D∗ thresholds, respectively, making them prime candidates for hadronic molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' In the heavy quark limit, these hidden charm pentaquarks are related to the scattering between the (Σc, Σ∗ c) and the ( ¯D, ¯D∗) heavy quark spin doublets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' By constructing the interactions between these two doublets in the ef- fective field theory (EFT) respecting the heavy quark spin symmetry (HQSS), Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' [49, 50] extracted the pole positions of the seven hidden charm pentaquarks by fit- ting the J/ψp invariant mass distribution for the pro- cess Λb → J/ψpK−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Following the same procedure, we consider the Σ(∗) c ¯D(∗) and J/ψp, ηcp channels as elastic and inelastic channels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The poten- tial quantum numbers of hidden charm pentaquarks are 1 2 −, 3 2 −, 5 2 − which correspond to the scattering of the Σc ¯D−Σc ¯D∗−Σ∗ c ¯D∗, Σ∗ c ¯D−Σc ¯D∗−Σ∗ c ¯D∗, Σ∗ c ¯D∗, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' In the HQSS limit, these scatterings are described by two parameters C 1 2 and C 3 2 where the subscripts refer 3 to the light degrees of freedom [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The coupling for the S-wave and D-wave inelastic channels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' the J/ψp and ηcp channels, are described by two parameters gS and gD, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The bare production amplitudes per JP channel is parameterized by FJ i with the subscript i indicating the ith channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Putting pieces together, the inelastic amplitude of the J/ψp for the decay process Λ0 b → J/ψpK− can be expressed as [49, 50] U J i (E, k) = − � β � d3q (2π)3 VJ iβ(k)Gβ(E, q)U J β (E, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' (1) Here, U J i is the ith J/ψp inelastic channel amplitude with the quantum number J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' VJ iβ is the transition vertex bet- ween the βth elastic and the ith inelastic channel, and described by the coupling constants gS for the S-wave and gD for the D-wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Gβ is the two-body propagator of the βth elastic channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The physical production am- plitude of the βth elastic channel U J β for total spin J is obtained from the Lippmann-Schwinger equation (LSE) U J α(E, p) = P J α − � β � d3q (2π)3 V J αβ(E, p, q)Gβ(E, q)U J β (E, q), (2) with P J α , constructed by seven parameters FJ n , the pro- duction amplitude for the αth elastic channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Note that above integral equations can be reduced to algebric equa- tions in the framework of a pionless EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' For a general introduction to this type of EFT, see [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' V J αβ is the effective potential, which contains both the scattering between the elastic channels described by two parame- ters C1/2, C3/2 and that between the elastic and inelas- tic channels described by the two parameters gS, gD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' In total, eleven parameters are combined in a vector P = � gS, gD, C 1 2 , C 3 2 , F 1 2 1 , F 1 2 2 , F 1 2 3 , F 3 2 1 , F 3 2 2 , F 3 2 3 , F 5 2 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' (3) To describe the invariant mass distribution M(J/ψp), a composite model is constructed via a probability dis- tribution function (PDF) as: PDF(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' P) = α � J � |U J|2p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' (E)G(E′ − E)dE′ + (1 − α)Chebyshev6(E) , (4) with U J the Λb → J/ψpK− amplitude (2) for total spin J = 1 2, 3 2, 5 2 and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' (E) is the phase space of the corre- sponding process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' A Gaussian function G(E′ −E) which represents the experimental detector resolution is con- voluted with the physical invariant mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Here, we take take the energy resolution as a constant of σ ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='3 MeV [3] without considering its energy depen- dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Further, Chebyshev6 is the sixth-order Chebyshev polynomial which represents the background distribu- tion, and 1 − α its fraction, α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The background fraction in the data is determined to be (96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='8%) as discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The coefficients (c0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=', c6) for the background component are obtained by fitting the Chebyshev polynomials to data [53], as listed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Note that the area enclosed by the signal (background) distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' the remnant of the first term after re- moving the factor α (the second term after removing the factor 1 − α ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' (4) are normalized to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' MONTE CARLO SIMULATION Based on the PDF given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' (4), 240184 samples corresponding to physical states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' with poles not too far away from the real axis in the considered energy range, are selected among 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='85 million samples uniform- ly generated in the space of P, distributed in the mass window from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='25 GeV to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='55 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' These samples are represented as a set: {Hj, Lj} where the j index refers to a given sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' A histogram H with 150 bins (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 2) denotes the invariant mass spectrum of the Pc states (background included) and the label L indicates the state label defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The values of P are 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='40 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='45 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='55 mJ/ p[Gev] 200 400 600 800 1000 1200 Monte Carlo Data Monte Carlo Data FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' An example of the simulated invariant mass distribu- tion M(J/ψp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' uniformly sampled in the following ranges, gS ∈ [0, 10] GeV−2, F 1 2 3 ∈ [−3600, −3300], gD ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='5] × gS, F 3 2 1 ∈ [−3900, −3600], C1/2 ∈ [−20, 0]GeV−2, F 3 2 2 ∈ [−1900, −1600], (5) C3/2 ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='5] × C1/2, F 3 2 3 ∈ [−4800, −4500], F 1 2 1 ∈ [0, 300], F 5 2 1 ∈ [600, 900].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' F 1 2 2 ∈ [700, 1000], The ranges are empirically taken to produce a mass spec- trum similar to the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Note that gS and gD as well as C1/2 and C3/2 are strongly correlated, so the second of these coefficients is related by a factor in 4 the range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='5] to the corresponding first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Note that these strong correlations were found in [49, 50], and is also found if one uses much larger set of samples to train the NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The MC production is performed with the open source software ROOT [53] and GSL [54], as categorized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Considering different background fractions, the set of MC samples produced at the back- ground fraction 1 − α = 90% is denoted as {S90}, so are the other sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Distribution of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The state label of the second column is defined as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The first label “0” means JPc(4440) = 3 2 and JPc(4457) = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The first label “1” means JPc(4440) = 1 2 and JPc(4457) = 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The last column represents the number of samples for this label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Mass Relation Label State Label Number of Samples 0 000 46951 1 000 4283 1 001 1260 1 002 4360 0 100 3740 0 110 4320 0 111 7520 1 111 360 0 200 9590 1 200 280 1 210 3980 1 211 2690 1 220 50240 1 221 50512 1 222 50098 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' TRAINING AND VERIFICATION The neural network (NN) is implemented with an in- frastructure of ResNet [55] as discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' [52], and solved with the Adam [56] optimizer parametrized by the cross-entropy loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' A reasonable solution can be obtained after training 500 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Note that 70% of MC simulation samples are used for training and the rest 30% for a benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The training details are summarized in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' In short, the NNs successfully retrieve the state label with an accuracy of 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='29%, 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='94%, 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='89%, 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='64% for MC simulation samples {S90}, {S92}, {S94}, {S96}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The prediction accuracy decreases as the back- ground increases, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Turning to the mass relation, we verified 100 {S90} samples corresponding to the label “1xxx”, and another 100 samples corresponding to the label “0xxx”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 3 (top)/(bottom) shows the predicted probability distribu- tion for the label 1xxx/0xxx samples, in which all labels are successfully retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' In other words, the NN can ac- curately predict the mass relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' For comparison, we also perform an analysis with the χ2-fitting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 4 (top)/(bottom) shows the normalized χ2/dof dis- tributions corresponding to the label 1xxx/0xxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' A 3% misidentification is observed for the label 1xxx samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='03 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='00 Predicted Probability Number of MC data samples Predicted label:1000 Predicted label:0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='04 0 5 10 15 20 25 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='00 Predicted Probability Number of MC data samples Predicted label:1000 Predicted label:0000 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The normalized χ2/dof distributions of the predicted probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' (top) samples corresponding to the “1000” label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' (bottom) samples corresponding to the “0000” label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' APPLICATION TO DATA We now use the well-trained NNs to analyse the exper- imental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The results are listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' To reduce the systematic uncertainties arising from the NNs, we trained five NN models which have an identical struc- ture under different initialization, for each group of sam- ples with different background fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The probabili- ty of the sum of all the other labels is smaller than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The labels with top three probabilities are 1000, 1001, 1002, which means that the NNs favors Solution A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' JPc(4440) = 1 2 and JPc(4457) = 3 2, as the first label is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The second label 0 and the third label 0 mean that the poles for the JP = 1 2 − and JP = 3 2 − channels behave as “bound states”, which is different from the virtual state conclusion for the Pc(4312) in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' [20] also using ma- chine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The pole situation for the JP = 5 2 − is undetermined, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' all the three labels 0, 1, 2 appearing for the forth label, as the structure around the Σ∗ c ¯D∗ is not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' To resolve this issue, precise data in this energy region would be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='8 ²/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 16 Number of MC data samples Success Failure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0 ²/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 16 Number of MC data samples Success Failure FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The normalized χ2/dof distributions for (top) sam- ples corresponding to the “1000” label, (bottom) samples cor- responding to the “0000” label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Of importance is to understand how the NN learns in- formation from the input mass spectrum H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' To achieve this goal, the neuron weights between the input and a NN are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 5, where the neuron weight a1,i projects the first bin of H onto the ith neuron of the NN’s hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The neuron weight is usually inter- preted as an impact power (I) of the input on a neuron, or a response power of a neuron on the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' A positive larger (negative smaller) weight implies that an input is more important than the others in activating (deactivat- ing) the neuron, or transferring more decision informa- tion from the input to the neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' If we treat a NN as a unity, the sum Ij ≡ �N i=1 |aj,i| (here, N denotes the number of neurons in the first layer of the NN) means an overall impact power of the jth bin of H on the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Half of the bins correspond to the counts and half of the bins to the statistical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Figure 6 shows the I dis- tributions before and after training the NN with {S90} samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' After training, the four bumps clearly seen in the I distribution are just around the peaks in the mass spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' This can also be explained by the theoreti- cal model which contains four thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' It is worth to TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Predicted probability for the 15 states correspond- ing to the different labels L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The bold numbers in each line are the maximum probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The probability of the sum of other labels is listed in the last column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Network Output Label 0000 1000 1001 1002 others prediction of NN trained with S90 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Network 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='69% 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='13% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='42% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='75% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='01% Network 2 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='00% point out that a peak around the right-most bump cor- responding to the 5/2− state threshold does not appear in the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Thus the NN makes an inac- curate prediction for the 5/2− state as mentioned before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' A comparison is performed by checking the traditional χ2-fitting approach, which minimizes the χ2 = N � k �N(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' P) − N d(k) σN d(k) �2 (6) between the theoretical prediction N(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' P) and experi- mental data N d(k) by summing over all bins of H, where P represents the model parameters to be determined and k the bin index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' We check how the jth bin of H impacts on the parameters P and the pole positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' For this purpose, we define the uncertainty ∆Ui ≡ | Pi(on)−Pi(off) Pi(on) | for the ith parameter of P, which is determined with the central values obtained by switching on/off each bin of H in the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The bigger uncertainty observed by ex- cluding a bin means a larger impact power of the bin on a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 7 illustrates the ∆U distributions for two of the eleven parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The distributions for other parameters can be found in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The bins near the 6 Input Layer Next Layer Hidden Layer and Output Layer FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' A schematic representation between the input data to the first layer of the NN, where a1,i denotes a weight of impacting the first input on the first neutron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Blue (red) circles represent binned values (errors) of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' peaks in the mass spectrum do not show stronger con- straints on the uncertainties of the parameters C1/2 and F1/2 1 than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' That is because that the model parameters are correlated to each other and do not re- flect the underlying physics directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' In addition, to check the impact of the jth experimental point on the pole positions of the JP channel, the uncertainty ∆U JP i ≡ ���� Re[Polei](on) − Re[Polei](off) Re[Polei](on) ���� (7) for the real part of the ith pole position is defined by switching on/off each bin of H in the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 8, 9, 10 illustrate the ∆U JP i distributions for the JP = 1 2 −, 3 2 −, 5 2 − channels, respectively, without considering the correlation among the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Although those values are of the order 10−5, one can still see that the ex- perimental data around the Σc ¯D, Σc ¯D∗ and Σ∗ c ¯D thresh- olds are more important than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The reason why the data around the Σ∗ c ¯D∗ threshold is not important is due to the small production amplitude of the Σ∗ c ¯D∗ chan- nel and the experimental data have little constraint about the corresponding poles [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' One can also obtain the same conclusion from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 10, where the experimental data around the JP = 5 2 − dynamic channel Σ∗ c ¯D∗ do not show any significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Due to the coupled-channel effect, the experimental data around the coupled chan- nels still have strong constraints on the physics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' the seven poles, around the other coupled channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Taking the first pole of the JP = 1 2 − channel as an example, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 8(a), the data around the Σc ¯D∗ threshold are as significant as those around the Σc ¯D threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' As the sample data of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 8, 9, 10 corresponds to Solution A, the data around Pc(4440) (Pc(4457)) are more important 0 20 40 60 80 100 120 140 Neuron Number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='5 Neuron Weight Neuron Weight 0 200 400 600 800 1000 1200 LHCb Data LHCb Data 0 20 40 60 80 100 120 140 Neuron Number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='2 Neuron Weight Neuron Weight 0 200 400 600 800 1000 1200 LHCb Data LHCb Data FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Upper pannel: Distribution of data weights in the neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The points are the 150 experimental data, and the bars are the normalized neuron weights after random initial- ization of the neural network without training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Lower pannel: Distribution of the data weights in the neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The points are the 150 experimental data, and bars are the normalized neuron weights after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' than those around the Pc(4457) (Pc(4440)) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 8(b) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 9(b)) as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' CONCLUSION We have investigated the nature of the famous hid- den charm pentaquarks with a NN-based approach in a pionless EFT, which strongly favors JPc(4440) = 1 2 and JPc(4457) = 3 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' solution A in Refs [48–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' By com- paring the training of 100 “1xxx” and 100 “0xxx” sam- ples, we find that solution A is systematically preferred over solution B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Furthermore, we also performed checks on both the NN-based approach and the χ2-fitting ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Our conclusion is that both approaches work well on the MC simulation samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' In the NN-based approach, the role of each data bin on the underlying physics is well reflected by the impact power I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' For the χ2-fitting approach, such a direct relation does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' This further explains why the two solutions can be bet- 7 0 20 40 60 80 100 120 140 Number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0035 Weight C1/2 0 200 400 600 800 1000 1200 Sample Data Sample Data 0 20 40 60 80 100 120 140 Number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='40 Weight 1/2 1 0 200 400 600 800 1000 1200 Sample Data Sample Data FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The weight distribution of the data points regarding parameter C1/2 and F 1/2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 0 20 40 60 80 100 120 140 Number 0 1 2 3 4 5 Weight 1e 5 Pole1/2 1 0 200 400 600 800 1000 1200 Sample Data (a) Sample Data 0 20 40 60 80 100 120 140 Number 0 1 2 3 4 5 6 Weight 1e 5 Pole1/2 2 0 200 400 600 800 1000 1200 Sample Data (b) Sample Data 0 20 40 60 80 100 120 140 Number 0 1 2 3 4 5 6 7 Weight 1e 5 Pole1/2 3 0 200 400 600 800 1000 1200 Sample Data (c) Sample Data FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The ∆U JP i distributions for the three pole positions of the JP = 1 2 − channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' (a), (b), (c) are for the poles from lower to higher energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The data is a simulated sample for Solution A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The distributions do not consider the correlation among the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' ter distinguished in the NN-based approach than in the χ2-fitting approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' At the same time, the effect of the background fraction on the accuracy of the network is accurately obtained, which provides a paradigm for neu- ral networks to study exotic hadrons by first extracting the background fraction from the data, and then analyz- ing the physics of the possible states from the data with the network trained from the corresponding background fraction data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' This study provides more insights about how the NN-based approach predicts the nature of exotic states from the mass spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Acknowledgements: We are grateful to Meng- Lin Du for the helpful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' This work is partly supported by the National Natural Science Foundation of China with Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 12035007, Guangdong Provincial funding with Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 2019QN01X172, Guangdong Major Project of Basic and Applied Basic Research No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 2020B0301030008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' are also supported by the NSFC and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the funds provided to the Sino-German Collaborative Research Center TRR110 “Symmetries and the Emergence of Structure in QCD” (NSFC Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 12070131001, DFG Project-ID 196253076-TRR 110).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The work of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' is further supported by the Chinese Academy of Sciences (CAS) President’s International Fellowship Initiative (PIFI) (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 2018DM0034) and Volkswagen Stiftung (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 93562).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Aaij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' (LHCb), Observation of J/ψp Resonances Consistent with Pentaquark States in Λ0 b → J/ψK−p Decays, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 115, 072001 (2015), arX- iv:1507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='03414 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Aaij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' (LHCb), Model-independent evidence for J/ψp contributions to Λ0 b → J/ψpK− decays, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 117, 082002 (2016), arXiv:1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='05708 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Aaij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' (LHCb), Observation of a narrow pen- taquark state, Pc(4312)+, and of two-peak structure of the Pc(4450)+, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 122, 222001 (2019), arX- iv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='03947 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 8 0 20 40 60 80 100 120 140 Number 0 1 2 3 4 5 6 7 Weight 1e 5 Pole3/2 1 0 200 400 600 800 1000 1200 Sample Data (a) Sample Data 0 20 40 60 80 100 120 140 Number 0 1 2 3 4 5 6 Weight 1e 5 Pole3/2 2 0 200 400 600 800 1000 1200 Sample Data (b) Sample Data 0 20 40 60 80 100 120 140 Number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='75 Weight 1e 4 Pole3/2 3 0 200 400 600 800 1000 1200 Sample Data (c) Sample Data FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Analogous plot as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 8 but for JP = 3 2 −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 0 20 40 60 80 100 120 140 Number 0 1 2 3 4 5 6 7 Weight 1e 5 Pole5/2 1 0 200 400 600 800 1000 1200 Sample Data (a) Sample Data FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Analogous plot as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 8 but for JP = 5 2 −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' [4] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Aaij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' (LHCb), Observation of structure in the J/ψ -pair mass spectrum, Sci.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Guo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Hanhart, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Meißner, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Zhao, Y(4260) as the first s-wave open charm vector molecular state?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' D 90, 074039 10 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 11 Supplemental Material for “Revealing the nature of hidden charm pentaquarks with machine learning” Appendix A: Fitting to the Background Distributions The background, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 11, is described by sixth-order Chebyshev polynomials (A2), with the coefficients listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' III, by fitting the experimental background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The template recursive functions (A1) for defining the Chebyshev polynomials is, T0(x) = 1, T1(x) = x, T2(x) = 2x2 − 1, (A1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' TN(x) = 2xTN−1(x) − TN−2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' This allows for the implementation of the Chebyshev polynomials as Chebyshev0(x) = c0, Chebyshev1(x) = c0 + c1x, Chebyshev2(x) = c2T2(x) + Chebyshev1(x), (A2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Chebyshev6(x) = c6T6(x) + Chebyshev5(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' where the c0, c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' are coefficients, as given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' III for the problem at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The coefficients of the sixth-order Chebyshev polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Coefficients value error c0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='96 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='18 c1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='13 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='18 c2 −16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='60 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='15 c3 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='55 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='15 c4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='27 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='14 c5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='88 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='14 c6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='66 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='40 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='45 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='55 mJ/ p[Gev] 200 400 600 800 1000 Experimental Background Experimental Background Error of Background FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The experimental background distribution of the invariant mass M(J/ψp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 12 Appendix B: Determination of the Background Fraction in Data We produce a group of samples for different background fractions from 0% to 90% every 10%, as well as (92%, 94%, 96%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' We trained a ResNet-based NN which employs the MSELoss function to measure the Euclidean distance between the predicted background fractions and the truth values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The NN successfully retrieves the background fraction, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 12 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The right plot shows the difference between the predicated values and the background values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The bias and uncertainty values are -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0583 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='71, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' We then use this NN to determine the background fraction for experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The background fraction for data is determined to be (96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='76)% in case of training the NN with samples {S50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='S90, S92, S94, S96}, and is determined to be (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='72)% in case of training the NN with samples {S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='S90, S92, S94, S96}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The background fraction is determined to be (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='76)% predicted with a different training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' In short, the background fraction is taken as their average value (96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='8)%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 0 20 40 60 80 100 120 140 160 180 200 220 3 10 × 0 20 40 60 80 100 Label 0 10 20 30 40 50 60 70 80 90 100 Prediction 5 − 0 5 R ∆ 0 100 200 300 400 500 600 3 10 × Entry bkg Entries 2401840 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0583 − Std Dev 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='71 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' (left) Predicted background fractions versus the background values for samples: {S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='S90}, (right) the difference distributions between predicted background fractions and ground-truth values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Appendix C: The weights of neurons with respect to other parameters We define the uncertainty ∆Ui ≡ ���� Pi(on) − Pi(off) Pi(on) ���� (C1) for the ith parameter of P, which is determined with the central values obtained by switching on/off each bin of H in the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' We use ∆Ui to measure the influence of each data point on a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Figure 13 illustrates ∆U distributions w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='t eleven parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The larger ∆U points correspond to data points deviating strongly from the interpolation of its neighbour data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 13 0 20 40 60 80 100 120 140 Number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0015 0.' metadata={'source': 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+page_content=' The ∆U distributions for each parameter (from left to right and top to bottom): C1/2, C3/2, gS, g′ D, F 1/2 1 , F 1/2 2 , F 1/2 3 , F 3/2 1 , F 3/2 2 , F 3/2 3 , F 5/2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 14 Appendix D: Detailed Predictions To reduce the systematic uncertainties arising from the NNs, we trained several NN models with identical structure but different initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' In particular, for the background fractions {S90}, {S92}, {S94}, and {S96}, we train nine, five, nine, five NNs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Their predicted results are presented in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' IV with maximum probabilities in boldface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' As shown in the table, the first labels of all the maximum probabilities are “1”, which indicates that all the NNs favor solution A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' JPc(4440) = 1 2 and JPc(4457) = 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' As these 9+5+9+5 = 28 NNs can distinguish solution A from solution B, we do not train more NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Predicted probability of 15 state labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The bold numbers in each line are the maximum probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Network Output Label 0000 1000 1001 1002 0100 0110 0111 1111 0200 1200 1210 1211 1220 1221 1222 prediction of NN trained with samples at a background level 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Network 1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='00% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='00% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='00% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='00% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='00% 15 Appendix E: Neural Network Comparison We have compared a MLP-based [58] NN to a ResNet-based [55] NN, which are implemented using PyTorch [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Both NN models are trained with the {S90} samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The Adam [56] optimizer is used to solve these models with a combination of the momentum algorithm with the RMSProp algorithm [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' A reasonable solution can be obtained after 500 training epochs using an initial learning rate value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' At the beginning of training, the weights of the neurons are randomly initialized with a normal distribution, and the biases of neurons are set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The threshold values of dropout layers are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='2 for MLP-based NN and to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='3 for ResNet-based NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' In summary, the CrossEntropyLoss function converges as the training epochs increase, and the predicted accuracy increases as training epochs increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The ResNet-based NN achieves a better solution, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The ResNet-based NN is structured as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 0 100 200 300 400 500 Epochs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0 Loss ResNet(600-900-600_5) ResNet(800-1500-800_5) ResNet(800-1500-800_10) ResNet(1000-1500-1000_15) DNN(200-150-100-50) DNN(600-900-600-300) DNN(1000-1500-1000-600) 0 100 200 300 400 500 Epochs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content='0 Loss ResNet(600-900-600_5) ResNet(800-1500-800_5) ResNet(800-1500-800_10) ResNet(1000-1500-1000_15) DNN(200-150-100-50) DNN(600-900-600-300) DNN(1000-1500-1000-600) 0 100 200 300 400 500 Epochs 10 20 30 40 50 60 70 Accuracy(%) ResNet(600-900-600_5) ResNet(800-1500-800_5) ResNet(800-1500-800_10) ResNet(1000-1500-1000_15) DNN(200-150-100-50) DNN(600-900-600-300) DNN(1000-1500-1000-600) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' (left) For training datasets, the loss functions used in different NNs converge as training epochs increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Here, “ResNet(600-900-600 5)” represents a ResNet-based NN consists of five ResBlock, and each block consists of a layer of 600 neurons, a layer of 900 neurons, and a layer of 600 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' “DNN(200-150-100-50)” represents a MLP-based NN which consists of four fully-connected layers, and each layer consists of 200, 150, 100, 50 neurons respectively (the input and output layers are excluded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' (middle) For testing datasets, the loss functions used in different NNs converge as training epochs increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' (right) The predicted accuracy increases as training epochs increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' Dropout Dropout Dropout 1000 neurons 2000 neurons 1000 neurons 1000 neurons 2000 neurons Dropout Dropout 1000 neurons Relu Relu Relu Relu Relu Relu Dropout 2000 neurons 1000 neurons 1000 neurons Auxiliary Classifier Inputs: 300 neurons Outputs: 15 neurons Auxiliary Classifier Relu Relu Relu FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E4T4oBgHgl3EQf-g5_/content/2301.05364v1.pdf'} +page_content=' The structure of the ResNet-based NN used in this work.' metadata={'source': 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sha256:63d50f3039e11c658e1d4b0a0f941fc76f4d13546c7cec1fb4697d9376d02466 +size 1400760 diff --git a/TtAzT4oBgHgl3EQfJfvP/content/tmp_files/2301.01082v1.pdf.txt b/TtAzT4oBgHgl3EQfJfvP/content/tmp_files/2301.01082v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..dd4fa602248095556f6d2631d51e112afd241d4c --- /dev/null +++ b/TtAzT4oBgHgl3EQfJfvP/content/tmp_files/2301.01082v1.pdf.txt @@ -0,0 +1,1785 @@ +MNRAS 000, 1–15 (2020) +Preprint 4 January 2023 +Compiled using MNRAS LATEX style file v3.0 +A search for variable subdwarf B stars in TESS Full Frame Images +III. An update on variable targets in both ecliptic hemispheres – +contamination analysis and new sdB pulsators +S. K. Sahoo1,2★, A. S. Baran2,3,4, H.L. Worters5, P. Németh2,6,7 and D. Kilkenny8 +1Nicolaus Copernicus Astronomical Centre of the Polish Academy of Sciences, ul. Bartycka 18, 00-716 Warsaw, Poland +2ARDASTELLA Research Group +3Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland +4Department of Physics, Astronomy, and Materials Science, Missouri State University, Springfield, MO 65897, USA +5South African Astronomical Observatory, Observatory 7935, South Africa +6Astronomical Institute of the Czech Academy of Sciences, Fričova 298, CZ-251 65 Ondřejov, Czech Republic +7Astroserver.org, Fő tér 1, 8533 Malomsok, Hungary +8Department of Physics and Astronomy, University of the Western Cape, Private Bag X17, Bellville 7535, South Africa +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +We present an update on the variable star survey performed on the TESS 30 min Full Frame +Image (FFI) data reported by our first two papers in this series. This update includes a +contamination analysis in order to identify false positives and analysis of the TESS 10 min +FFI data collected during Years 3 and 4 of the mission. We clarify the variability status of +2 995 targets identifying 1 403 variable stars. In addition, we spectroscopically classify 24 pre- +filtered targets sampled with the 10 min FFI data and discover 11 new sdB pulsators. Future +follow-up space- and/or ground-based data of variables reported here, to identify the nature +of their variability and reveal spectroscopic parameters of the stars, would complement this +work. +Key words: Stars: subdwarfs – Stars: oscillations (including pulsations) – asteroseismology +1 +INTRODUCTION +Sahoo et al. (2020) (Paper I) and Baran et al. (2021) (Paper II) +presented their results of variability checks of the most promising +subdwarf B (sdB) candidates found in the Geier et al. (2019) and +Geier (2020) catalogs. The former authors pre-selected 45 674 tar- +gets and used the Full Frame Images (FFI) collected by the TESS +mission in the Southern Ecliptic Hemisphere (SEH) during Year 1 +and in the Northern Ecliptic Hemisphere (NEH) during Year 2. As a +result, 2 313 new variable targets in both ecliptic hemispheres were +listed. +It is a well known feature of the TESS CCDs that an individual +pixel has a 21 arcsec square projection on the sky. This makes con- +tamination a serious problem. A contaminating star contributes by +either increasing the average flux level in an aperture, thus lowering +the apparent amplitude of flux variation in the target, or – if variable +– makes a constant flux target star a false positive variable. The +former issue can be corrected by removing the estimated amount +of a contaminating star’s flux and this is fairly well achieved in the +Pre-search Data Conditioning fluxes. The latter issue can only be +resolved by checking the variability of all contaminators around +★ E-mail: sumanta.kumar27@gmail.com +the target of interest. We also found this method useful to resolve +variability in case both the target of interest and contaminator are +variables. +In this work, we aimed to remove false positives from our +list of 2 313 new variable objects. We searched all pixels within +the target masks of all 2 313 objects listed in Papers I and II. In +addition, for the most promising pulsating subdwarf candidates, +we retrieved the new light curves from the FFI collected during +Years 3 and 4, which are sampled at a 10 min cadence. This cadence +extends the Nyquist frequency up to 833.3 μHz, which allows us to +cover the entire gravity mode region in an amplitude spectrum1. By +promising candidates, we mean objects that in Papers I and II show +signals close to the Nyquist frequency of 277.7 μHz resulting from +the 30 min cadence of observation used during Years 1 and 2. +To summarize the findings reported in Paper I and II, there +are 15 pulsating subdwarf B star candidates, 79 variable (other than +pulsating) sdB stars, 33 variable subdwarf candidates and 2 186 +other variable stars, including 123 stars with non-sd classification. +1 In hot subdwarfs, pressure or p-mode pulsations typically have periods of +a few minutes; gravity or g-modes have periods of the order of an hour. See, +for example, Heber (2016). +© 2020 The Authors +arXiv:2301.01082v1 [astro-ph.SR] 3 Jan 2023 + +2 +S.K. Sahoo et al. +Data are downloaded and processed in the same way as ex- +plained in Papers I and II. For a convenient comparison, the Tables +presented here will preserve the listing order from the previous +papers of this series, however some of the tables from Paper I are +merged to be consistent with those in Paper II. They also contain +the most important information from Papers I and II with additional +information resulting from this work. In the following sections, we +explain the details of the contamination analysis and provide the +results of this work. +2 +CONTAMINATION ANALYSIS +We defined each target mask as a square of 11 × 11 pixels, which +provides enough pixels to clarify contamination, if any. We used +our custom PYTHON scripts to retrieve fluxes separately from each +pixel in a 3 × 3 pixel square centered at a given target. Then, for +each of the nine pixels, we calculated an amplitude spectrum and +checked if the signals reported in Papers I and II is detected in +pixels that overlap with the location of the target. We employed +either PanSTARRS or DSS images and overlapped them with target +pixel files (TPF) created from the TESSCUT tool (Brasseur et al. +2019). In specific cases, that is, bright stars in target masks, even +though far enough from our targets not to contaminate them, we +calculated the amplitude spectra in pixels covering these stars to +check for their variability. As a result of our contamination analysis, +we derived a few common cases, listed below (the columns refer to +Tables in the online materials): +• A signal comes only from the target star, which means no +contamination and the target listed in Papers I and II is the source of +the variability (Figure 1). In the variable contaminator column +of Tables 1-14, we marked these cases with none. +• A signal comes from a contaminating object (Figure 2). These +cases have the name of the contaminator listed in the variable +contaminator column. In parentheses, we added additional infor- +mation on the contaminators collected from the simbad database. If +these contaminators are new variable stars, then they are also listed +in Table 16. This is a false positive case. +• A signal comes from both the target and contaminating ob- +ject(s) (Figure 3). These cases will have frequencies assigned to +either our target (marked as none) or contaminator(s) listed in the +variable contaminator column. This is the case of a variable +target showing an additional signal, which is a false positive. +• No signal is detected in single pixels (Figure 4). These cases +are caused by low S/N in merged pixels defined in Papers I and II. In +the variable contaminator column we marked these cases with +no signal in individual pixels. This case is not verified, however +remarks on nearby stars listed in tables in Papers I and II give some +clue on the possible contamination. +• A signal is detected in all pixels across a target mask (Figure 5). +In the variable contaminator column we marked these cases with +signal in all pixels. This is a false positive case. The source of the +signal is either a nearby bright object that shines over a large area +or an instrumental artifact. +• A nearby non-contaminating bright object (within the target +mask) has been verified positively for its variability (Figure 6). +These new variable stars are listed in Table 16. +The full tables with remarks of our contamination analysis are +presented in the online materials. The most interesting conclusions +of our contamination analysis are the following: +• Among two sdBV candidates from Paper I, TIC 237597052, is +no longer considered a candidate. It turned out to be a main sequence +B8 star (Table 1). Our fit to a spectrum which we collected from the +LAMOST survey provides the following atmospheric parameters, +Teff = 12 460(300) K, log (𝑔/ cm s−2) = 4.17(12). The other candi- +date is a confirmed sdB star, so TIC 262753627 is a new sdB pul- +sator. +• Out of 13 sdBV candidates listed in Paper II, five are no +longer considered the sources of the signal. Apart from these +five contaminated targets, TIC 363766470 is also partially contam- +inated by ATO J265.8117+21.5538. However, our target is still the +source of the 116.44 𝜇Hz frequency. The contamination of one case +could not be verified, while the remaining seven sdBs including +TIC 363766470 are confirmed pulsators. ATO J265.8117+21.5538 +is listed in the simbad database as an eclipsing binary candidate +and we confirm this type by detecting signal at a frequency of +41.09 𝜇Hz along with its harmonics. We determined an orbital pe- +riod of 0.28169 days (Table 2). +• Among 66 other (than the above) variable sdBs listed in Pa- +pers I and II, we found 14 not to be the sources of the variable +signals. Six cases are not verified. The remaining 46 are confirmed +variable sdBs (Tables 1 and 2). +• Out of 10 sdV candidates listed in Paper II, four are no longer +considered to be the sources of the signals, while TIC 194781979 +was identified as a main sequence B6 star. One case is not veri- +fied. The remaining four sdVs are confirmed pulsating candidates +(Table 3). +• Among 23 other (than the above) variable sdVs listed in Pa- +per II, we found seven targets not to be the sources of the signal. +Four cases are not verified. The remaining 12 sdVs are confirmed +variables (Table 3). +• Among 113 spectroscopically unclassified pulsators, we found +68 targets not to be the sources of the signal. Seven cases are not con- +firmed. The remaining 38 original targets are confirmed pulsators. +In the case of TIC 311792028, the 95.37 𝜇Hz frequency along with +its harmonics are native to TIC 311792021, while the 34.14 𝜇Hz +frequency originating in our target does not seem to be related to +pulsations (Tables 4 and 5). +• Among 106 candidate eclipsing binaries, we found 65 targets +not to be the sources of the signal. We do not confirm the variability +of TIC 847473488. We found 38 original targets to be confirmed +eclipsing binaries. In the case of TIC 1509561926, the eclipses +originate in TYC 9289-2657-1, while our target itself shows only +the 102.89 𝜇Hz frequency. In the case of TIC 159448831, the target +is contaminated by an eclipsing binary TIC 159448824, however our +target shows 158.68 𝜇Hz along with harmonics. These frequencies, +though, are not responsible for the eclipses we plot in Figure 9 in +Paper II (Tables 6 and 7). +• Among 248 binaries showing one symmetric maximum, we +found 118 targets not to be the sources of the signal. Five cases +are not verified. We found 125 original targets that are confirmed +variables. TIC 377658867 shows a significant signal at 54.28 𝜇Hz +and its harmonics, though the contaminator TIC 378037013 shows +76.27 𝜇Hz frequency. Likewise in TIC 388622589, which shows +13.77 𝜇Hz frequency, while a contaminator TIC 388622573 is +responsible for the 161.92 𝜇Hz frequency. On the other hand, +TIC 463006021 is heavily contaminated by two sources, even +though the target still shows 53.01 𝜇Hz frequency and its harmonics. +TIC 463006006 shows the 9.03 𝜇Hz frequency presented in Paper I, +while TIC 463006054 shows a 6.71 𝜇Hz frequency and its harmon- +ics. In the case of TIC 2040326958, it is actually TIC 10596964 +that shows the 7.87 𝜇Hz frequency and its harmonics reported in +MNRAS 000, 1–15 (2020) + +TESS sdBVs in both ecliptic hemispheres +3 +Table 1. Southern Ecliptic Hemisphere (SEH) – 28 objects classified as sdBs. Only the first five objects are listed while the full table can be found in the online +materials. +G +Period +Variable +No. +Gaia DR2 +TIC +Name +[mag] +[d] +contaminator +sdBVs +1 +3129751228471383808 +237597052 +TYC 161-49-1 +11.14 +0.05-0.1 +none +2 +3159937564294110080 +262753627 +TYC 770-941-1 +12.46 +0.04-0.08 +none +Phased lightcurves +1 +2333936291513550336 +12379252 +Ton S138 +16.01 +0.2648 +none +2 +2385348183917624448 +9035375 +PHL 460 +12.21 +0.4734 +none +3 +2969438206889996160 +139397815 +- +13.61 +0.2746 +none +Table 2. Northern Ecliptic Hemisphere (NEH) – 53 objects classified as sdBs. Only the first five objects are listed while the full table can be found in the online +materials. +G +Period +Variable +No. +Gaia DR2 +TIC +Name +[mag] +[d] +contaminator +sdBVs +1 +1028374599849118976 +802232206 +SDSSJ082428.41+512601.6 +18.72 +0.1734 +CRTS J082433.5+512441 (EB) +2 +127674641678296704 +353892824 +KUV02281+2730 +15.15 +0.0487 +none +3 +1345049483546987904 +159850392 +GALEXJ17566+4125 +14.28 +0.08 - 0.13 +none +4 +1469357759922416256 +321423000 +SDSSJ132432.37+320420.9 +16.64 +0.0907 +TIC 321422994 +5 +1495329392800826624 +23746001 +PG1350+372 +14.31 +0.06 - 0.09 +none +Table 3. NEH – 33 objects classified as sds. Only the first five objects are listed while the full table can be found in the online materials. +G +Period +Variable +No. +Gaia DR2 +TIC +Name +[mag] +[d] +contaminator +sdVs +1 +1906485375099435136 +259091223 +FBS2209+354 +14.30 +0.07-0.13 +no signal in individual pixels +2 +1952553606634620928 +407657360 +LAMOSTJ214600.31+372119.7 +14.66 +0.045 - 0.1 +TIC 407657373 +3 +2041883531914920064 +20688004 +GALEXJ18578+3048 +13.73 +0.1726 +ATO J284.4865+30.8044 (EB candidate) +4 +2128012018629286144 +1882679963 +KeplerJ19352+4555 +17.16 +0.1766 +ATO J293.8168+45.8972 (EB candidate) +5 +237650985848157312 +194781979 +LAMOSTJ032717.71+410344.5 +10.19 +0.1594 +none +Paper II, though our target still shows a 131.94 𝜇Hz frequency (Ta- +bles 8 and 9). +• Among 52 binaries showing one asymmetric maximum, we +found 42 targets not to be the sources of the signal. The remaining +10 original targets are confirmed variables. TIC 79689537, which +shows a 82.64 𝜇Hz frequency and its harmonics, is heavily con- +taminated by two objects. TIC 79689505 shows the 3.7 𝜇Hz fre- +quency and its harmonics reported in Paper I, while ATO J104.6541- +23.0081 shows a 5.56 𝜇Hz frequency and its harmonics (Tables 8 +and 9). +• Among 66 binaries showing two maxima, we found 49 targets +not to be the source of the signal. The remaining 17 targets are +confirmed binaries (Tables 8 and 9). +• The two novae listed in Paper I are not contaminated (Table 10). +• Among 1 490 spectroscopically unclassified targets showing +signal in their amplitude spectra, we found 719 not to be the sources +of the signal while 460 cases are not verified. Such a high number of +these cases is a consequence of a low signal that is not detectable in +individual pixels. The remaining 311 original targets are confirmed +variables. In five cases, our original targets show a signal, although +it may not be the one reported in Papers I or II (Tables 11 and 12). +• Among 122 spectroscopically classified non-sdB stars, we +found 14 targets not to be the sources of the signal. Nine cases are +not verified, while the remaining 99 original targets are confirmed +variables (Tables 13 and 14). +To summarize our contamination analysis, we found 1 141 tar- +gets not to be the sources of the signal, while 451 targets were not +verified. This leaves us with 721 variable sdB candidates remaining, +including both pulsating and binary stars. +MNRAS 000, 1–15 (2020) + +4 +S.K. Sahoo et al. +Figure 1. Contamination analysis for TIC 323174439, representing an uncontaminated object. Top panel: target mask overlapped with a DSS2 image. The blue +star indicates the target. The red box defines the pixels for which the amplitude spectra were calculated. Bottom panel: Amplitude spectra of the 16 pixels +within the red box in the top panel. +MNRAS 000, 1–15 (2020) + +DSS2 color +E +1' +5.292'x 5.3924 +10.0 +2.0 +15 +3 +7.5 +1.5 +10 +2 +5.0 +1.0 +5 +0.5 +2.5 +0 +0 +0.0 +0.0 +2.5 +2.5 +12.5 +6 +2.0 +10.0 +2.0 +1.5 +1.5 +7.5 +4 +1.0 +1.0 +5.0 +Amplitude [ppt] +0.5 +0.5 +2.5 +0.0 +0.0 +0.0 +3 +2.5 +w +20 +2.0 +2 +15 +2 +1.5 +10 +1.0 +0.5 +0 +0.0 +0 +1.5 +200 +6 +15 +150 +1.0 +4 +10 +100 +0.5 +50 +0.0 +0160 320 480 640 800 +0160 320 480 640 800 +0 160 320 480 640 800 +160 320 480 640 800 +Frequency [μHz]TESS sdBVs in both ecliptic hemispheres +5 +TIC 323890455 +Figure 2. Contamination analysis for TIC 1514267365, representing a false positive case. The marks and colors are the same as in Figure 1. The blue rectangle +represents the aperture used in Paper I. The dominant signal comes from pixels centered on TIC 323890455 (magenta circle in the top panel) and overlaps with +the aperture used in Paper I. +MNRAS 000, 1–15 (2020) + +5 +8 +20 +3 +4 +6 +15 +3 +2 +4 +10 +N +5 +0 +0 +0 +2.5 +2.0 +3 +6 +2.0 +1.5 +2 +1.5 +4 +1.0 +1.0 +Amplitude [ppt] +0.5 +0.5 +0.0 +0.0 +4 +2.0 +3 +3 +1.5 +2 +2 +1.0 +2 +0.5 +0.0 +0 +0 +80 +5 +5 +10.0 +4 +60 +4 +7.5 +3 +3 +40 +5.0 +2 +2 +20 +2.5 +O E +0.0 +0 +160 +320 +480 +640 +0 +160 +320480640 +0 +160 +320 +480 +640 +0 +160 +320 +480 +640 +Frequency [uHz]DSS2 color +4.629' x 4.6696 +S.K. Sahoo et al. +ATO J124.4930-17.9750 +Figure 3. Contamination analysis for TIC 218791808, representing the signal coming from both the target and a neighboring object. The marks and colors are +the same as in Figure 1. The blue dashed box represents the aperture used in Paper I. +MNRAS 000, 1–15 (2020) + +50 +2.0 +12.5 +15 +40 +10.0 +1.5 +30 +7.5 +10 +1.0 +5.0 +20 +5 +0.5 +2.5 +10 +0.0 +0.0 +0 +6 +1.25 +20 +15 +1.00 +15 +4 +10 +0.75 +10 +0.50 +Amplitude [ppt] +2 +5 +5 +0.25 +0 +0 +0.00 +0.8 +1.25 +0.8 +1.00 +0.6 +0.6 +0.75 +0.4 +0.4 +0.50 +0.2 +0.2 +0.25 +0.0 +0.0 +0.00 +0.25 +0.25 +0.20 +0.20 +0.20 +0.3 +0.15 +0.15 +0.15 +0.2 +0.10 +0.10 +0.10 +0.1 +0.05 +0.05 +0.05 +0.00 +0.00 +0.0 +0.00 +0 +40 80 120 160 200 +0 +40 80 120 160 200 +0 40 80 120 160 200 +0 +40 +80 120 160 200 +Frequency [μHz]DSS2 color +11 +5.074'x4.687TESS sdBVs in both ecliptic hemispheres +7 +Figure 4. Contamination analysis for TIC 275358553, representing an unconfirmed case. The marks and colors are the same as in Figure 1 but a PanSTARRS +DR1 image is used instead. No significant signal is detected in any of the individual pixels. +MNRAS 000, 1–15 (2020) + +PanSTARRS DRl color-z-zg-g +4.211x4.19715 +20 +8 +8 +15 +10 +6 +8 +6 +4 +Amplitude [ppt] +6 +2.5 +2.0 +2.0 +1.5 +1.5 +( +1.0 +0.5 +0.5 +0.0 +0.0 +.25 +4 +1.00 +1.00 +3 +0.75 +0.75 +0.50 +0.25 +0.25 +0.00 +0 160 320 480 640 800 +0 160 320 480 640 800 +0 160 320 480 640 800 +0 160 320 480 640 800 +Frequency [μHz]8 +S.K. Sahoo et al. +01 +02 +03 +04 +15 +10 +09 +08 +07 +06 +05 +14 +13 +12 +11 +16 +11 +12 +10 +09 +16 +15 +14 +13 +08 +07 +06 +05 +04 +03 +02 +01 +Figure 5. Contamination analysis for TIC 1314011445, representing a false positive case. The marks and colors are the same as in Figure 1. Magenta numbers +in selected pixels correspond to amplitude spectra shown in the bottom panel. The location of the source of the signal is unconstrained. +MNRAS 000, 1–15 (2020) + +DSS2 color +4.514'x 4.5215 +40 +20 +3 +15 +10 +30 +2 +20 +10 +5 +10 +0 +10 +2.5 +20 +100 +8 +2.0 +15 +75 +6 +1.5 +10 +50 +1.0 +Amplitude [ppt] +25 +0.5 +0 +0 +0.0 +6 +1.5 +1.0 +3 +0.8 +1.0 +2 +0.6 +0.4 +0.5 +0.2 +OF +0.0 +0 +0.0 +1.5 +12.5 +0.5 +0.6 +10.0 +0.4 +1.0 +7.5 +0.4 +0.3 +5.0 +0.2 +0.5 +0.2 +2.5 +0.1 +0.0 +0.0 +0.0 +0.0 +0 +80 +160 +240 +0 +80 +160 +240 +0 +80 +160 +240 +0 +80 +160 +240 +Frequency [μHz]TESS sdBVs in both ecliptic hemispheres +9 +TIC 372181882 +Figure 6. Contamination analysis for TIC 372181885 (the blue star mark).It is an example of discovering a new non-contaminating variable star (magenta +circle) within the target mask during the contamination check of target of interest. The marks and colors are the same as in Figure 2. +MNRAS 000, 1–15 (2020) + +0.08 +0.08 +0.08 +0.06 +0.06 +0.06 +0.06 +0.04 +0.04 +0.04 +0.04 +0.02 +0.02 +0.02 +0.02 +0.00 +0.00 +0.00 +0.00 +0.08 +0.10 +0.05 +0.06 +0.08 +0.04 +0.06 +0.06 +0.04 +0.03 +0.04 +0.04 +0.02 +0.02 +0.02 +0.01 +e0.00 +0.00 +0.00 +0.00 +0.05 +0.05 +0.08 +0.05 +0.04 +0.06 +0.04 +A +0.03 +0.03 +0.03 +0.04 +0.02 +0.02 +0.02 +0.02 +0.01 +0.01 +0.01 +0.00 +0.00 +0.00 +0.00 +0.05 +0.05 +0.04 +0.04 +0.04 +0.04 +0.03 +0.03 +0.03 +0.03 +0.02 +0.02 +0.02 +0.02 +0.01 +0.01 +0.01 +0.01 +0.00 +0.00 +0.00 +0.00 +0 +60 120 180 240 +0 +60 120 180 240 +0 +60 120 180 240 +0 +60 120 180 240 +Frequency [uHz]PanSTARRS DRl color-z-zg-g +4.83x4.8310 +S.K. Sahoo et al. +Table 4. SEH – 83 pulsator candidates. Only the first five objects are listed while the full table can be found in the online materials. +G +Variable +No. +Gaia DR2 +TIC +[mag] +contaminator +1 +2921500461998485248 +744231977 +18.31 +TYC 6526-2198-1 +2 +2927637764107094272 +744958933 +18.67 +ATO J106.3086-23.5750 +3 +3062196993541803904 +754827446 +17.45 +TYC 4817-751-1 +4 +3087146252404755584 +257068255 +15.08 +none +5 +3111790534231122944 +284329074 +15.26 +TYC 163-370-1 +Table 5. NEH – 30 pulsator candidates. Only the first five objects are listed while the full table can be found in the online materials. +G +Variable +No. +Gaia DR2 +TIC +[mag] +contaminator +1 +1422182595056481536 +320525680 +13.87 +no signal in individual pixels +2 +1943952161530528256 +431548978 +15.61 +no signal in individual pixels +34.14 μHz: none; +3 +1974973679520560896 +311792028 +15.88 +95.37 μHz+harmonics: TYC 3605-1317-1 (V) +4 +1988552407605096320 +66784300 +14.88 +TIC 66784249 +5 +1991879937806406656 +2044241813 +18.47 +NSVS 1502401 (EB) +Table 6. SEH – 83 eclipsing binaries. Only the first five objects are listed while the full table can be found in the online materials. +G +Period +Variable +No. +Gaia DR2 +TIC +[mag] +[d] +contaminator +32 eclipsing binaries that show both primary and secondary eclipses +1 +2938186341221700480 +60523137 +16.23 +1.2532 +none +2 +3056677303432024960 +753916356 +17.97 +2.4282 +TIC 68060528 +3 +4037952609036313728 +1556986400 +18.86 +4.2844 +TIC 368875977 +4 +4038037855601783296 +1557298522 +17.18 +2.9735 +TYC 7404-5579-1 +5 +4044609357370901632 +1569961982 +16.46 +2.415 +RS Sgr (B3/4IV/V, EB) +Table 7. NEH – 23 eclipsing binaries. Only the first five objects are listed while the full table can be found in the online materials. +G +Period +Variable +No. +Gaia DR2 +TIC +[mag] +[d] +contaminator +Eclipsing binaries that show only primary eclipses +49.77 μHz+harmonics: none; +1 +1131845039229607680 +459182998 +16.16 +0.2344 +6.94 μHz+harmonic: TIC 459183003 +2 +1417117518648285056 +1400704733 +17.03 +0.3637 +none +3 +1816806183083980288 +1943324398 +17.22 +1.3135 +HD 195052 (F8) +4 +1840900601716813440 +1951174238 +18.92 +1.0329 +TIC 126684646 +5 +1846629538332584960 +15040115 +11.83 +0.8099 +none +Table 8. SEH – 273 spectroscopically unclassified variables with phased light curves. Only the first five objects are listed while the full table can be found in +the online materials. +G +Period +Variable +No. +Gaia DR2 +TIC +[mag] +[d] +contaminator +One symmetric maximum +1 +2896588449084891136 +49547169 +13.28 +0.3086 +IS CMa (F3V) +2 +2905822663130146688 +31353391 +14.03 +0.8929 +none +3 +2909497952544966272 +37118148 +14.28 +0.2681 +none +4 +2911497105202950400 +37004041 +15.16 +0.2833 +none +5 +2921050693020996864 +63113578 +11.45 +0.4854 +none +MNRAS 000, 1–15 (2020) + +TESS sdBVs in both ecliptic hemispheres +11 +Table 9. NEH – 93 spectroscopically unclassified variables with phased light curves. Only the first five objects are listed while the full table can be found in the +online materials. +G +Period +Variable +No. +Gaia DR2 +TIC +[mag] +[d] +contaminator +One symmetric maximum +1 +1000519267329142144 +444946935 +15.92 +1.6725 +none +2 +1086341235118052096 +85158691 +12.85 +0.3546 +none +3 +1099487030500185344 +743328948 +16.90 +0.3297 +none +4 +1133795950814826240 +841399917 +17.31 +1.5134 +none +5 +1141625057721183616 +138400883 +16.24 +0.0680 +none +Table 10. SEH – two novae. This table is also included in the online materials. +G +Variable +No. +Gaia DR2 +TIC +Name +[mag] +contaminator +1 +5207384891323130368 +735128403 +AH Men +13.51 +none +2 +6544371342567818496 +121422158 +RZ Gru +12.63 +none +Table 11. SEH – 1262 spectroscopically unclassified variables with amplitude spectra. Only the first five objects are listed while the full table can be found in +the online materials. +G +Variable +No. +Gaia DR2 +TIC +[mag] +contaminator +1 +2326333512204996992 +380826878 +15.69 +none +2 +2342907791000463232 +610076106 +17.18 +[SHM2017] J013.19449-26.56892 (RR Lyr) +3 +2342907962798690944 +610077229 +16.68 +[SHM2017] J013.19449-26.56892 (RR Lyr) +4 +2409630520260038784 +2052262357 +18.09 +Cl* NGC 7492 C 1306 (RR Lyr) +5 +2410677839445234944 +111183765 +14.48 +none +Table 12. NEH – 228 spectroscopically unclassified variables with amplitude spectra. Only the first five objects are listed while the full table can be found in +the online materials. +G +Variable +No. +Gaia DR2 +TIC +[mag] +contaminator +1 +1082306439760979840 +743148169 +18.81 +TIC 284473271 +2 +1107705772542003200 +705157619 +18.09 +no signal in individual pixels +3 +1108642968765677696 +743476657 +18.74 +V486 Cam (RR Lyr) +4 +1112770367915047424 +705166070 +17.41 +none +5 +1113516073020001152 +705175423 +18.32 +TIC 468921975 +Table 13. SEH – 76 non-sdB classified variables. Only the first five objects are listed while the full table can be found in the online materials. +G +Period +Variable +No. +Gaia DR2 +TIC +[mag] +spT +[d] +contaminator +Pulsators +1 +2969201399574096128 +708596809 +11.30 +A0IV/V +- +no signal in individual pixels +2 +3109409266919646976 +168595004 +10.70 +A5 +- +none +3 +3115125211261708032 +293137161 +11.01 +A0 +- +none +4 +3344114626761364224 +437889214 +10.02 +B5 +- +none +5 +3396397877830881792 +247513086 +8.17 +A0 +- +none +Table 14. NEH – 46 non-sdB classified variables. Only the first five objects are listed while the full table can be found in the online materials. +G +Period +Variable +No. +Gaia DR2 +TIC +[mag] +spT +[d] +contaminator +Pulsators +1 +1625627602365544832 +198209459 +13.53 +B2 +- +none +2 +1713032695100155904 +298091568 +15.00 +B4.1 +- +none +3 +1861191062326013696 +1955410399 +10.31 +A0 +- +none +11 μHz+harmonics: none; +4 +2077737678383889408 +270610177 +11.77 +Be +- +46 - 70 μHz: UCAC4 663-077912 (PulV) +5 +2132171608553758336 +279919275 +13.46 +B3V +- +none +MNRAS 000, 1–15 (2020) + +12 +S.K. Sahoo et al. +3 +UPDATED AMPLITUDE SPECTRA AND NEW +PULSATING SDB STARS +We took advantage of access to the 10 min FFI data collected during +Years 3 and 4 to discover new variable sdB stars and to improve the +amplitude spectra for known sdBV stars. Not all variable stars listed +in Papers I and II were subject to this updated analysis. Binaries +would surely benefit from a three times better sampling, which +would yield much better definition of eclipses and more precise +orbital period estimation, but it would not affect their variability +type. The contaminated stars, which turned out not to be sources +of the signal, were also excluded. We focused only on targets that +show rich signal close to the 277.7 μHz Nyquist frequency, which +appear to be quite convincing pulsations. In these cases shifting the +Nyquist frequency to a three times higher value would uncover the +entire g-mode region of possible pulsating sdB stars. We ended up +with a final list of 78 targets. +The light curve extraction process and data reduction of the +10 min FFI data, are the same as for the 30 min FFI data described +in Papers I and II, where we refer the reader for details. Out of 78 pre- +selected targets we found 24 that show multiple frequencies, which +we interpret as pulsations, and the number of frequencies were +significantly increased or more frequencies beyond the 277.7 μHz +Nyquist frequency were detected. We list these targets in Table 15 in +the online material. The targets are confirmed as the original sources +of the detected signals. We show the amplitude spectra of these +24 targets in Figure 7. We marked the typical g-mode frequency +range (i.e. 100 – 400 μHz) with dashed vertical lines. This region +overlaps with the typical p-mode frequency range of 𝛿 Scuti stars, +so it is not generally straightforward to claim the detection of a +pulsating sdB star based only on the amplitude spectrum content. +Alternatively, if the frequencies detected are outside the indicated +range, we might well doubt the detection of an sdB pulsator. For +instance, TIC 224284872 is an A star and the frequencies are outside +the expected range. Similarly, TIC 181142865 turned out to be a +main sequence star. TIC 437889214 shows frequencies below the +expected region – its spectral type is B5. There are other targets +which are not classified as hot subdwarfs, yet show frequencies in +the range characteristic of pulsating sdB stars. These cases may be +either 𝛿 Scuti pulsators or misclassified hot subdwarfs. We confirm a +detection of g-mode pulsations in 11 sdB pulsators. TIC 442750342 +seems to be an exception in our sample being a low gravity and cool +sdB pulsator. +MNRAS 000, 1–15 (2020) + +TESS sdBVs in both ecliptic hemispheres +13 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Amplitude [ppt] +0.2 +0.4 +0.6 +TIC 1955410399 [late B type] +0.2 +0.4 +0.6 +TIC 749940934 [late B type] +0.5 +1.0 +TIC 262753627 [sdB] +0.3 +0.5 +TIC 437889214 [B5] +0.2 +0.4 +0.6 +TIC 247513086 [A0] +1.0 +2.0 +TIC 201251043 [sdB] +0.2 +0.4 +TIC 686449995 [A5] +1.0 +2.0 +TIC 323174439 [sdB] +2.0 +4.0 +TIC 362098036 [early B type] +0.5 +1.0 +1.5 +TIC 442128473 [early B type] +1.0 +2.0 +TIC 141684783 [sdB] +0.2 +0.4 +0.6 +TIC 442750342 [sdB] +0.2 +0.4 +0.6 +TIC 126486580 [late B type] +0.5 +1.0 +1.5 +TIC 178621334 [sdB] +0.5 +1.0 +1.5 +TIC 4595563 [early A type] +0.3 +0.5 +TIC 25836205 [sdB] +0.3 +0.5 +TIC 181142865 [late B or early A type] +0.2 +0.4 +0.6 +TIC 190720627 [B or B+A type] +1.0 +2.0 +TIC 401975322 [B1] +2.0 +4.0 +6.0 +TIC 446919722 [sdB] +2.0 +4.0 +6.0 +TIC 1035773311 [sdB] +1.0 +2.0 +TIC 388832080 [sdB] +0 +80 +160 +240 +320 +400 +480 +560 +640 +720 +800 +Frequency [ Hz] +5.0 +10.0 +TIC 22217594 [sdB] +0 +80 +160 +240 +320 +400 +480 +560 +640 +720 +800 +Frequency [ Hz] +2.5 +5.0 +TIC 224284872 [A0] +Figure 7. The amplitude spectra of pulsators analyzed with 10 min FFI data. Vertical dashed lines indicate the typical g-mode region of pulsating hot subdwarfs. +The horizontal lines mark the detection thresholds. +MNRAS 000, 1–15 (2020) + +14 +S.K. Sahoo et al. +Table 15. Pulsators confirmed with the 10 min FFI data. We show amplitude spectra of these objects in Figure 7. This table is also included in the on-line +materials. +G +Variable +No. +Gaia DR2 +TIC +[mag] +contaminator +Remarks +1 +1861191062326013696 +1955410399 +10.66 +none +late B type +2 +3032890473180888192 +749940934 +12.46 +none +late B type +3 +3159937564294110080 +262753627 +12.46 +none +sdB +4 +3344114626761364224 +437889214 +10.16 +none +B5 +5 +3396397877830881792 +247513086 +8.44 +none +A0 +6 +4923853724788504192 +201251043 +11.93 +none +sdB +7 +5090382015016433920 +686449995 +6.99 +none +A5 +8 +5196271513123121152 +323174439 +13.30 +none +sdB +9 +5250674622612902912 +362098036 +11.48 +none +early B type +10 +5257747299878049152 +442128473 +10.84 +none +early B type +11 +5266133451162548864 +141684783 +14.53 +none +sdB +12 +5307946881949072000 +442750342 +12.82 +none +sdB +13 +5326745919424790656 +126486580 +9.30 +none +late B type +14 +5362558250096941056 +178621334 +13.33 +none +sdB +15 +5429254969036524416 +4595563 +10.11 +none +early A type +16 +5439887654492064256 +25836205 +13.14 +none +sdB +17 +5525342630213336448 +181142865 +11.11 +none +late B or early A type +18 +5622554881336253824 +190720627 +11.07 +none +B or B+A type +19 +5796399012705196800 +401975322 +10.21 +none +B1 +20 +5823403087017696384 +446919722 +12.96 +none +sdB +21 +5872410931625754624 +1035773311 +17.46 +none +sdB +22 +5922070855307705472 +388832080 +12.58 +none +sdB +23 +6143764182206682112 +22217594 +15.16 +none +sdB +24 +6534581776366266752 +224284872 +13.53 +none +A0 +4 +NEW VARIABLES +As a by-product of the contamination analysis we report the true +sources of variability preliminarily assigned to the targets listed in +Papers I and II. In Tables 1-14 we provide a variable contamina- +tor column which, in the case of a positive variability contamina- +tion, contains a name of a contaminator. In addition, we detected +new variables that do not contaminate our pre-selected targets but +are located within the target masks of our targets. For a practical +reason, all these variable contaminators and new non-contaminating +variables are listed in Table 16 in the online materials. In total, we +report detection of 682 new variable stars, including two, listed +last in Table 16, that have no Gaia (Gaia Collaboration et al. 2018) +designation yet. To be precise, the discovery of the variability of +the majority of these stars was presented in Papers I and II, so only +97 stars are found to be new variables (accounting for Papers I and +II) while the remaining 585 stars now have the variability properly +assigned (as compared to Paper I and II). +MNRAS 000, 1–15 (2020) + +TESS sdBVs in both ecliptic hemispheres +15 +Table 16. Basic information of the 682 new variable stars found during the contamination analysis in both SEH and NEH. Only first five objects are listed +while the full table can be found in the online materials. +G +No. +Gaia DR2 +TIC +Name +[mag] +1 +1000847845211000960 +14196021 +- +16.65 +2 +1030011910101662336 +467154863 +- +12.56 +3 +1082306439760980224 +284473271 +- +16.95 +4 +1113516077316307328 +468921975 +- +17.32 +5 +1131845245388039296 +459183003 +- +14.38 +5 +SUMMARY +We presented the results of our contamination analysis of stars +included in Papers I and II. We identified 1 141 false positives, while +451 variables were not verified because, in most cases, the signal is +of too low amplitude to be detected in individual pixels. The total +number of targets, which are the sources of the signal we presented +in Papers I and II is 721. As a by-product of our contamination +analysis we found 97 new variables that happened to be within +target masks of our original stars listed in Papers I and II. In total, +we analysed 2 995 targets in TESS fields, where 2313 targets were +presented in Papers I and II and the remaining 682 variable targets +were found during the contamination check. Out of the 2 313 targets, +we confirmed 721 as variable after contamination analysis. Hence +the total number of variable targets we found is 1 403. +We pre-selected 78 uncontaminated targets that are pulsator +candidates (that is, they show rich pulsation content close to the +277.7 μHz Nyquist frequency) for additional analysis using the +10 min FFI data collected during Years 3 and 4. We ended up with +24 targets for which those new data turned out to be beneficial – that +is, more peaks either below, or especially beyond, the 277.7 μHz +frequency were detected. For any of these 24 stars without spectral +type, we used publicly available data and/or spectroscopic data col- +lected with the 1.9 m telescope at the South African Astronomical +Observatory to identify hot subdwarfs. In total, we found 11 new +sdB pulsators. Details of the spectroscopic analysis will be provided +in Worters et al. (in preparation). +One of the pulsator candidates, TIC 362098036, was a subject +of a pulsation mode identification and the result was reported in +Paper 1. Our analysis confirmed that the target is a main sequence +B star, which makes the mode identification irrelevant. +ACKNOWLEDGEMENTS +Financial +support +from +the +National +Science +Center +in +Poland under projects No. UMO-2017/26/E/ST9/00703 and UMO- +2017/25/B/ST9/02218 is acknowledged. PN acknowledges sup- +port from the Grant Agency of the Czech Republic (GAČR 22- +34467S). The Astronomical Institute in Ondřejov is supported by +the project RVO:67985815. This paper includes data collected by +the TESS mission. Funding for the TESS mission is provided by +the NASA Explorer Program. This work has made use of data +from the European Space Agency (ESA) mission Gaia (https: +//www.cosmos.esa.int/gaia), processed by the Gaia Data Pro- +cessing and Analysis Consortium (DPAC, https://www.cosmos. +esa.int/web/gaia/dpac/consortium). Funding for the DPAC +has been provided by national institutions, in particular, the institu- +tions participating in the Gaia multilateral agreement. This research +has used the services of www.Astroserver.org. +DATA AVAILABILITY +The datasets were derived from MAST in the public domain +archive.stsci.edu. +REFERENCES +Baran A. S., Sahoo S. K., Sanjayan S., Ostrowski J., 2021, Monthly Notices +of the Royal Astronomical Society, 503, 3828 +Brasseur C. E., Phillip C., Fleming S. W., Mullally S. E., White R. L., 2019, +Astrocut: Tools for creating cutouts of TESS images (ascl:1905.007) +Gaia Collaboration et al., 2018, A&A, 616, A1 +Geier S., 2020, A&A, 635, A193 +Geier S., Raddi R., Gentile Fusillo N. P., Marsh T. R., 2019, A&A, 621, A38 +Heber U., 2016, PASP, 128, 2001 +Sahoo S. K., Baran A. S., Sanjayan S., Ostrowski J., 2020, Monthly Notices +of the Royal Astronomical Society, 499, 5508 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–15 (2020) + diff --git a/TtAzT4oBgHgl3EQfJfvP/content/tmp_files/load_file.txt b/TtAzT4oBgHgl3EQfJfvP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f3f137c3841455f5e87b674c61009dfc39a8fe81 --- /dev/null +++ b/TtAzT4oBgHgl3EQfJfvP/content/tmp_files/load_file.txt @@ -0,0 +1,879 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf,len=878 +page_content='MNRAS 000, 1–15 (2020) Preprint 4 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 A search for variable subdwarf B stars in TESS Full Frame Images III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' An update on variable targets in both ecliptic hemispheres – contamination analysis and new sdB pulsators S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Sahoo1,2★, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Baran2,3,4, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Worters5, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Németh2,6,7 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Kilkenny8 1Nicolaus Copernicus Astronomical Centre of the Polish Academy of Sciences, ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Bartycka 18, 00-716 Warsaw, Poland 2ARDASTELLA Research Group 3Astronomical Observatory, University of Warsaw, Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Ujazdowskie 4, 00-478 Warszawa, Poland 4Department of Physics, Astronomy, and Materials Science, Missouri State University, Springfield, MO 65897, USA 5South African Astronomical Observatory, Observatory 7935, South Africa 6Astronomical Institute of the Czech Academy of Sciences, Fričova 298, CZ-251 65 Ondřejov, Czech Republic 7Astroserver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='org, Fő tér 1, 8533 Malomsok, Hungary 8Department of Physics and Astronomy, University of the Western Cape, Private Bag X17, Bellville 7535, South Africa Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' in original form ZZZ ABSTRACT We present an update on the variable star survey performed on the TESS 30 min Full Frame Image (FFI) data reported by our first two papers in this series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' This update includes a contamination analysis in order to identify false positives and analysis of the TESS 10 min FFI data collected during Years 3 and 4 of the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' We clarify the variability status of 2 995 targets identifying 1 403 variable stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' In addition, we spectroscopically classify 24 pre- filtered targets sampled with the 10 min FFI data and discover 11 new sdB pulsators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Future follow-up space- and/or ground-based data of variables reported here, to identify the nature of their variability and reveal spectroscopic parameters of the stars, would complement this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Key words: Stars: subdwarfs – Stars: oscillations (including pulsations) – asteroseismology 1 INTRODUCTION Sahoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' (2020) (Paper I) and Baran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' (2021) (Paper II) presented their results of variability checks of the most promising subdwarf B (sdB) candidates found in the Geier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' (2019) and Geier (2020) catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The former authors pre-selected 45 674 tar- gets and used the Full Frame Images (FFI) collected by the TESS mission in the Southern Ecliptic Hemisphere (SEH) during Year 1 and in the Northern Ecliptic Hemisphere (NEH) during Year 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' As a result, 2 313 new variable targets in both ecliptic hemispheres were listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' It is a well known feature of the TESS CCDs that an individual pixel has a 21 arcsec square projection on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' This makes con- tamination a serious problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' A contaminating star contributes by either increasing the average flux level in an aperture, thus lowering the apparent amplitude of flux variation in the target, or – if variable – makes a constant flux target star a false positive variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The former issue can be corrected by removing the estimated amount of a contaminating star’s flux and this is fairly well achieved in the Pre-search Data Conditioning fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The latter issue can only be resolved by checking the variability of all contaminators around ★ E-mail: sumanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='kumar27@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='com the target of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' We also found this method useful to resolve variability in case both the target of interest and contaminator are variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' In this work, we aimed to remove false positives from our list of 2 313 new variable objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' We searched all pixels within the target masks of all 2 313 objects listed in Papers I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' In addition, for the most promising pulsating subdwarf candidates, we retrieved the new light curves from the FFI collected during Years 3 and 4, which are sampled at a 10 min cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' This cadence extends the Nyquist frequency up to 833.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='3 μHz, which allows us to cover the entire gravity mode region in an amplitude spectrum1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' By promising candidates, we mean objects that in Papers I and II show signals close to the Nyquist frequency of 277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='7 μHz resulting from the 30 min cadence of observation used during Years 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' To summarize the findings reported in Paper I and II, there are 15 pulsating subdwarf B star candidates, 79 variable (other than pulsating) sdB stars, 33 variable subdwarf candidates and 2 186 other variable stars, including 123 stars with non-sd classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' 1 In hot subdwarfs, pressure or p-mode pulsations typically have periods of a few minutes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' gravity or g-modes have periods of the order of an hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' See, for example, Heber (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' © 2020 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='01082v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='SR] 3 Jan 2023 2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Sahoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Data are downloaded and processed in the same way as ex- plained in Papers I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' For a convenient comparison, the Tables presented here will preserve the listing order from the previous papers of this series, however some of the tables from Paper I are merged to be consistent with those in Paper II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' They also contain the most important information from Papers I and II with additional information resulting from this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' In the following sections, we explain the details of the contamination analysis and provide the results of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' 2 CONTAMINATION ANALYSIS We defined each target mask as a square of 11 × 11 pixels, which provides enough pixels to clarify contamination, if any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' We used our custom PYTHON scripts to retrieve fluxes separately from each pixel in a 3 × 3 pixel square centered at a given target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Then, for each of the nine pixels, we calculated an amplitude spectrum and checked if the signals reported in Papers I and II is detected in pixels that overlap with the location of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' We employed either PanSTARRS or DSS images and overlapped them with target pixel files (TPF) created from the TESSCUT tool (Brasseur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' In specific cases, that is, bright stars in target masks, even though far enough from our targets not to contaminate them, we calculated the amplitude spectra in pixels covering these stars to check for their variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' As a result of our contamination analysis, we derived a few common cases, listed below (the columns refer to Tables in the online materials): A signal comes only from the target star, which means no contamination and the target listed in Papers I and II is the source of the variability (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' In the variable contaminator column of Tables 1-14, we marked these cases with none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' A signal comes from a contaminating object (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' These cases have the name of the contaminator listed in the variable contaminator column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' In parentheses, we added additional infor- mation on the contaminators collected from the simbad database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' If these contaminators are new variable stars, then they are also listed in Table 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' This is a false positive case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' A signal comes from both the target and contaminating ob- ject(s) (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' These cases will have frequencies assigned to either our target (marked as none) or contaminator(s) listed in the variable contaminator column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' This is the case of a variable target showing an additional signal, which is a false positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' No signal is detected in single pixels (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' These cases are caused by low S/N in merged pixels defined in Papers I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' In the variable contaminator column we marked these cases with no signal in individual pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' This case is not verified, however remarks on nearby stars listed in tables in Papers I and II give some clue on the possible contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' A signal is detected in all pixels across a target mask (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' In the variable contaminator column we marked these cases with signal in all pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' This is a false positive case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The source of the signal is either a nearby bright object that shines over a large area or an instrumental artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' A nearby non-contaminating bright object (within the target mask) has been verified positively for its variability (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' These new variable stars are listed in Table 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The full tables with remarks of our contamination analysis are presented in the online materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The most interesting conclusions of our contamination analysis are the following: Among two sdBV candidates from Paper I, TIC 237597052, is no longer considered a candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' It turned out to be a main sequence B8 star (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Our fit to a spectrum which we collected from the LAMOST survey provides the following atmospheric parameters, Teff = 12 460(300) K, log (𝑔/ cm s−2) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='17(12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The other candi- date is a confirmed sdB star, so TIC 262753627 is a new sdB pul- sator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Out of 13 sdBV candidates listed in Paper II, five are no longer considered the sources of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Apart from these five contaminated targets, TIC 363766470 is also partially contam- inated by ATO J265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='8117+21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='5538.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' However, our target is still the source of the 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='44 𝜇Hz frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The contamination of one case could not be verified, while the remaining seven sdBs including TIC 363766470 are confirmed pulsators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' ATO J265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='8117+21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='5538 is listed in the simbad database as an eclipsing binary candidate and we confirm this type by detecting signal at a frequency of 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='09 𝜇Hz along with its harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' We determined an orbital pe- riod of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='28169 days (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Among 66 other (than the above) variable sdBs listed in Pa- pers I and II, we found 14 not to be the sources of the variable signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Six cases are not verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The remaining 46 are confirmed variable sdBs (Tables 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Out of 10 sdV candidates listed in Paper II, four are no longer considered to be the sources of the signals, while TIC 194781979 was identified as a main sequence B6 star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' One case is not veri- fied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The remaining four sdVs are confirmed pulsating candidates (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Among 23 other (than the above) variable sdVs listed in Pa- per II, we found seven targets not to be the sources of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Four cases are not verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The remaining 12 sdVs are confirmed variables (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Among 113 spectroscopically unclassified pulsators, we found 68 targets not to be the sources of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Seven cases are not con- firmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The remaining 38 original targets are confirmed pulsators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' In the case of TIC 311792028, the 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='37 𝜇Hz frequency along with its harmonics are native to TIC 311792021, while the 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='14 𝜇Hz frequency originating in our target does not seem to be related to pulsations (Tables 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Among 106 candidate eclipsing binaries, we found 65 targets not to be the sources of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' We do not confirm the variability of TIC 847473488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' We found 38 original targets to be confirmed eclipsing binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' In the case of TIC 1509561926, the eclipses originate in TYC 9289-2657-1, while our target itself shows only the 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='89 𝜇Hz frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' In the case of TIC 159448831, the target is contaminated by an eclipsing binary TIC 159448824, however our target shows 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='68 𝜇Hz along with harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' These frequencies, though, are not responsible for the eclipses we plot in Figure 9 in Paper II (Tables 6 and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Among 248 binaries showing one symmetric maximum, we found 118 targets not to be the sources of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Five cases are not verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' We found 125 original targets that are confirmed variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' TIC 377658867 shows a significant signal at 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='28 𝜇Hz and its harmonics, though the contaminator TIC 378037013 shows 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='27 𝜇Hz frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Likewise in TIC 388622589, which shows 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='77 𝜇Hz frequency, while a contaminator TIC 388622573 is responsible for the 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='92 𝜇Hz frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' On the other hand, TIC 463006021 is heavily contaminated by two sources, even though the target still shows 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='01 𝜇Hz frequency and its harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' TIC 463006006 shows the 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='03 𝜇Hz frequency presented in Paper I, while TIC 463006054 shows a 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='71 𝜇Hz frequency and its harmon- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' In the case of TIC 2040326958, it is actually TIC 10596964 that shows the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='87 𝜇Hz frequency and its harmonics reported in MNRAS 000, 1–15 (2020) TESS sdBVs in both ecliptic hemispheres 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Southern Ecliptic Hemisphere (SEH) – 28 objects classified as sdBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Only the first five objects are listed while the full table can be found in the online materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' G Period Variable No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Gaia DR2 TIC Name [mag] [d] contaminator sdBVs 1 3129751228471383808 237597052 TYC 161-49-1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='05-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='1 none 2 3159937564294110080 262753627 TYC 770-941-1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='04-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='08 none Phased lightcurves 1 2333936291513550336 12379252 Ton S138 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='2648 none 2 2385348183917624448 9035375 PHL 460 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='4734 none 3 2969438206889996160 139397815 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='2746 none Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Northern Ecliptic Hemisphere (NEH) – 53 objects classified as sdBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Only the first five objects are listed while the full table can be found in the online materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' G Period Variable No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Gaia DR2 TIC Name [mag] [d] contaminator sdBVs 1 1028374599849118976 802232206 SDSSJ082428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='41+512601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='1734 CRTS J082433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='5+512441 (EB) 2 127674641678296704 353892824 KUV02281+2730 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0487 none 3 1345049483546987904 159850392 GALEXJ17566+4125 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='08 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='13 none 4 1469357759922416256 321423000 SDSSJ132432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='37+320420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='9 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0907 TIC 321422994 5 1495329392800826624 23746001 PG1350+372 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='06 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='09 none Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' NEH – 33 objects classified as sds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Only the first five objects are listed while the full table can be found in the online materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' G Period Variable No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Gaia DR2 TIC Name [mag] [d] contaminator sdVs 1 1906485375099435136 259091223 FBS2209+354 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='07-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='13 no signal in individual pixels 2 1952553606634620928 407657360 LAMOSTJ214600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='31+372119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='045 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='1 TIC 407657373 3 2041883531914920064 20688004 GALEXJ18578+3048 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='1726 ATO J284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='4865+30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='8044 (EB candidate) 4 2128012018629286144 1882679963 KeplerJ19352+4555 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='1766 ATO J293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='8168+45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='8972 (EB candidate) 5 237650985848157312 194781979 LAMOSTJ032717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='71+410344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='1594 none Paper II, though our target still shows a 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='94 𝜇Hz frequency (Ta- bles 8 and 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Among 52 binaries showing one asymmetric maximum, we found 42 targets not to be the sources of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The remaining 10 original targets are confirmed variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' TIC 79689537, which shows a 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='64 𝜇Hz frequency and its harmonics, is heavily con- taminated by two objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' TIC 79689505 shows the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='7 𝜇Hz fre- quency and its harmonics reported in Paper I, while ATO J104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='6541- 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0081 shows a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='56 𝜇Hz frequency and its harmonics (Tables 8 and 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Among 66 binaries showing two maxima, we found 49 targets not to be the source of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The remaining 17 targets are confirmed binaries (Tables 8 and 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The two novae listed in Paper I are not contaminated (Table 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Among 1 490 spectroscopically unclassified targets showing signal in their amplitude spectra, we found 719 not to be the sources of the signal while 460 cases are not verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Such a high number of these cases is a consequence of a low signal that is not detectable in individual pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The remaining 311 original targets are confirmed variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' In five cases, our original targets show a signal, although it may not be the one reported in Papers I or II (Tables 11 and 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Among 122 spectroscopically classified non-sdB stars, we found 14 targets not to be the sources of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Nine cases are not verified, while the remaining 99 original targets are confirmed variables (Tables 13 and 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' To summarize our contamination analysis, we found 1 141 tar- gets not to be the sources of the signal, while 451 targets were not verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' This leaves us with 721 variable sdB candidates remaining, including both pulsating and binary stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' MNRAS 000, 1–15 (2020) 4 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Sahoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Contamination analysis for TIC 323174439, representing an uncontaminated object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Top panel: target mask overlapped with a DSS2 image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The blue star indicates the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The red box defines the pixels for which the amplitude spectra were calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Bottom panel: Amplitude spectra of the 16 pixels within the red box in the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=" MNRAS 000, 1–15 (2020) DSS2 color E 1' 5." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content="292'x 5." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='3924 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='5 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 0160 320 480 640 800 0160 320 480 640 800 0 160 320 480 640 800 160 320 480 640 800 Frequency [μHz]TESS sdBVs in both ecliptic hemispheres 5 TIC 323890455 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Contamination analysis for TIC 1514267365, representing a false positive case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The marks and colors are the same as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The blue rectangle represents the aperture used in Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The dominant signal comes from pixels centered on TIC 323890455 (magenta circle in the top panel) and overlaps with the aperture used in Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' MNRAS 000, 1–15 (2020) 5 8 20 3 4 6 15 3 2 4 10 N 5 0 0 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 3 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='00 0 40 80 120 160 200 0 40 80 120 160 200 0 40 80 120 160 200 0 40 80 120 160 200 Frequency [μHz]DSS2 color 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content="074'x4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='687TESS sdBVs in both ecliptic hemispheres 7 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Contamination analysis for TIC 275358553, representing an unconfirmed case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The marks and colors are the same as in Figure 1 but a PanSTARRS DR1 image is used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' No significant signal is detected in any of the individual pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' MNRAS 000, 1–15 (2020) PanSTARRS DRl color-z-zg-g 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='211x4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='19715 20 8 8 15 10 6 8 6 4 Amplitude [ppt] 6 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Sahoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' 01 02 03 04 15 10 09 08 07 06 05 14 13 12 11 16 11 12 10 09 16 15 14 13 08 07 06 05 04 03 02 01 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Contamination analysis for TIC 1314011445, representing a false positive case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The marks and colors are the same as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Magenta numbers in selected pixels correspond to amplitude spectra shown in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The location of the source of the signal is unconstrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' MNRAS 000, 1–15 (2020) DSS2 color 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content="514'x 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='5215 40 20 3 15 10 30 2 20 10 5 10 0 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='5 20 100 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 15 75 6 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 0 80 160 240 0 80 160 240 0 80 160 240 0 80 160 240 Frequency [μHz]TESS sdBVs in both ecliptic hemispheres 9 TIC 372181882 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Contamination analysis for TIC 372181885 (the blue star mark).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='It is an example of discovering a new non-contaminating variable star (magenta circle) within the target mask during the contamination check of target of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The marks and colors are the same as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' MNRAS 000, 1–15 (2020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='08 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='8310 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Sahoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' SEH – 83 pulsator candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Only the first five objects are listed while the full table can be found in the online materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' G Variable No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Gaia DR2 TIC [mag] contaminator 1 2921500461998485248 744231977 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='31 TYC 6526-2198-1 2 2927637764107094272 744958933 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='67 ATO J106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='3086-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='5750 3 3062196993541803904 754827446 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='45 TYC 4817-751-1 4 3087146252404755584 257068255 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='08 none 5 3111790534231122944 284329074 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='26 TYC 163-370-1 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' NEH – 30 pulsator candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Only the first five objects are listed while the full table can be found in the online materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' G Variable No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Gaia DR2 TIC [mag] contaminator 1 1422182595056481536 320525680 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='87 no signal in individual pixels 2 1943952161530528256 431548978 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='61 no signal in individual pixels 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='14 μHz: none;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' 3 1974973679520560896 311792028 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='88 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='37 μHz+harmonics: TYC 3605-1317-1 (V) 4 1988552407605096320 66784300 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='88 TIC 66784249 5 1991879937806406656 2044241813 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='47 NSVS 1502401 (EB) Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' SEH – 83 eclipsing binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Only the first five objects are listed while the full table can be found in the online materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' G Period Variable No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Gaia DR2 TIC [mag] [d] contaminator 32 eclipsing binaries that show both primary and secondary eclipses 1 2938186341221700480 60523137 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='2532 none 2 3056677303432024960 753916356 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='4282 TIC 68060528 3 4037952609036313728 1556986400 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='86 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='2844 TIC 368875977 4 4038037855601783296 1557298522 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='9735 TYC 7404-5579-1 5 4044609357370901632 1569961982 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='415 RS Sgr (B3/4IV/V, EB) Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' NEH – 23 eclipsing binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Only the first five objects are listed while the full table can be found in the online materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' G Period Variable No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Gaia DR2 TIC [mag] [d] contaminator Eclipsing binaries that show only primary eclipses 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='77 μHz+harmonics: none;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' 1 1131845039229607680 459182998 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='2344 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='94 μHz+harmonic: TIC 459183003 2 1417117518648285056 1400704733 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='3637 none 3 1816806183083980288 1943324398 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='3135 HD 195052 (F8) 4 1840900601716813440 1951174238 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='92 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0329 TIC 126684646 5 1846629538332584960 15040115 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='8099 none Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' SEH – 273 spectroscopically unclassified variables with phased light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Only the first five objects are listed while the full table can be found in the online materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' G Period Variable No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Gaia DR2 TIC [mag] [d] contaminator One symmetric maximum 1 2896588449084891136 49547169 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='3086 IS CMa (F3V) 2 2905822663130146688 31353391 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='8929 none 3 2909497952544966272 37118148 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='2681 none 4 2911497105202950400 37004041 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='2833 none 5 2921050693020996864 63113578 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='4854 none MNRAS 000, 1–15 (2020) TESS sdBVs in both ecliptic hemispheres 11 Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' NEH – 93 spectroscopically unclassified variables with phased light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Only the first five objects are listed while the full table can be found in the online materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' G Period Variable No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Gaia DR2 TIC [mag] [d] contaminator One symmetric maximum 1 1000519267329142144 444946935 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='92 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='6725 none 2 1086341235118052096 85158691 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='3546 none 3 1099487030500185344 743328948 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='3297 none 4 1133795950814826240 841399917 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='5134 none 5 1141625057721183616 138400883 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0680 none Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' SEH – two novae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' This table is also included in the online materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' G Variable No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Gaia DR2 TIC Name [mag] contaminator 1 5207384891323130368 735128403 AH Men 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='51 none 2 6544371342567818496 121422158 RZ Gru 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='63 none Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' SEH – 1262 spectroscopically unclassified variables with amplitude spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Only the first five objects are listed while the full table can be found in the online materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' G Variable No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Gaia DR2 TIC [mag] contaminator 1 2326333512204996992 380826878 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='69 none 2 2342907791000463232 610076106 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='18 [SHM2017] J013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='19449-26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='56892 (RR Lyr) 3 2342907962798690944 610077229 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='68 [SHM2017] J013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='19449-26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='56892 (RR Lyr) 4 2409630520260038784 2052262357 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='09 Cl* NGC 7492 C 1306 (RR Lyr) 5 2410677839445234944 111183765 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='48 none Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' NEH – 228 spectroscopically unclassified variables with amplitude spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Only the first five objects are listed while the full table can be found in the online materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' G Variable No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Gaia DR2 TIC [mag] contaminator 1 1082306439760979840 743148169 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='81 TIC 284473271 2 1107705772542003200 705157619 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='09 no signal in individual pixels 3 1108642968765677696 743476657 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='74 V486 Cam (RR Lyr) 4 1112770367915047424 705166070 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='41 none 5 1113516073020001152 705175423 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='32 TIC 468921975 Table 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' SEH – 76 non-sdB classified variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Only the first five objects are listed while the full table can be found in the online materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' G Period Variable No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Gaia DR2 TIC [mag] spT [d] contaminator Pulsators 1 2969201399574096128 708596809 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='30 A0IV/V no signal in individual pixels 2 3109409266919646976 168595004 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='70 A5 none 3 3115125211261708032 293137161 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='01 A0 none 4 3344114626761364224 437889214 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='02 B5 none 5 3396397877830881792 247513086 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='17 A0 none Table 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' NEH – 46 non-sdB classified variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Only the first five objects are listed while the full table can be found in the online materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' G Period Variable No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Gaia DR2 TIC [mag] spT [d] contaminator Pulsators 1 1625627602365544832 198209459 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='53 B2 none 2 1713032695100155904 298091568 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='00 B4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='1 none 3 1861191062326013696 1955410399 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='31 A0 none 11 μHz+harmonics: none;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' 4 2077737678383889408 270610177 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='77 Be 46 - 70 μHz: UCAC4 663-077912 (PulV) 5 2132171608553758336 279919275 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='46 B3V none MNRAS 000, 1–15 (2020) 12 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Sahoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' 3 UPDATED AMPLITUDE SPECTRA AND NEW PULSATING SDB STARS We took advantage of access to the 10 min FFI data collected during Years 3 and 4 to discover new variable sdB stars and to improve the amplitude spectra for known sdBV stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Not all variable stars listed in Papers I and II were subject to this updated analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Binaries would surely benefit from a three times better sampling, which would yield much better definition of eclipses and more precise orbital period estimation, but it would not affect their variability type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The contaminated stars, which turned out not to be sources of the signal, were also excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' We focused only on targets that show rich signal close to the 277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='7 μHz Nyquist frequency, which appear to be quite convincing pulsations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' In these cases shifting the Nyquist frequency to a three times higher value would uncover the entire g-mode region of possible pulsating sdB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' We ended up with a final list of 78 targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The light curve extraction process and data reduction of the 10 min FFI data, are the same as for the 30 min FFI data described in Papers I and II, where we refer the reader for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Out of 78 pre- selected targets we found 24 that show multiple frequencies, which we interpret as pulsations, and the number of frequencies were significantly increased or more frequencies beyond the 277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='7 μHz Nyquist frequency were detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' We list these targets in Table 15 in the online material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The targets are confirmed as the original sources of the detected signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' We show the amplitude spectra of these 24 targets in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' We marked the typical g-mode frequency range (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' 100 – 400 μHz) with dashed vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' This region overlaps with the typical p-mode frequency range of 𝛿 Scuti stars, so it is not generally straightforward to claim the detection of a pulsating sdB star based only on the amplitude spectrum content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Alternatively, if the frequencies detected are outside the indicated range, we might well doubt the detection of an sdB pulsator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' For instance, TIC 224284872 is an A star and the frequencies are outside the expected range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Similarly, TIC 181142865 turned out to be a main sequence star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' TIC 437889214 shows frequencies below the expected region – its spectral type is B5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' There are other targets which are not classified as hot subdwarfs, yet show frequencies in the range characteristic of pulsating sdB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' These cases may be either 𝛿 Scuti pulsators or misclassified hot subdwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' We confirm a detection of g-mode pulsations in 11 sdB pulsators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' TIC 442750342 seems to be an exception in our sample being a low gravity and cool sdB pulsator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' MNRAS 000, 1–15 (2020) TESS sdBVs in both ecliptic hemispheres 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 Amplitude [ppt] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='6 TIC 1955410399 [late B type] 0.' 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+page_content='5 TIC 437889214 [B5] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='6 TIC 247513086 [A0] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 TIC 201251043 [sdB] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='2 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 TIC 446919722 [sdB] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 TIC 1035773311 [sdB] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 TIC 388832080 [sdB] 0 80 160 240 320 400 480 560 640 720 800 Frequency [ Hz] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 TIC 22217594 [sdB] 0 80 160 240 320 400 480 560 640 720 800 Frequency [ Hz] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='0 TIC 224284872 [A0] Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The amplitude spectra of pulsators analyzed with 10 min FFI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Vertical dashed lines indicate the typical g-mode region of pulsating hot subdwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The horizontal lines mark the detection thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' MNRAS 000, 1–15 (2020) 14 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Sahoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Table 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Pulsators confirmed with the 10 min FFI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' We show amplitude spectra of these objects in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' This table is also included in the on-line materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' G Variable No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Gaia DR2 TIC [mag] contaminator Remarks 1 1861191062326013696 1955410399 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='66 none late B type 2 3032890473180888192 749940934 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='46 none late B type 3 3159937564294110080 262753627 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='46 none sdB 4 3344114626761364224 437889214 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='16 none B5 5 3396397877830881792 247513086 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='44 none A0 6 4923853724788504192 201251043 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='93 none sdB 7 5090382015016433920 686449995 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='99 none A5 8 5196271513123121152 323174439 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='30 none sdB 9 5250674622612902912 362098036 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='48 none early B type 10 5257747299878049152 442128473 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='84 none early B type 11 5266133451162548864 141684783 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='53 none sdB 12 5307946881949072000 442750342 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='82 none sdB 13 5326745919424790656 126486580 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='30 none late B type 14 5362558250096941056 178621334 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='33 none sdB 15 5429254969036524416 4595563 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='11 none early A type 16 5439887654492064256 25836205 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='14 none sdB 17 5525342630213336448 181142865 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='11 none late B or early A type 18 5622554881336253824 190720627 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='07 none B or B+A type 19 5796399012705196800 401975322 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='21 none B1 20 5823403087017696384 446919722 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='96 none sdB 21 5872410931625754624 1035773311 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='46 none sdB 22 5922070855307705472 388832080 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='58 none sdB 23 6143764182206682112 22217594 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='16 none sdB 24 6534581776366266752 224284872 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='53 none A0 4 NEW VARIABLES As a by-product of the contamination analysis we report the true sources of variability preliminarily assigned to the targets listed in Papers I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' In Tables 1-14 we provide a variable contamina- tor column which, in the case of a positive variability contamina- tion, contains a name of a contaminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' In addition, we detected new variables that do not contaminate our pre-selected targets but are located within the target masks of our targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' For a practical reason, all these variable contaminators and new non-contaminating variables are listed in Table 16 in the online materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' In total, we report detection of 682 new variable stars, including two, listed last in Table 16, that have no Gaia (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' 2018) designation yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' To be precise, the discovery of the variability of the majority of these stars was presented in Papers I and II, so only 97 stars are found to be new variables (accounting for Papers I and II) while the remaining 585 stars now have the variability properly assigned (as compared to Paper I and II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' MNRAS 000, 1–15 (2020) TESS sdBVs in both ecliptic hemispheres 15 Table 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Basic information of the 682 new variable stars found during the contamination analysis in both SEH and NEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Only first five objects are listed while the full table can be found in the online materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' G No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Gaia DR2 TIC Name [mag] 1 1000847845211000960 14196021 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='65 2 1030011910101662336 467154863 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='56 3 1082306439760980224 284473271 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='95 4 1113516077316307328 468921975 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='32 5 1131845245388039296 459183003 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='38 5 SUMMARY We presented the results of our contamination analysis of stars included in Papers I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' We identified 1 141 false positives, while 451 variables were not verified because, in most cases, the signal is of too low amplitude to be detected in individual pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The total number of targets, which are the sources of the signal we presented in Papers I and II is 721.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' As a by-product of our contamination analysis we found 97 new variables that happened to be within target masks of our original stars listed in Papers I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' In total, we analysed 2 995 targets in TESS fields, where 2313 targets were presented in Papers I and II and the remaining 682 variable targets were found during the contamination check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Out of the 2 313 targets, we confirmed 721 as variable after contamination analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Hence the total number of variable targets we found is 1 403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' We pre-selected 78 uncontaminated targets that are pulsator candidates (that is, they show rich pulsation content close to the 277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='7 μHz Nyquist frequency) for additional analysis using the 10 min FFI data collected during Years 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' We ended up with 24 targets for which those new data turned out to be beneficial – that is, more peaks either below, or especially beyond, the 277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='7 μHz frequency were detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' For any of these 24 stars without spectral type, we used publicly available data and/or spectroscopic data col- lected with the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='9 m telescope at the South African Astronomical Observatory to identify hot subdwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' In total, we found 11 new sdB pulsators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Details of the spectroscopic analysis will be provided in Worters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' (in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' One of the pulsator candidates, TIC 362098036, was a subject of a pulsation mode identification and the result was reported in Paper 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Our analysis confirmed that the target is a main sequence B star, which makes the mode identification irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' ACKNOWLEDGEMENTS Financial support from the National Science Center in Poland under projects No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' UMO-2017/26/E/ST9/00703 and UMO- 2017/25/B/ST9/02218 is acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' PN acknowledges sup- port from the Grant Agency of the Czech Republic (GAČR 22- 34467S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' The Astronomical Institute in Ondřejov is supported by the project RVO:67985815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' This paper includes data collected by the TESS mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Funding for the TESS mission is provided by the NASA Explorer Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' This work has made use of data from the European Space Agency (ESA) mission Gaia (https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='int/gaia), processed by the Gaia Data Pro- cessing and Analysis Consortium (DPAC, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='int/web/gaia/dpac/consortium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' Funding for the DPAC has been provided by national institutions, in particular, the institu- tions participating in the Gaia multilateral agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' This research has used the services of www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='Astroserver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' DATA AVAILABILITY The datasets were derived from MAST in the public domain archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' REFERENCES Baran A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=', Sahoo S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAzT4oBgHgl3EQfJfvP/content/2301.01082v1.pdf'} +page_content=', Sanjayan S.' metadata={'source': 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b/TtE0T4oBgHgl3EQfVAA3/content/tmp_files/2301.02257v1.pdf.txt @@ -0,0 +1,4581 @@ +Draft version January 9, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +GalCEM I – An Open-Source Detailed Isotopic Chemical Evolution Code ∗ +Eda Gjergo +,1, 2, 3 Aleksei G. Sorokin +,4 Anthony Ruth +,5 Emanuele Spitoni +,6 +Francesca Matteucci +,7, 8, 9, 10 Xilong Fan +,1 Jinning Liang +,1 Marco Limongi +,11, 12 Yuta Yamazaki +,13, 14 +Motohiko Kusakabe +,15, 13 and Toshitaka Kajino +15, 13, 14 +1School of Physics and Technology, Wuhan University, Wuhan 430072, China. +2School of Astronomy and Space Science, Nanjing University, Nanjing 210093, People’s Republic of China +3Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), +Ministry of Education, Nanjing 210093, People’s Republic of China +4Department of Applied Mathematics, Illinois Institute of Technology, RE 220, 10 W. 32nd St., Chicago IL 60616, USA +5Cubic PV, 1807 Ross Ave, Ste 333 Dallas, Texas 75201, USA +6Universit´e Cˆote d’Azur, Observatoire de la Cˆote d’Azur, CNRS, Laboratoire Lagrange, +Bd de l’Observatoire, CS 34229, 06304 Nice cedex 4, France +7Dipartimento di Fisica, Sezione di Astronomia, Universit`a di Trieste, Via G. B. Tiepolo 11, I-34143 Trieste, Italy +8INAF, Osservatorio Astronomico di Trieste, Via Tiepolo 11, I-34143 Trieste, Italy +9INFN, Sezione di Trieste, Via A. Valerio 2, I-34127 Trieste, Italy +10IFPU—Institute for the Fundamental Physics of the Universe, Via Beirut, 2, I-34151 Trieste, Italy +11Istituto Nazionale di Astrofisica - Osservatorio Astronomico di Roma, Via Frascati 33, I-00040, Monteporzio Catone, Italy +12Kavli Institute for the Physics and Mathematics of the Universe, Todai Institutes for Advanced Study, the University of Tokyo, +Kashiwa, Japan 277-8583 (Kavli IPMU, WPI) +13National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan +14Graduate School of Science, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 11-0033, Japan +15School of Physics, and International Research Center for Big-Bang Cosmology and Element Genesis, Beihang University, Beijing +100083, P. R. China +(Received January 9, 2023; Revised; Accepted) +Submitted to ApJS +ABSTRACT +This is the first of a series of papers that will introduce a user-friendly, detailed, and modular +GALactic Chemical Evolution Model, GalCEM, that tracks isotope masses as a function of time in a +given galaxy. The list of tracked isotopes automatically adapts to the complete set provided by the +input yields. The present iteration of GalCEM tracks 86 elements broken down in 451 isotopes. The +prescription includes massive stars, low-to-intermediate mass stars, and Type Ia supernovae as enrich- +ment channels. We have developed a preprocessing tool that extracts multi-dimensional interpolation +curves from the input yield tables. These interpolation curves improve the computation speeds of the +full convolution integrals, which are computed for each isotope and for each enrichment channel. We +map the integrand quantities onto consistent array grids in order to perform the numerical integration +at each time step. The differential equation is solved with a fourth-order Runge-Kutta method. +We constrain our analysis to the evolution of all the light and intermediate elements from carbon +to zinc, and lithium. Our results are consistent up to the extremely metal poor regime with Galactic +abundances. We provide tools to track the mass rate change of individual isotopes on a typical spiral +galaxy with a final baryonic mass of 5 × 1010 M⊙. Future iterations of the work will extend to the full +periodic table by including the enrichment from neutron-capture channels as well as spatially-dependent +treatments of galaxy properties. GalCEM is publicly available at https://github.com/egjergo/GalCEM. +eda.gjergo@gmail.com +∗ Released on MM, DDth, YYYY +arXiv:2301.02257v1 [astro-ph.GA] 5 Jan 2023 + +ID2 +Gjergo et al. +Keywords: Publicly available software(1864) — Galaxy chemical evolution(580) — Stellar nucleosyn- +thesis(1616) — Chemical enrichment(225) +1. INTRODUCTION +Galactic Chemical Evolution (GCE) is the field that +investigates how the chemical makeup of a galaxy +changes with time. +To do that, GCE must first con- +sider the astrophysical events where each isotope may +be synthesized. These events are point-like sources of +enrichment, which occur with varying rates throughout +the galaxy and across time. To model how such events +are distributed in a variety of galactic morphologies, the +field of GCE has developed rate equations which track +the abundance of chemical species. +A galactic mor- +phology is reconstructed by means of stellar birthrate +functions. +Stellar lifetimes will then determine when +the chemical enrichment will contaminate the galactic +media. Hereafter we will refer to“astrophysical events” +and“enrichment channels” interchangeably (for a recent +review, see Matteucci 2021). +The theory of GCE dates its origins to the mid 1950’s +with the pioneering works of Salpeter (1955, 1959), and +Schmidt (1959), who laid the foundations to the formal- +ism best explained in Schmidt (1963) and subsequently +in Tinsley (1980). +A schematic representation of the +formalism can be seen in Fig. 1 as it applies to one- +zone models (e.g., Talbot & Arnett 1971; Timmes et al. +1995), i.e., the treatment of galaxies as homogeneous +environments where the gas is mixed instantaneously. +In order to formulate the luminosity function evolu- +tion of main-sequence stars, Salpeter (1955) proposed +the existence of an “original mass function”, now called +initial mass function (IMF) – or, the distribution, with +respect to stellar mass M∗, of stars at their birth in a +single stellar population. This IMF distribution is color- +coded in Fig. 1 according to main-sequence colors. The +IMF is one of the two components of the birthrate func- +tion B(M∗, t), the other being the star formation rate +(SFR, +Salpeter 1959; Schmidt 1959). +The ansatz is +that the birthrate function is separable with respect to +stellar mass and time and is represented by the product +of SFR and IMF. This is a simplistic ansatz, already +challenged by numerous works (Yan et al. 2017, 2021), +but it nonetheless proved to be valuable in GCE. +We must also stress the difference between backward +and forward approaches. Cosmological simulations gen- +erally follow a forward approach, wherein stellar birth +is tracked first, then the chemical enrichment is pro- +jected onto future time steps. A classic GCE approach, +instead, follows a backward approach, namely it re- +constructs past stellar distributions by tracking stellar +death rates at every time-step (Matteucci 2012). +The interstellar medium (ISM) will generally be en- +riched by the nucleosynthesis products (yields) gener- +ated in individual astrophysical sites – be them super- +novae (SNe), asymptotic giant branch (AGB) winds, or +coalescence events, to name a few. GCE must integrate +the occurrence of such events over the lifetime of the +galaxy. To do that, GCE must couple the birthrate func- +tion with stellar lifetimes to extrapolate a “death-rate +function” and hence learn when the enrichment events +will occur. These, coupled with the yields, define how +the abundance of each isotope evolves with time. +In +Fig. 1, at each time step tn, GCE models reconstruct +the birth time and properties of all stars that die within +that time step and compute their integrated contribu- +tion to the enrichment from each astrophysical site. A +first exhaustive review connecting nuclear physics to as- +trophysical sites undergoing nucleosynthesis is the cor- +nerstone work by Burbidge et al. (1957), famously re- +ferred to as B2FH. +Over the decades, chemical evolution has provided +useful insights on galaxy properties. For example, Eggen +et al. (1962) first discovered that halo stars are old and +metal poor while disk stars are young and metal-rich +– hence leading to the understanding that disk stars +formed along with gas infall – and that such stars offer +a snapshot to the chemical composition of their parent +star-forming cloud. +Based on the cornerstone work by Salpeter (1955), +Salpeter (1959) and Schmidt (1959) independently de- +veloped a very similar GCE formalism that laid the foun- +dation for all subsequent works (reviews throughout the +years include Tinsley 1980; Prantzos 2008; Pagel 2009; +Matteucci 2012, 2021). In particular, Talbot & Arnett +(1971) developed a numerical solution to the full con- +volution equations in a work that did not require the +metallicity to grow monotonically. +Pagel & Patchett +(1975) developed a modified GCE model based on the +simple model which included prompt initial enrichment, +the early version of metal-enhanced star formation, and +inhomogeneous collapse and infall. Chiosi (1980) pro- +posed an open model with gas infall and outflow. Chiosi +& Matteucci (1980) and later also Portinari & Chiosi +(2000) developed a radially-dependent disk GCE model, +which was later improved and expanded in Spitoni & +Matteucci (2011) where they found that an inside-out +formation scenario (and/or a variable flow) in spiral + +GalCEM method presentation +3 +Figure 1. Diagram of the GCE rationale. At each time step, the SFR, ψ(t), determines how much total gas mass is converted +into stellar mass. The distribution of stellar masses M∗ follows the given IMF, φ(M∗). At each subsequent time step, the GCE +convolution integral determines how many stars die at tn, and therefore how much enrichment these stars return to the gas +mass. The stellar birth-time t′, the baby blue and magenta grids, and lifetimes τ(M∗, Z∗), i.e. the curves connecting birth-time +with tn, are updated at each time step. Each enrichment channel will cover a mass/lifetime range specific to its process (SNCC +– or equivalently for our purposes, SNII – and LIMs are included in the graphic). Onto these enrichment channel grids we +compute the integrals, as explained in Fig. 5. +disks is necessary to reproduce Galactic abundance gra- +dients. The effect of stellar migration in the Milky way +was investigated by Sch¨onrich & Binney (2009); Spitoni +et al. (2015) and Johnson et al. (2021). Gas radial flow +as well as stellar mixing were considered in a multi- +zone GCE model in Chen et al. (2022). +A Bayesian +approach was undertaken in Cˆot´e et al. (2017); Rybizki +et al. (2017) and Spitoni et al. (2020), this last work in +particular applied Markov chain Monte Carlo methods +to a two-infall Milky Way formation scenario. +Spitoni et al. (2017) proposed new analytical solutions +to a GCE model that describes SDSS galaxies only as +a function of infall timescales, infall masses, and mass +loading factors. +A handful of GCE models have also been released as +public codes. Andrews et al. (2017) presented flexCE, +a flexible one-zone chemical evolution code which was +recently used in Hasselquist et al. (2021). Cˆot´e et al. +(2017) presented the One-zone Model for the Evolu- +tion of GAlaxies (OMEGA) code within the NuGrid +Python Chemical Evolution Environment (NuPyCEE). +OMEGA is paired with the Stellar Yields for Galac- +tic Modeling Applications (SYGMA, Ritter et al. 2018) +which computes the ejecta of single stellar populations +(SSP) to reconstruct the enrichment of a galaxy. Yan +et al. (2017) proposed a one-zone model (GalIMF) in +which they adopted a variable integrated galactic ini- +tial mass function (IGIMF). Rybizki et al. (2017) devel- +oped Chempy, which parametrizes open one-zone mod- +els within a Bayesian framework. And more recently, +Johnson & Weinberg (2020) developed the Versatile In- +tegrator for Chemical Evolution (VICE), an efficient and +user-friendly code that shortens computation times by + +SFR(t) +(stellar birth) + (stellar death) +galacticCEM@gmail.com +IMF +(Galaxy age) +SNil enrichment +(stellar birth-time) +t(M) = t - t' (stellar lifetime) +LIMs enrichment +(stellar +birth-time) +t(M) = t - t' +(stellar lifetime)4 +Gjergo et al. +Symbol +Description +i +index of the tracked isotope (the chemical species) +k +mass grid integration index +n +time step index +P +the label that identifies an astrophysical process +that is a channel of chemical enrichment +tG +final age of the galaxy +t +age vector of the galaxy +t′(M∗, Z∗) +birth time of a star of mass M∗ +τ(M∗, Z∗) = t − t′(M∗, Z∗) +lifetime of a star of mass M∗ and metallicity Z∗ +M∗ +mass of a single star +MP,l, MP,u +lower and upper mass limits for stars in the integrals +(mass limits are process-specific) +Minf(t) +baryonic mass of the galaxy as a function of time +(determined by the infall rate) +Mgas(t) +gas mass of the galaxy as a function of time +Mi,gas(t) +gas mass of the isotope i in the galaxy as a function of time +s.t. � +i Mi,gas(t) = Mgas(t) +M∗,tot(t) +star mass of the galaxy as a function of time +Minf(t) = Mgas(t) + M∗,tot(t) +ν +star formation efficiency +ψ(t) +star formation rate (SFR) equivalent to +˙M∗,tot(t) +φ(M∗) +initial mass function (IMF) +Z(t) +the metallicity, or the mass fraction of all of the chemical +elements with the exclusion of H and He +YP (i, M∗, Z∗) +Yields: tabulations of the mass fractions Mi/M∗,R, +where M∗,R is the total mass returned to +the interstellar medium by a star of mass M∗ +and of initial metallicity Z∗, for the astrophysical process P +Table 1. Glossary of GCE Symbols and Quantities of Interest, as They Have Been Used throughout the Present Article. +applying IMF-averaged yields to the SN enrichment by +core collapse SNe. +There has been a fortunate surge in high precision +surveys and instruments — some concluded very re- +cently, some ongoing still — that are providing unprece- +dented levels of precision to constrain GCE. These in- +clude LAMOST (Large Sky Area Multi-Object Fiber +Spectroscopic Telescope, Cui et al. 2012; Deng et al. +2012; Zhao et al. 2012), Gaia-ESO (Gilmore et al. 2012; +Randich et al. 2022), Gaia DR3 (Recio-Blanco et al. +2022) whose α-element abundances have already been +investigated with the two-infall model (Spitoni et al. +2022), RAVE (Kordopatis et al. 2013; Steinmetz et al. +2020), APOGEE (Majewski et al. 2017; Ahumada et al. +2020; Abdurro’uf et al. 2022) with spectra in the IR, +SDSS with photometric optical-nearIR bands, GALAH +(De Silva et al. 2015; Buder et al. 2021) with the goal of +obtaining abundances for 30 elements within the Galaxy, +Vista Variables in the Via Lactea (VVV Minniti et al. +2010) with a focus on the Galactic Bulge, MUSE (Bacon +et al. 2010) with a highly resolved integrated field spec- +troscopy unit and a broader scope out to high redshifts, +4MOST (de Jong et al. 2019; Walcher et al. 2019) which +will measure spectroscopic redshifts of X-ray-identified +groups and galaxy clusters, and other upcoming instru- +ments such as JWST (Gardner et al. 2006). Notably, the +Gaia-ESO mission was devised with the purpose of pre- +cisely mapping the 3D locations and motions of billions + +GalCEM method presentation +5 +of stars within our galaxy; the low-resolution LAMOST +spectrograph has been able to collect simultaneously up +to 4000 spectra over huge volumes of the Galaxy, with +both a large aperture and a large field of view. The anal- +ysis of these data which in most cases is coupled with +stellar dynamics make it so that this 70-year-old field +is mature enough to properly constrain multi-zone sta- +tistical analysis and provide further information on the +evolutionary history of the galaxy. +Before investigat- +ing such pursuits, we lay our foundations by reaching a +benchmark with previous studies by means of one-zone +modeling. +With GalCEM we offer a public code that adapts to +the complete nuclide list of the chosen yields. GalCEM +can solve the GCE integrodifferential equation including +infall, outflow, SFR and fully convolved returned ejecta +across the whole main-sequence. By default, low-mass +stars, massive stars, and Type Ia supernova yields are +always included. In this work we limit our analysis to +these three enrichment channels only. +The article is structured as follows: in Section 2 we +introduce the GCE formalism and we present our nu- +merical solution to our adopted general GCE equations. +In Section 3 present some preliminary results, as well as +the data products available at this stage in GalCEM, and +we evaluate aspects of the code performance. Finally in +Section 4 we summarize the article and we discuss fu- +ture aims and scientific goals we expect to achieve with +this tool. +2. METHOD +In this section we present both the theoretical frame- +work on which we base our computation and the nu- +merical solution we developed. At present, GalCEM is +available as a one-zone code, i.e., we adopt an instan- +taneous mixing approximation and the whole galaxy is +treated as a single homogeneous zone. The solution has +been implemented in the Python code GalCEM, publicly +available on GitHub1. +We follow the conventional formalism, so the general +GCE equation can be succinctly expressed as (Pagel +2009): +˙Mi,gas(t) = I(t) − Xi,gasψ(t) + +� +P +RP,i(t) − O(t), (1) +where +˙Mi,gas(t) is the mass rate of change of the gas- +phase component of a chemical species or isotope i, +I(t) represents the infall term while O(t) is the outflow +term. Xi,gasψ(t) is the mass fraction of the i-th isotope +1 https://github.com/egjergo/GalCEM/galcem +(Xi,gas = Mi,gas/Mtot,gas) that is subtracted from the to- +tal gas component due to star formation. ψ(t) refers to +the SFR, i.e., of the total gas mass that forms stars at a +given time step. RP,i(t) is the rate of the i-th component +returned to the gas phase by each enrichment channel +P. At present, such enrichment channels include the fi- +nal stages of low-to-intermediate mass stars (LIMs) and +their enrichment of the ISM through asymptotic giant +branch (AGB) winds, the death of massive stars as core +collapse supernovae (SNCC), and Type Ia supernovae +(SNIa). In Table 1 is the synopsis all of the main sym- +bols involved in the GCE formalism. The symbols which +have not been introduced in the present paragraph will +be explained in due course within this section2. +The galaxy begins by having no mass. The only source +of mass growth is provided by the infall term that ac- +cretes gas from a primordial (or otherwise very metal +poor) intergalactic medium3. +We conform our solar +metallicity to the value derived in Asplund et al. (2009) +of Z⊙ = 0.0134. An environment, star, or system is de- +fined as very metal poor whenever < 10−2Z⊙, and ex- +tremely metal poor for < 10−3Z⊙ (e.g., Nomoto et al. +2013). +The presence of gas triggers star formation, described +by the SFR. We take the SFR to be a function of the +total galaxy mass and the total gas mass at any given +time. We assume that the mass distribution of stars is +described at every time step by the IMF. The mass of +newly synthesized isotopes at any time will be given by +the convolution of the SFR with the IMF and yields, as +described by Eq. 18, which is the full integrodifferen- +tial version of Eq. 1 and the equation being solved by +GalCEM. The majority of the calibrations in this section +are taken from Molero et al. (2021a,b). In the rest of +this Methods section we will provide detailed informa- +tion about every necessary GCE ingredient. +2.1. General workflow in GalCEM +Fig. 2 represents the GalCEM flowchart. Its design fol- +lows a simple principle, namely that GCE considers in- +dividual nucleosynthetic enrichment channels and inte- +2 The present article follows the common square bracket notation +of the logarithmic abundance: for an element or isotope A, the +abundance relative to another element B will be: +[A/B] = log (MA/MB) − log (MA/MB)⊙ += log (µAXA/µBXB) − log +� +µAXA,⊙/µBXB,⊙ +� += log (XA/XB) − log (XA/XB)⊙ , +(2) +where µ is the atomic weight of a chemical species and ⊙ marks +solar values. +3 Here we follow the gas-phase definition of metallicity Z(t) = +(MZ(t) = Mgas(t) − MH(t) − MHe(t))/Mgas(t)., where “metals” +are all elements aside from H and He + +6 +Gjergo et al. +Figure 2. GalCEM flowchart. A description of the figure is provided in Section 2.1. +grates the occurrence of each event across both time and +across the distribution of such events in a galaxy, as con- +structed through the combination of birth-rate functions +and stellar lifetimes. Each event associated with each +enrichment channel is represented by individual points +for each isotope in yield tabulations. These point sources +of enrichment are treated in the yield class. Birth rate +and stellar lifetimes are instead treated in the morphol- +ogy class. While GalCEM follows well-established liter- +ature prescriptions on the GCE theory, the design and +numerical solution to the integrodifferential equation is +original to this work. +GalCEM contains a preprocessing yield tool, and re- +quires the input from three classes at setup: inputs.py, +yields.py, and morphology.py. +Regarding the prepro- +cessing yield tool, it extracts the interpolation surfaces +for each yield tabulation selected for a run. +morphology.py contains the ingredients that charac- +terize the galaxy properties (namely, IMF, SFR, infall, +and stellar lifetimes). yields.py imports the necessary +yield properties, including the preprocessed interpola- +tions, and provides the tools to combine them within +the setup class in Main. +yields.py also contains the solar normalization class, +and the class that extracts a combined list of unique +isotopes from all of the selected yields. +The One- +Zone class in Main opens and writes the outputs and +runs the evolve function, which computes the time- +step-dependent GCE quantities. +The global quanti- +ties (namely, total gas and total stellar mass) are +computed by the total evolution function, +while +the isotope-dependent quantities are computed by +isotopes evolution. +Light-gray font colors repre- +sent GalCEM features that will be presented in the near +future + +pre-processing +Jets +Dynamic Symmetric +Dynamic Asymmetric +BBN +Massive stars +LIMs +SNla +NSM +MHDJ +Collapsars +AGNS +Handle Isotopes +Solarnormalization +Yields +Dust +Yield classes +parameter values +options +custom functions +inputs +auxiliary functions +Cosmology,hierarchy +multi-zones +Stellar lifetimes +Infall +IMF +SFR +input class +Morphology classes +Setup +OneZone +output +Maininstantiation +Evolve +inputsetup +total quantities +isotopegquantities +total evolution +Plots +integration grid +Integration classes +isotopes evolution +integrals computation +observational databaseGalCEM method presentation +7 +GalCEM runs on Python3. After importing the pack- +age, an inputs object must be defined. The inputs object +is customizable4. A minimum working example has the +following form5: +import galcem as gc +inputs = gc . Inputs () +oz = gc . OneZone( inputs , outdir=MYDIR) +oz . main () +By defining the oz object, the user initializes the setup, +including a series of properties like the time vector, the +total gas mass, and the complete list of isotopes gener- +ated from the chosen yields. The actual run is launched +by calling the main function onto the oz object. +The parameters of the simulation can either be set +within the inputs.py parameters or they can be edited +manually in a script before executing a run. +For ex- +ample, the size of the time step, in Gyr, can be easily +changed with the following: +inputs . ntime +step = .2 +GalCEM has been published as a Python Package In- +dex Project, and for pip users can be installed with the +terminal or conda command: +pip +i n s t a l l +galcem +Alternatively, the interested user who prefers to work +on the cloud can request a JupyterHub account.6 +2.2. Infall rate and Outflows +Infall as well as outflow rates play an essential role in +GCE models, in that they can characterize the dynamics +of galaxy formation (e.g., see Pagel 2009). +Concerning +infall rates, declining models are favored because they +predict metallicity distributions more closely consistent +with those of G-dwarf Milky Way disk stars (Larson +1974, 1976; Matteucci 1996). +Spitoni et al. (2021) recently investigated the assump- +tion of a declining infall, as well as the impact of inflow +and outflow on the cumulative metallicity distribution +of galaxies as a function of their stellar mass. In this +work we therefore consider a simple single exponential +infall rate described by: +I(t) = ˙Minf(t) = I0e−t/τinf , +(3) +where +˙Minf(t) = dMinf(t)/dt is the rate of gas mass +from the extragalactic medium falling into the gravita- +4 The full list of input parameters is in https://github.com/ +egjergo/GalCEM/blob/main/galcem/classes/inputs.py +5 The minimum working example script is located in https:// +github.com/egjergo/GalCEM/blob/main/examples/mwe.py +6 Available at https://galcem.space/. +tional potential of a galaxy with time. We assume that +only gas falls into a galaxy’s potential. In this prelim- +inary work, we ignore the outflow term in Eq. 1. This +simplification is justified by the results of Spitoni et al. +(2009), where they found that the outflow timescales as +inferred from the orbit times of clouds subject to the +Galactic gravitational potential should be of the order +of 0.1 Gyr. The impact of such outflow timescales on +GCE models is negligible. +The infall timescale τinf varies depending on the for- +mation history of a galaxy. +Shorter timescales imply +more rapid star formation histories and are associated +with elliptical galaxies (Pipino & Matteucci 2004), while +more relaxed timescales are associated with spiral or +dwarf galaxies. In Molero et al. (2021a), the timescale +that reproduces the Milky Way thin disk is fine-tuned +to 7 Gyr, while 0.2 Gyr is suitable for elliptical galaxies +(or 0.05 Gyr for some dwarf galaxies in +Molero et al. +2021b). +I0 has units of [M⊙ / Gyr], with M⊙ being the +solar mass. It is normalized so that: +� tG +0 +� +˙Minf(t) − ˙Mout(t) +� +dt = Mtot, +(4) +where tG and Mtot (e.g., Minf(tG) = Mtot) represent the +present-day age and total baryonic mass of the galaxy, +respectively. Given that we do not treat dynamics, grav- +itational components including dark matter are beyond +the scope of this work. +Similarly, in the specific case of no outflow, the total +baryonic mass of a galaxy as a function of time is given +by: +Minf(t) = Mtot(t) = +� t +0 +˙Minf(t)dt. +(5) +Infall rate in GalCEM — GalCEM computes the infall mass +Minf(t) as well as the total mass Mtot(t) at the setup +stage when the one-zone object oz is defined. We take +the final total baryonic mass to be Mtot(tG) = 5 × 1010 +M⊙, with tG = 13.7 Gyr being the present age of the +Galaxy, and the infall timescale τinf = 7 Gyr. +Outflows —Galactic outflows are an integral component +galaxy evolution modeling. +They have been invoked +since the late 70s (e.g., De Young 1978) as the source +of metal-enrichment in the intracluster medium. +Re- +cent Chandra X-ray observations of M82 outflows sug- +gest that the hot halo gas is being mass-loaded with +cold-phase material (Lopez et al. 2020). This material +can be very metal-rich, like in the case of the edge-on +star-forming galaxy Mrk 1485, whose outflows are about +1.6 times the ISM metallicity (Cameron et al. 2021). +How these outflows impact the chemical evolution of a +galaxy is not yet a settled issue. A number of papers + +8 +Gjergo et al. +which do incorporate outflows find that it significantly +impacts the evolution of chemical abundances (e.g., An- +drews et al. 2017; Weinberg et al. 2017; Trueman et al. +2022). +However, studies on Galactic fountains triggered by +core-collapse SNae associated with the solar annulus Me- +lioli et al. (2008); Spitoni et al. (2008) find that that out- +flow ejecta should fall back into the close proximity to +the same Galactocentric regions. Furthermore, the typ- +ical fall-back delay of such clouds is found to be ∼ 0.1 +Gyr (Spitoni et al. 2009). Despite the dynamical com- +plexity of such phenomena, their impact on the chemi- +cal evolution appears to be negligible. In fact, a number +of GCE models which assume no outflow are able to +reproduce observations (e.g., Minchev et al. 2013; Ro- +mano et al. 2019). But the issue is further complicated +by another consideration: if the gas is ejected onto the +circumgalactic medium, semi-analytic models find that +this gas will become available again for cooling on much +longer timescales, of the order of several Gyr (Faerman +et al. 2022). +Outflows in GalCEM —To set a benchmark for com- +parison with other works, in this first paper we set +the outflow to zero, but the user can easily edit +inputs.wind efficiency to implement a dimension- +less parameter 0 < ω < 1 so that the outflow may +be proportional to the SFR, O(t) = ωψ(t). +The +proportionality of the outflow to the SFR follows +Bradamante et al. (1998), where the galactic winds orig- +inate whenever the thermal energy of the gas exceeds +its binding energy. +Alternatively, the user can com- +ment out the self.wind efficiency = 0 override in +classes/inputs.py to restore the default parameter for +the given morphology. +2.3. Initial Mass Function (IMF) +The initial mass function, IMF, is the number distri- +bution of stars in a galaxy as a function of stellar mass. +Its original single-power-law formulation dates back to +Salpeter (1955): +φ(M∗) = dN∗/dM∗ = φ0(M∗/M⊙)−x, +(6) +where M∗ is the stellar mass of individual stars, normal- +ized to the solar mass M⊙, and the best fit for the power +law index was found to be x = 2.35. The IMF is one +component of the birthrate function, the other being the +SFR. Given that we already express the SFR in units of +solar masses over time, we normalize the mass-weighted +IMF to 1: +� Mu +Ml M∗φ(M∗)dM∗ = 1. +A more appropriate IMF which is representative of +the stellar distribution in the Milky Way is the universal +IMF (Kroupa 2001): +φ(M∗) = +� +� +� +� +� +2φ0M −α1 +∗ +, +Ml ≤ M∗/M⊙ < 0.50 , +φ0M −α2 +∗ +, +0.50 ≤ M∗/M⊙ < 1.00 , +φ0M −α3 +∗ +, +1.00 ≤ M∗/M⊙ < Mu . +(7) +This IMF is now commonly referred to as canonical IMF, +and it is a piecewise power law that accounts for the fact +that the low-mass end of the stellar mass distribution +is not as steep as the high-mass end. For a historical +overview of the field, see Kroupa et al. (2013). +IMF in GalCEM —For the present paper we adopted +the Kroupa (2001) canonical IMF, with α1 = 1.3 and +α2 = α3 = 2.3, according to the canonical values. In +Eq. +7 we left the break at M∗/M⊙> 1 because it is +representative of GalCEM’s implementation, i.e., the user +may readily explore top-light or top-heavy IMFs. We +chose Ml = 0.08 to Mu = 120 M⊙to be the span of the +mass limits for the normalization, in accordance with the +upper mass limit of our chosen SNCC yields (Limongi & +Chieffi 2018). The Salpeter IMF is also an available op- +tion (inputs.IMF option = ’Salpeter55’). +GalCEM +can take custom IMF laws, as long as they are only de- +pendent on the stellar mass. Variable IMFs that depend +e.g. on the metallicity and SFR, such as the integrated +galaxy-wide IMF (IGIMF Yan et al. 2017; Jeˇr´abkov´a +et al. 2018; Yan et al. 2021), will be investigated in the +near future. +2.4. The Star Formation Rate (SFR) +One of the prevalent SFR prescriptions, available by +default in GalCEM, is: +ψ(t) = ˙M∗,tot(t) = ν +�Mgas(t) +Mtot(t) +�κ +, +(8) +is a Kennicutt-Schmidt definition (Kennicutt 1998) of +the SFR, where ˙M∗,tot(t) is the rate of total stellar mass +formed as a function of time, Mgas(t) is the gas mass +at time t, and Mtot(t) is the total baryonic mass fallen +within the galaxy by time t. Mtot(t) is initially added to +the gas component. This gas mass will be partly com- +posed of the pristine infall gas mass, but it will also be +enriched over time by the returned ejecta of the astro- +physical sites P. This returned mass is computed by the +convolution integrals shown in Eq. 19. +The star formation efficiency (SFE, ν) has units of +[Gyr−1] for a SFR defined in terms of surface densities, +while κ is a dimensionless parameter that varies depend- +ing on the morphology. κ normally ranges between 1 and +2. +The best fit found in Kennicutt (1998) is 1.4. +This single SFR power law is however a simplifica- +tion. Kennicutt & De Los Reyes (2021) find breaks in + +GalCEM method presentation +9 +both the power-law index and zero-points of the star for- +mation law between starbursting and non-starbursting +galaxies. Ellison et al. (2021) demonstrate that there +is significant galaxy-to-galaxy variations in this trend, +suggesting that a single relation is not universally appli- +cable. Much of the debate on this topic can be traced +back to the uncertainties in the CO-to-H2 conversion +factor (e.g., Kennicutt & Evans 2012; Liu et al. 2015). +de los Reyes & Kennicutt (2019) argue that this is still +a viable prescription for GCE models, so it is reason- +able to use the single power-law. Nonetheless, there is +sufficient reason to expect the nuances of the star for- +mation law may turn out to be important for chemical +evolution. +The SFR prescription adopted in GalCEM —We follow the +SFR prescription as in Portinari et al. (1998): +ψ(t) = ν +�σ(tG) +σ(t) +�κ−1 +Gκ(t), +(9) +where G(t) = σg(t)/σ(tG) is the normalized surface gas +density: +ψ(t) = ν +�σ(tG) +σ(t) +�κ−1 � σg(t) +σ(tG) +�κ += ν +[σg(t)]κ +[σ(t)]κ−1 σ(tG) += ν +[Mgas(t)/V (t)]κ +[Mtot(t)/V (t)]κ−1 (Mtot(tG)/V (tG)) +, +(10) +where V (t) is the surface area. +Mtot(tG) = Mfinal is +the final baryonic galaxy mass. Given that the effective +radius of the galaxy does not change with time, V (tG) = +V (0) = V (t) = const. +Let us define the gas fraction +fg = σg(t)/σ(t), so that: +ψ(t) = +ν +Mfinal +M κ +gas(t) +M κ−1 +tot (t) ≡ +ν +Mfinal +�Mgas(t) +Mtot(t) +�κ−1 +Mgas(t) += νf κ−1 +g +(t)Mgas(t) +Mfinal +, +(11) +this latter expression being the equation implemented +in the code. Notice that for κ = 1, the SFR is simply +proportional to Mgas(t). The results in this article are +presented assuming that κ = 1. +2.5. Stellar lifetimes +Stellar lifetimes is the crucial component that brings +together SFR and IMF. Specifically, lifetimes provide +the means of connecting stellar masses with time, and +they make the integration component of GCE equations +possible. +100 +101 +102 +10 +3 +10 +1 +101 +(M * , Z) +9M +6M +3M + + +Z = 0.0004 +Z = 0.004 +Z = 0.008 +Z = 0.02 +Z = 0.05 +100 +101 +102 +Mass +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +(X)/ (Z = 0.0004) +9M +6M +3M + +Z = 0.004 +Z = 0.008 +Z = 0.02 +Z = 0.05 +7.0 +7.5 +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +(M * ) +Figure 3. The adopted (Portinari et al. 1998) metallicity- +dependent stellar lifetimes, τ(M∗, Z∗), as a function of stellar +mass (top panel) and the ratio of the top 4 metallicity bins +normalized by the lowest one (Z = 0.0004, bottom panel). +The 4 next metallicity bins on the bottom panel are 0.004, +0.008, 0.02, and 0.05 (represented in dotted, dashed, and +solid blue lines, respectively). The data points of the metal- +licity tabulation are color-coded by the actual stellar lifetime +as shown in the color bar (where the units are in log10 yr). +In both the top and bottom panels, three arbitrary stellar +masses are highlighted with red vertical lines (3, 6, and 9 +M⊙, solid, dashed, and dotted, respectively). The top-left +panels in Fig. 7 represents this same color-coded grid points +in a full 3D graphic. +Stellar lifetimes in GalCEM —In the present version of +the code, we adopt the metallicity-dependent lifetimes +τ(M∗, Z) from Portinari et al. (1998). The lifetimes are +reported on the top panel of Fig. 3, where the stellar +mass is on the x-axis, the lifetime on the y-axis, and the +various curves display varying metallicities. The three +red vertical lines aid with tracking 3 reference masses +(9 to 3 M⊙) with lifetimes ranging from ∼ 3 × 107 and +3 × 108 yr. +The lifetimes in Portinari et al. (1998) account for +the H- and He-burning timescales and are computed +with stellar evolution models in the Padua library (Bres- +san et al. 1993; Fagotto et al. 1994a,b). +The compu- +tation of these lifetimes also follow the instantaneous +mixing approximation, a common assumption in one- +zone GCE models. Other burning stages would anyway +be shorter than the resolution of these lifetimes (e.g., +Limongi 2017) In the bottom panel of Fig. 3 we show +the linear ratio between the three lifetimes computed +on larger metallicities divided by the lowest metallicity +(Z = 0.0004) lifetime, to better highlight trends. For + +10 +Gjergo et al. +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +Atomic Mass A +0 +20 +40 +60 +80 +Proton (Atomic) Number Z +Tracked isotopes +0 +20 +40 +60 +80 +100 +Isotope + abundance % +Atomic Mass A +0 +50 100 +150 +200 +Atomic Number Z +0 +20 +40 +60 +80 + isotopic % +0 +20 +40 +60 +80 +100 +Figure 4. All the isotopes (451) tracked in the present run, including the ones coming from LIMs (Cristallo et al. 2015), massive +stars dying as core-collapse SNe (Limongi & Chieffi 2018), Type Ia SNe (Iwamoto et al. 1999), and BBN (Galli & Palla 2013). +The color bar represents the percentage of each isotope that composes each element in terms of solar abundances (Asplund et al. +2009). The subplot is a 3D projection of the figure’s histogram. +masses ≳ 6 M⊙ and up to solar metallicities, so no life- +time varies by more than 10%. +The lifetime at a super-solar metallicity of Z = 0.05 +does not follow the same trends as the ones at lower +metallicities. This is caused by the assumed increased +relative ratio of He abundance. While in the other life- +times the dependence on metallicity strongly depends on +a higher opacity, for super-solar metallicities two states +are at play: a lower hydrogen abundance and a higher +average molecular weight µ – the latter of which causes +a higher luminosity of L ∝ µ7.4 (Portinari et al. 1998). +The combination of these two conditions causes the life- +times to fall for super-solar metallicities in the massive +star regime. In the lower mass regime (∼ 6 M⊙) the +lowest metallicity leads to the shortest lifetime. Having +noted these trends, we notice that at all stellar masses, +the lifetimes never varies by more than a factor of two. +We will show in future works that GCE models are not +particularly sensitive to these variations, so that two-Z- +bins approaches like Schaller et al. (1992) or analytical +approaches such as Padovani & Matteucci (1993) are +still viable. +2.6. Yields adopted in GalCEM +In Fig. 4 we plot the isotopic solar abundance in the +Sun of all the isotopes included in the present run of the +simulation, which are 451. We will however limit the +discussion to the first 118 isotopes, i.e. from hydrogen +to 30Zn. We will leave the remaining isotopes for the +second paper (Gjergo et al., in prep.) where GCE will +be explored thoroughly through the introduction of a +multitude of r-process enrichment channels. +For LIMs, we submitted a nucleosynthesis query to +FUll-Network Repository of Updated Isotopic Tables & +Yields (F.R.U.I.T.Y, Cristallo et al. 2011, 2015)7. We +requested the total isotopic yields for the full stellar mass +range (1.3 ≤ M∗ ≤ 6.0 M⊙) the full metallicity range +(0.00002 ≤ Z ≤ 0.02), a standard 13C pocket, and all +initial rotational velocities, even though in this work we +only consider zero initial rotational velocities (IRV = +0). The F.R.U.I.T.Y. yields were calibrated on Lodders +(2003) solar metallicities, who reported a present-day +photospheric metal abundance of Z = 0.0133. In our +paper we adopt Asplund et al. (2009) with Z = 0.0134. +For consistency, we rescale F.R.U.I.T.Y. to the same +metallicities. +7 Query submitted to http://fruity.oa-teramo.inaf.it/ in July 2021. + +GalCEM method presentation +11 +The lowest 4 metallicity bins in F.R.U.I.T.Y. are re- +ported to be Z = 3 × 10−4 to 2 × 10−5. However, these +metallicities are α-enhanced due to a prevalence of early +SN core enrichment. Therefore the authors enhance the +α isotopes (12C, 16O, 20Ne, 24Mg, 28Si, 32S, 36Ar and +40Ca) by a factor of 3 ([α/Fe]=0.5) and that leads to +a metallicity for these lower metallicity bins enhanced +by a factor of 2.4 compared to the solar-scaled nominal +values. +In GalCEM we leave the raw input untouched, however +we follow the repository’s instructions: wherever [α/Fe] +is explicitly indicated, we associate the yields not to the +reported metallicity but to a metallicity rescaled by a +factor of 2.4.8 +The SNCC yields are taken from the R set of Limongi +& Chieffi (2018, O.R.F.E.O.)9. +Set R is the recom- +mended set. +All stars with M∗ > 25 M⊙ fully col- +lapse into a black hole, so the ejecta in this range come +only from the wind component. The mass range is 13 +to 120 M⊙ divided into 9 bins, while there are 4 initial +metallicity bins expressed as [Fe/H]=0 to -3. Set R is +computed assuming mixing and fall-back, and the mass +cut is chosen so that 0.07 M⊙ of 56Ni is ejected for every +supernova event. To set up a common baseline, also in +this case we consider the zero rotational velocity case. +We select the table containing the total final explosive +yields with stable isotopes. The assumption on the un- +stable nuclei is that they fully decay to their closest sta- +ble daughter. +We however note that theoretical models +of massive stars with physically motivated explosion cri- +teria do not predict a black hole landscape with a simple +mass cutoff (Ertl et al. 2016; Sukhbold et al. 2016) as +adopted by Limongi & Chieffi (2018). This could impact +massive star yield ratios based on the mass dependence +of explosive yields for different nuclear species. +The SNIa yields come from Iwamoto et al. (1999)10, +specifically, the favored WDD2 model where the mass of +synthesized 56Ni is set to 0.69 and the explosion energy +to 1.40×1051 ergs. This model works within a deflagra- +tion to detonation transition framework. They assume a +central density of 2.12×109 g cm−3, a slow deflagration +speed of vdef/vs = 0.015 after the thermonuclear run- +away, where vdef is the speed of the deflagration wave +while vs is the local sound speed. +8 ”(e.g. +the +case +”0.0001 +[α/Fe]=0.5” +has +a +metallicity +Z=0.00024)”, +from +the +(1) +HowTo +note +in +http://fruity. +oa-teramo.inaf.it/modelli.pl +9 http://orfeo.iaps.inaf.it/, set R, tab yieldstot iso exp.dec +10 Table 3, +formatted in https://github.com/egjergo/GalCEM/ +tree/main/galcem/input/yields/snia/i99 +In GalCEM it is possible to select the flags for the se- +lection of the yields or the desired enrichment channel. +A constant primordial infall (BBN) is included at setup +by default. +inputs . include channel = +[ ’SNCC’ , +’LIMs ’ , +’ SNIa ’ ] +inputs . yields LIMs option = ’ c15 ’ +inputs . yields SNCC option = ’ lc18 ’ +inputs . yields SNIa option = ’ i99 ’ +inputs . yields BBN option = ’ gp13 ’ +The Karakas (2010) yields have already been pro- +cessed in GalCEM +as well as all the tables from +F.R.U.I.T.Y. (Cristallo et al. 2011, 2015) and Limongi +& Chieffi (2018). In the near future we plan to explore +the yields computed for stars whose initial rotational ve- +locity is different from zero, and to further expand the +yield library to other popular tabulations. +2.7. Rates +Rates return at any given time the newly synthesized +isotopes by an enrichment channel in the whole galaxy. +In order to carry out the computation, rates reconstruct +the occurrence of astrophysical events and pair them +with their respective yields. In the case of LIMs and +SNCC, the occurrence is marked by the death of indi- +vidual main-sequence stars. SNIa, on the other hand, +are linked to the evolution of a binary system that con- +tains at least one white dwarf. All rates are expressed +in units of [M⊙/Gyr]. +2.7.1. LIMs and SNCC rates +The rates of LIMs and SNCC are given by the follow- +ing convolution (a historical definition whose first nu- +merical solution dates back to Talbot & Arnett 1971): +RP,i(t) = αP +� MP,u +MP,l +ψ (t − τM∗,Z∗) YP,i,M∗,Z∗φ(M∗)dM∗, +(12) +where ψ and φ are the SFR and IMF respectively. αP +is the fraction of the stars in the integration limits that +undergoes the astrophysical process P. For the sake of +decluttering, hereafter a subscript indicates a variable +dependence (e.g., τM∗,Z∗ = τ(M∗, Z∗)). +YP,i,M∗,Z∗ are the mass- and metallicity-dependent +yields for the i-th isotope. The P subscript stands for +either LIMs or SNCC. Alternative notations as well as +reviews can be found in a variety of sources, includ- +ing Tinsley (1980); Prantzos (2008); Pagel (2009); Mat- +teucci et al. (2006). +A classic convolution has the form +� a +b f(t − x)g(x)dx, +while the convolved SFR is a function of time, lifetime, + +12 +Gjergo et al. +metallicity, and stellar mass, i.e. +� a +b f(t−τ(x, y))g(x)dx. +The stellar lifetimes, τM∗,Z∗, are treated as discussed in +Section 2.5. +MP,u and MP,l are the upper and lower +mass integration limits. The integration is carried with +respect to stellar mass. +LIMs and SNCC are associ- +ated to individual stars. Technically, LIMs integrations +should not extend above 6 M⊙ and SNCC should not +fall below 13 M⊙, as those are the largest and smallest +bin, respectively, for the F.R.U.I.T.Y. and O.R.F.E.O. +yields. Instead of adopting yield computations specific +to this mass range, we extrapolate the yields linearly to +a mass of 10 M⊙ on both ends. +Rates for single-star enrichment channels in GalCEM —The +integration limits assume that enough time has passed +so that stars may die. In GalCEM we impose this condi- +tion by ensuring that the birth-time is always positive. +Birth-time t′ is defined as the difference between galaxy +time and stellar lifetime. This is reflected in the integra- +tion limits of Eq. 19 – after a change of variables that +switches the independent variable of the integral M∗ to +its birth-time t′. Birth-time is univocal only to a given +stellar population. +In the case of LIMs, there will be an overlap with the +binary white-dwarf progenitor that will produce SNIa +(the very next Section). +By defining αSNIa the frac- +tion of binaries that will produce SNIa, the LIMs rate +which overlap the SNIa rate will be rescaled by a factor +αLIMs = 1 − αSNIa. +2.7.2. Type Ia supernova rates +There is an extensive body of literature investigating +SNIa rates (for a review, see Maoz & Mannucci 2012). +SNIa have been shown to be responsible for the produc- +tion of about 2/3 of the Fe content of a galaxy (Mat- +teucci & Chiappini 2005), although we note that this +fraction is sensitive to the adopted IMF. +SNIa may +occur either when a white dwarf accretes mass from a +close binary companion (single-degenerate scenario, SD) +or when two white dwarfs in a binary system coalesce +(double-degenerate scenario, DD). Gonz´alez Hern´andez +et al. (2012) finds that the SD scenario should not occur +in more than 20% of the SNIa events. Studies on solar +neighborhood abundances do not show a strong prefer- +ence for either a SD or DD scenario (Matteucci et al. +2009). What they do require is a large delay, i.e., of the +total number of SNIa, a fraction not larger that 30% +(and preferably smaller than 20%) should have occurred +within timescales shorter than 100 Myr (Matteucci et al. +2006, 2009). Even larger delay times are predicted in +Totani et al. (2008) but in that prescription, a double- +degenerate scenario is favored. The necessity for signifi- +cant delay times is also apparent through other empirical +findings. For example, Holoien et al. (2019) finds that in +the ASAS-SN bright supernova catalog, over 10% of the +events occurs over 10 kpc away from their host galaxy, +suggesting that the binary progenitors migrated consid- +erable distances before exploding. +SNIa rates in GalCEM —For SNIa rates we follow the SD +scenario proposed by Greggio (eq. 16, 2005), which is +an improvement on the SD models employed in Greggio +& Renzini (1983) and Matteucci & Greggio (1986). +In this scenario, the SNIa rate is given by: +RSNIa,i(t) = αSNIa YSNIa,i +� min(t,τx) +τi +ψ(t − τ)DTDSNIa(τ)dτ, +(13) +where αSNIa absorbs kα, the number of stars per unit +mass in one stellar generation, which is equivalent to +1.55 for the Kroupa (2001) IMF, as well as the realiza- +tion probability of the SNIa scenario ASNIa, set to 10−3, +according to Greggio (2005), for our adopted canoni- +cal IMF. The yields YSNIa are assumed to not vary as +a function of time or mass. +The delay time distribu- +tion of SNIa, DTDSNIa, after being normalized to 1 +( +� τx +τi DTDSNIa(τ)dτ = 1) is then convolved with the +SFR ψ and integrated across delay times. τi corresponds +to the lifetime of a progenitor of ∼ 8M⊙, i.e., the most +massive star capable of producing a WD. In the case of +the SD model, τx = min(m2,e) is set by a limit on, the +envelope mass m2,e of the mass of the secondary com- +ponent of the binary system, m211, which should not +fall below m2,e > 0.15/ϵ. ϵ is an efficiency parameter +that determines how much of the mass of the secondary +companion is accreted onto the WD, and it will appear +again at the end of this subsection. +The DTDSNIa in the SD scenario is proportional to +two quantities that depend on m2. +Specifically, it is +proportional to the absolute value of the time derivative +of the mass, | ˙m2|, and to the distribution function of the +secondary in a progenitor system, n(m2) : +DTDSNIa ∝ n(m2)| ˙m2|. +(14) +The mass of the secondary is well approximated by +the main-sequence lifetime τMS by Girardi et al. (2000) +in the mass range 0.8 ≲ m2/M⊙ ≲ 8, corresponding to +0.04 ≲ τMS/Gyr ≲ 25: +log m2 = 0.0471 (log τMS)2 − 1.2 log τMS + 7.3. +(15) +11 m2 is the companion with a smaller mass – m2 ≤ m1 where m1 +is the primary, more massive companion. + +GalCEM method presentation +13 +The distribution function of the secondary mass is +given by an integral over the mass of the primary mass, +and will depend on the slope of the power law distribu- +tion of the binary system, −α, and on the slope of the +power law distribution of the ratio between secondary +and primary masses, γ. The integral simplifies to (eq. +16 Greggio 2005): +n(m2) ∝ m−α +2 +� +(m2/m1,i)α+γ) − (m2/8)α+γ� +, +(16) +where m1,i is the minimum mass of the primary compan- +ion. It is constrained (m1,i = max(m2, m1,n)) so that +it is the largest between m2 and the remnant mass of +the primary (m1,n = max {2., 2. + 10.(mWD,n − 0.6)}), +which is constrained by the minimum acceptable mass +for the WD, i.e. mWD,n = 1.4 + ϵ m2,e. In this work +we take ϵ = 1, i.e., the solid curves in Fig.2 of Greggio +(2005). +The mass of the envelope of the secondary, m2,e = +m2 − m2,c, is constrained by its own secondary remnant +mass, derived in Nelemans et al. (2001) to be: +m2,c = max {0.3; 0.3 + 0.1(m2 − 2); 0.5 + 0.15(m2 − 4)} . +(17) +2.8. General integrodifferential GCE Equation +The full GCE equation solved in GalCEM is given by +Eq. 18. And it expresses the rate of change of the gas +mass of isotope i in units of [M⊙/Gyr]. +˙Mi,gas(t) = Xi,inf ˙Minf(t) − (1 − ω)Xi,gas(t)ψ(t) ++ +� +P +αP RP,i(t) +(18) +The first term on the RHS is the infall component, +while the second term is the SFR and outflow compo- +nent, with ω being the wind efficiency of the outflow. +The third and last term is the summation of the rate in- +tegrals of all the enrichment channels for the isotope i, +with the rate expressed in full in Eq. 19. αP is the frac- +tion of the stars in the enrichment channel mass range +which will undergo the given astrophysical event. +The first and second terms (RHS on the first row +of Eq. +18) are rescaled to the respective isotopic gas +mass fractions i (Xi,inf = Mi,inf/� +i Mi,inf), which in +the first term reflects the primordial composition of the +infall gas. In the second case, by applying the instan- +taneous mixing approximation, the fraction X refers +to the ith gas mass fraction as a function of time t, +Xi,gas(t) = Mi,gas(t)/� +i Mi,gas(t). +In the third term, the summation of the enrichment +channel rates, a change of variables has been applied, so +that the integral is expressed in terms of birth-time, t′ = +t − τM∗,Z∗. The following rate (or a variation thereof) +applies to any enrichment channel in which the yield is +either mass or metallicity dependent: +RP,i(t) = +� t−min(t,τ(Mu,ZMu )) +t−max(t,τ(Ml,ZMl )) +dt′ψ(t′)× +� +−dM∗ (t − t′, Zt′) +dτ +φ [M∗(t − t′, Zt′)] +YP,i [M∗ (t − t′, Zt′)] +� +M∗(t−t′,Zt′) +, +(19) +where +the +integral +is +computed +in +the +range +[t′(Ml,P), t′(Mu,P)], which undergoes the astrophysi- +cal process P. +t′(Ml,P) = t − max(t, τ(Ml,ZMl)), and +t′(Mu,P) = t − min(t, τ(Mu,ZMu)). That is to say, the +mass limits of integration of Eq. 12 correspond to the +respective birth-time limits t′(Ml,P) and t′(Mu,P). This +prescription is consistent with the Matteucci & Greggio +(1986) formalism, later expressed explicitly in Portinari +et al. (1998). +The integral will be computed when (1) more time has +passed than the lifetime of the most massive star, and +(2) the birth-time is positive, i.e. there are stars which +have died at time t. +The integrand terms within the +curly brackets of Eq. 19 are the stellar-mass-dependent +convolution pulses. +The terms consist of the IMF, +φ(M∗, t − t′, Zt′), the yields YP,i for any given enrich- +ment channel P and isotope i, and a chain rule element +emerging from the change of variables from M∗ to t′12. +The negative sign emerges due to dτ/dt′ = −1. The +integral is computed as a function of stellar birth time +t′, where t′ = t − τ(M∗, Z∗). Elsewhere in this article +written as τM∗,Z∗, the lifetime is the function described +in Section 2.5 that returns the span of existence in Gyr +of a star of mass M∗ and metallicity Z∗. Inside the inte- +gral, the metallicity of the star Z∗ is reconstructed from +the Galaxy metallicity Z(t) via the lifetime of the star of +mass M∗. So the term dM∗ can be converted to dt′ via +the first order derivative of the stellar mass with respect +to the stellar lifetime. +2.8.1. Integration of the Rates in GalCEM +The whole stellar mass range has to be split appropri- +ately to isolate only the stars associated with the yield +YP . The mass range can be denoted by the stellar lim- +its Ml,P and Mu,P for the lower and upper mass limit +respectively. +In GalCEM all of the components of the +12 The chain rule reads: +dM∗ = dM∗ +dt′ dt′ = dM +dτ +dτ +dt′ dt′ = − dM∗(τ∗,Z∗) +dτ(M∗,Z∗) dt′, + +14 +Gjergo et al. +Figure 5. Schematic diagram representing the solution to the integrodifferential GCE equation. A description of the diagram +can be found in Section 2.8.1 +integrand are evaluated on a mass grid constructed be- +tween t′(Ml,P) and t′(Mu,P) on a grid of size Nk with +k = 200 (Portinari et al. 1998), uniformly distributed in +log-space. +At each time step and for every process, GalCEM com- +putes a grid of quantities starting from the stellar mass, +the lifetime, and the birth-time. Onto these grid points +it evaluates all the functions in the integrand. This in- +cludes for example the SFR, ψ(t′), which is evaluated +with respect to the birth-time: i.e., the past ψ(t) is in- +terpolated in order to reconstruct the SFR at the time +of birth of all the stars that die at any given time step. +Fig. 5 is a schematic representation of the grid across +which the returned rate integral is computed. Galaxy +age t grows vertically downwards and has a size of n, +while stellar mass M∗ is shown horizontally on a grid +of length k. +At each time step, there are a series of +stored total physical quantities (whose growth is plot- +ted in Fig. 10). From right to left there is galaxy age +t, total baryonic mass Mtot(t), total gas mass Mgas(t), +total stellar mass Mstar(t), SFR(t), metallicity, and fi- +nally in dark green is the (Ni, Nn) matrix representing +the mass [M⊙] of every isotope as a function of time. +The boxed line indicates Mi(t) is a matrix instead of a +vector like the other quantities. On the horizontal axis +are the integral components for every enrichment chan- +nel, evaluated at every time step and for every isotope. +From bottom to top are the stellar mass M∗, the stellar +lifetime, the stellar birth-time, the IMF, the metallicity +evaluated at the stellar birth-times, Z(t′), and similarly +the SFR, ψ(t′), is also evaluated at the stellar birth- +times, and finally the yield interpolations Yi. There are +two quadrants shaded in blue and pink that represent +the SNCC and LIMs enrichment channels, respectively. +The only independent variables are t and M∗. +For each time step and for each enrichment channel, +an integral must be solved that returns the total yield +from all the events that occur at that given time step. +On the vertical axis are the time-dependent quantities +emerging from the solution of the differential equations. +We label a selection of grid points, with a sampling that +is not to scale. We have also shown plausible lifetimes for +the mass grid, evaluated on the metal poor regime. An +enrichment channel will activate only when the galaxy +time elapsed will be longer than the shortest lifetime of +the channel’s stars, and the N = 200 mass grid will be +cropped to exclude the lower mass stars which have not +died yet. +The analytical form of the GCE integrands, Fi, have +the following form: +Fi(t, M∗) = ψ(t − τ(M∗)) × [Yi, φ] (M∗) +(20) + +SNII +LIMs +V: +[Msun] +SFR +[Msun / yr] +SFR +Z +Z +IMF +IMF +[Gyr] +02 +0.04 +~0.003 +~0.07 +[Gyr] +60 +120 +[Msun] +M* +Mi +SFR +003 +n +13.8 +[Msun +[Msun] +[Msun][Msun] [Msun][Gyr] +/yrl +KGalCEM method presentation +15 +lifetime_Gyr +3 +2 +1 +0 +1 +2 +metallicity +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +mass +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2 +1 +0 +1 +2 +lifetime_Gyr +0.00 +0.05 +0.10 +0.15 +0.20 +metallicity +0.70 +0.35 +0.00 +0.35 +0.70 +1.05 +1.40 +1.75 +mass +lifetime_Gyr +0 +10 +20 +30 +40 +50 +60 +70 +80 +metallicity +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +mass +0 +20 +40 +60 +80 +100 +120 +10 +20 +30 +40 +50 +60 +70 +lifetime_Gyr +0.01 +0.02 +0.03 +0.04 +0.05 +metallicity +0 +12 +24 +36 +48 +60 +72 +84 +96 +mass +lifetime_Gyr +3 +2 +1 +0 +1 +2 +metallicity +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +mass +1.4 +1.2 +1.0 +0.8 +0.6 +0.4 +d(mass) / d(lifetime_Gyr) +2 +1 +0 +1 +2 +lifetime_Gyr +0.00 +0.05 +0.10 +0.15 +0.20 +metallicity +d(mass) / d(lifetime_Gyr) +1.48 +1.34 +1.20 +1.06 +0.92 +0.78 +0.64 +0.50 +0.36 +0.22 +mass +lifetime_Gyr +0 +10 +20 +30 +40 +50 +60 +70 +80 +metallicity +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +mass +40000 +30000 +20000 +10000 +d(mass) / d(lifetime_Gyr) +10 +20 +30 +40 +50 +60 +70 +lifetime_Gyr +0.01 +0.02 +0.03 +0.04 +0.05 +metallicity +d(mass) / d(lifetime_Gyr) +44800 +39200 +33600 +28000 +22400 +16800 +11200 +5600 +0 +mass +MassInterpolant +mass by ['lifetime_Gyr', 'metallicity'] +Transformed Domain (odd rows) | Original Domain (even rows) +Figure 6. Mass interpolant employed in this work and ex- +plained in Section 2.5. The even rows represent the origi- +nal domain, and the odd rows the transformed domain onto +which the interpolation is computed. The top half of the im- +age shows the actual mass interpolant, while the bottom half +shows the derivative of the mass with respect to the lifetime. +The 3D projections in the first column show the metallicity +and lifetime in Gyr for the bottom x-y plane, while mass +(or lifetime derivative of the mass for the bottom panels) is +shown on the vertical z-axis. The second column displays +the respective 2D contours. The color-coding of the scatter +points is consistent with Fig. 3. +with the change of variables we transform them into: +Fi(t, t′) = ψ(t′) × +� +−dM∗ +dτ , Yi, φ +� +(M∗(t − t′)) , +(21) +i.e., the stellar-mass-dependent component is a function +dependent on galaxy age, stellar birth-time w(t, t′) = +(dM/dτ × φ × Yi) (M∗(t − t′)). +mass +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +metallicity +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +lifetime_Gyr +2 +1 +0 +1 +2 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +mass +0.00 +0.05 +0.10 +0.15 +0.20 +metallicity +2.24 +1.68 +1.12 +0.56 +0.00 +0.56 +1.12 +1.68 +2.24 +lifetime_Gyr +mass +0 +20 +40 +60 +80 +100 +120 +metallicity +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +lifetime_Gyr +0 +20 +40 +60 +80 +20 +40 +60 +80 +100 +120 +mass +0.01 +0.02 +0.03 +0.04 +0.05 +metallicity +0 +9 +18 +27 +36 +45 +54 +63 +72 +81 +lifetime_Gyr +mass +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +metallicity +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +lifetime_Gyr +2 +1 +0 +1 +d(lifetime_Gyr) / d(mass) +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +mass +0.00 +0.05 +0.10 +0.15 +0.20 +metallicity +d(lifetime_Gyr) / d(mass) +2.80 +2.24 +1.68 +1.12 +0.56 +0.00 +0.56 +1.12 +1.68 +lifetime_Gyr +mass +0 +20 +40 +60 +80 +100 +120 +metallicity +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +lifetime_Gyr +0 +250 +500 +750 +1000 +1250 +1500 +1750 +d(lifetime_Gyr) / d(mass) +20 +40 +60 +80 +100 +120 +mass +0.01 +0.02 +0.03 +0.04 +0.05 +metallicity +d(lifetime_Gyr) / d(mass) +0 +240 +480 +720 +960 +1200 +1440 +1680 +1920 +lifetime_Gyr +LifetimeInterpolant +lifetime_Gyr by ['mass', 'metallicity'] +Transformed Domain (odd rows) | Original Domain (even rows) +Figure 7. Lifetime interpolant employed in this work and +explained in Section 2.5. +It follows the same structure as +Fig. 6, but for τ(M∗, Z); i.e., the even rows represent the +original domain, and the odd rows the transformed domain +onto which the interpolation is computed. The top half of the +image shows the lifetime interpolant, while the bottom half +shows the derivative of the lifetime with respect to the mass. +The 3D projections in the first column show the metallicity +and lifetime in Gyr for the bottom x-y plane, while mass +is shown on the vertical z-axis. In the second column are +the respective 2D contours. The color-coding of the scatter +points is consistent with Fig. 3. +Having mapped appropriately each item in this inte- +grand to its appropriate grid point as shown in Fig. 5, +the integrand is simply given by the following product: +F (t, t′) = ψ (t′) w(t, t′). +(22) + +16 +Gjergo et al. +metallicity +5.0 +4.5 +4.0 +3.5 +3.0 +2.5 +2.0 +1.5 +mass +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +2.2 +yield +6 +5 +4 +3 +2 +1 +0 +5.0 +4.5 +4.0 +3.5 +3.0 +2.5 +2.0 +1.5 +metallicity +1.2 +1.4 +1.6 +1.8 +2.0 +mass +6.00 +5.25 +4.50 +3.75 +3.00 +2.25 +1.50 +0.75 +0.00 +yield +metallicity +0.0000 +0.0025 +0.0050 +0.0075 +0.0100 +0.0125 +0.0150 +0.0175 +mass +20 +40 +60 +80 +100 +120 +yield +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +0.0025 +0.0050 +0.0075 +0.0100 +0.0125 +0.0150 +0.0175 +metallicity +20 +40 +60 +80 +100 +120 +mass +0.0 +0.3 +0.6 +0.9 +1.2 +1.5 +1.8 +2.1 +2.4 +yield +lc18_z8.a16.irv0.O16 +yield by ['metallicity', 'mass'] +Transformed Domain (odd rows) | Original Domain (even rows) +Figure 8. Example of interpolating yield tabulations. The current plot refers to 16O for the massive yields by Limongi & Chieffi +(2018). The two plots on the bottom show the fit to the raw data. On the top is shown the interpolation on the transformed +domain. This is the preprocessed interpolation curve which is read inside the SNCC rate integral. An equivalent computation +for LIMs is shown in Fig. 9. +So any convenient integration method compatible with +the Volterra Equations is capable of dealing with the +integrand F (t, t′). We apply the Simpson’s rule. +Solving the Differential Equation in GalCEM —At each time +step, the integrals of each isotope and each enrichment +channel is summed into a single quantitity to be added to +the total gas mass component. To solve the differential +equation we simply solve a classic fourth-order Runge- +Kutta method to the total galaxy quantities, namely +SFR, stellar mass, and gas mass. At this stage also the +metallicity is updated. +Currently, GalCEM runs on a uniform time step. The +output of the runs presented in this paper results form +a ∆t = 2 Myr. Adaptive time steps will be tested in the +future to improve the computation times. +2.9. Summary of the GalCEM framework +With the yield selection outlined in Section 2.6, +GalCEM runs on 451 isotopes i for 86 chemical elements. + +GalCEM method presentation +17 +metallicity +4.5 +4.0 +3.5 +3.0 +2.5 +2.0 +1.5 +mass +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +yield +3.5 +3.0 +2.5 +2.0 +1.5 +4.5 +4.0 +3.5 +3.0 +2.5 +2.0 +1.5 +metallicity +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +mass +3.56 +3.32 +3.08 +2.84 +2.60 +2.36 +2.12 +1.88 +1.64 +1.40 +yield +metallicity +0.0000 +0.0025 +0.0050 +0.0075 +0.0100 +0.0125 +0.0150 +0.0175 +0.0200 +mass +2 +3 +4 +5 +6 +yield +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +0.035 +0.040 +0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0200 +metallicity +2 +3 +4 +5 +6 +mass +0.0000 +0.0042 +0.0084 +0.0126 +0.0168 +0.0210 +0.0252 +0.0294 +0.0336 +0.0378 +yield +c15_z8.a16.irv0.O16 +yield by ['metallicity', 'mass'] +Transformed Domain (odd rows) | Original Domain (even rows) +Figure 9. Example of interpolating yield tabulations. The current plot refers to 16O for the LIMs yields by Cristallo et al. +(2015). Similarly to Fig. 8, the bottom plot shows the interpolation on the raw data while the top plot shows the interpolation +on the transformed domain. This latter curve is the preprocessed interpolation which is read inside the LIMs rate integral. +The variable t represents the age of the galaxy. +By +solving for Mi,gas(t) in Eq. +18, the goal is to obtain +an (i, t) matrix as output, with each entry representing +the mass of every isotope as a function of time. +Eq. +19 is, for each enrichment channel, a system of equa- +tions of size i, computed at every time step. +˙Minf(t) +is Eq. 3, fully and uniquely defined by the parameter +τinf, chosen at the start of the run, therefore we com- +pute both ˙Minf(t), the infall rate, and Minf(t), the total +mass of the system at t at the setup stage. +˙M∗,tot(t) is +the SFR defined in Eq. 8. It depends on +˙Minf(t) (Eq. +3) and on Mgas(t) = � +i Mi,gas(t) (that we are solving +for). we save both the one-dimentional vector of length +t for ψ(t) = ˙M∗,tot(t), the SFR, and M∗,tot(t), the total +stellar mass of the system at t. +There is not a single lifetime relation τZ(M∗), in +Portinari et al. (1998) there are 4, as seen in Fig. 7, +depending on the initial metallicity of the star (Z = +0.05, 0.02, 8×10−3, 4×10−4). These functions are inter- + +18 +Gjergo et al. +0 +5 +10 +Age [Gyr] +106 +107 +108 +109 +1010 +1011 +Masses [M ] +Mstar +Mgas +Mg, tot, i +MH, g +MZ, g +Mtot +Mg + Ms +MHe, g +Mgal, f +0 +5 +10 +Age [Gyr] +10 +3 +10 +2 +10 +1 +100 +101 +102 +Rates [M /yr] +RSNII +RSNIa +RLIMs +Infall +SFR +10 +3 +10 +2 +10 +1 +100 +101 +102 +SFRMW CP11 +RSNII, MW M05 +RSNIa, MW M05 +Figure 10. Evolution of the global quantities in a GCE solution. The figure on the left represents mass quantities, while the +figure on the right represents rate quantities. Mtot is the total time-dependent baryonic mass. Mstar is the total stellar mass, +or M∗,tot in the text. Mgas is the total gas mass. The sum of total stellar and total gas mass, Mgas + M∗,tot as a sanity check +coincides with Mtot. Mg,tot,i is the total gas mass as inferred from the isotopic evolution output. As another sanity check, it +should coincide with Mgas. MH,g, MHe,g, and MZ,g are the time-dependent masses of H, He, and all metals, respectively. Mgal,f +represents the final baryonic mass of the galaxy. When it comes to the rates, the infall is shown in the black solid line, the SFR +is in the dashed orange yellow line, and the dotted dark blue, light blue, and magenta lines represent the SNCC, SNIa and LIMs +rate respectively. The present-day Milky Way estimates of the rates are shown with the vertical segments at time 13.7 Gyr. The +SFR is taken from CP11, Chomiuk & Povich (2011), while the SNCC and SNIa rates come from M05, Mannucci et al. (2005). +polated and extrapolated in accordance with the method +described in the following section, Section 2.10. +The convolved integral is given by the product of two +functions: the SFR as a function of birth time t′, and +a product of quantities that depend on the lifetime and +metallicity-dependent stellar mass M∗(t − t′ +Zt′ ). +2.10. GalCEM interpolation tool +Solving detailed GCE equations requires interpolating +at several stages: the yield tabulations must be interpo- +lated over mass, metallicity, and occasionally other pa- +rameters such as initial rotational velocity. Inside the +integrals, also stellar lifetime, metallicity, and SFR need +to be interpolated. Lastly, an interpolation is required +for the derivative of the stellar mass with respect to +stellar lifetime. +Yields, mass, and derivatives can be +preprocessed with the GalCemInterpolant class in the +yield interpolation package. +With the aim of computing the derivative of the stel- +lar mass with respect to its lifetime, we derive the fits +as they appear in Fig. +6 where lifetimes and metal- +licities are interpolated to obtain stellar masses, and +Fig. 7 where masses and lifetimes are interpolated to +obtain lifetimes13. We implement the SciPy (Virtanen +et al. 2020) BivariateSpline interpolation which, while +not providing as good of a fit like other methods such +as LinearNDInterpolator and NearestNDInterpolator, is +smooth so it supports taking derivatives. +Bivariate +splines are ideal methods for the solution of boundary- +value problems by finite-element-type methods because +they consist of piecewise polynomials triangulated on a +polygonal domain (e.g., N¨urnberger & Zeilfelder 2000). +The even rows of both Fig. 6 and Fig. 7 are shown +in the transformed domain. +The transformations im- +plemented by the interpolator are: logarithmic mass, +square root metallicity, and logarithmic lifetime. The +odd rows depicts the original domain. The transforma- +tion was necessary to ensure stable fits, especially with +the derivatives and specifically the dτ/dM∗ fit which we +employ in the returned rate integral. +Fig. 8 and Fig. 9 are the interpolation surfaces com- +puted for SNCC and LIMs, respectively. Both the orig- +13 The +interpolation +code +can +be +found +in +https://github. +com/egjergo/GalCEM/tree/main/yield interpolation/ +lifetime mass metallicity/main.py + +GalCEM method presentation +19 +0 +2 +4 +6 +8 +10 +12 +Galaxy Age [Gyr] +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +metallicity +Z +Age +linear fit on [M/H] +Silva Aguirre et al. (2018) +0 +2 +4 +6 +8 +10 +12 +Galaxy Age [Gyr] +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +[Fe/H] +[Fe/H] +Age +linear fit on [Fe/H] +Silva Aguirre et al. (2018) +Figure 11. The age-metallicity relation with respect to the +metallicity (upper figure) and iron abundance (lower figure) +normalized to solar values. +The orange lines cross at the +solar age and abundance. The observational scatter comes +from the APOKASC sample. +Stellar age and abundances +are taken from Silva Aguirre et al. (2018), while the iron +abundance is taken from Pinsonneault et al. (2014). +inal and the transformed domain are shown in order to +display the fit comparison. The statistics for the good- +ness of the fit are reported in Section 4. Unlike Fig. 6 +and 7, Fig. 8 and 9 are obtained with a combination of +LinearNDInterpolator, NearestNDInterpolator. +These +two methods together give better recovery of known val- +ues. +However, the models are less smooth, especially +outside the hull where the derivative is 0 almost ev- +erywhere and undefined at points equidistant between +neighbors. Bivariate splines do not recover y values at +fitted x values while the linear nd interpolator, the near- +est neighbor does. +This comes at the cost of lacking +smoothness. The linear nd interpolator makes predic- +tions for points inside the hull and the nearest neigh- +bor outside. The errors in Listings 1 and 2 in the Ap- +pendix are computed with the data used to fit models +(in-sample). +The original domain plots shows predic- +tions from the model fit in the transformed domain i.e. +the models are not separate, rather they are viewed in +different scalings. +3. RESULTS: GALCEM APPLICATION TO A MILKY +WAY-LIKE ONE-ZONE GALAXY +We next present our results for a galaxy of final bary- +onic mass Mtot = 5×1010M⊙, that forms from an expo- +nentially decaying gas infall with an infall timescale of 7 +Gyr, no outflow, a Kennicutt (1998) SFR with κ = 1., an +invariant canonical IMF (Kroupa 2001), convolved en- +richment from AGB winds and core collapse SNae with +yields by Cristallo et al. (2011, 2015), and Limongi & +Chieffi (2018), respectively, and SNIa enrichment with +yields by Iwamoto et al. (1999) and a delay-time distri- +bution from Greggio (2005), specifically the novel single- +degenerate model with ϵ = 1. +We first present the results from the global time- +dependent quantities in Fig. +10, namely the time- +dependent evolution of total baryonic galaxy mass, to- +tal gas mass, total stellar mass, metallicity, hydrogen +and helium mass on the left-hand side; meanwhile in- +fall, SFR, and enrichment channel rates (SNCC, SNIa, +and LIMs) are on the right-hand side. The evolution is +tracked linearly with time expressed in Gyr. The trends +of the absolute mass quantities mirrors the slope of the +SFR. This is owed to the choice of linear SFR law as +illustrated in Section 2.4. The mass in metals is a factor +of 0.05 smaller than the gas mass, which is consistent +with super-solar metallicities measured in young stars. +The total stellar mass to gas mass fraction tends at the +present time to a value nearly 4 times larger than the +fiducial ≈ 10. +In Fig. 11 we show the evolution of the metallicity and +iron abundance as a linear function of time. The scatter +data are taken from the stellar ages in Silva Aguirre et al. +(2018), the ID of the KIC stars is matched to the data +from Pinsonneault et al. (2014) to get the iron abun- +dance of said stars. The red lines in each plot represents +a linear fit to the observational data. The black curve +and blue curve are the iron abundance and metallicity, +respectively. The model is in fairly good agreement with +the data. It reproduces the solar abundances at the so- +lar age. Given that the iron abundance is dominated +by SNIa enrichment, we notice that [Fe/H] comes at a +delay compared to the remaining metallicity (primarily +oxygen) coming from SNCC and LIMs, a delay which is +consistent with the SNIa rate. +Fig. 12 is the central result of this work. We restrict +our analysis to an atomic number of 30 with zinc, be- +cause heavier elements require r-process enrichment – to +be explored in a follow-up paper. Visible in the figure + +20 +Gjergo et al. +2 +0 +2 +3Li +Lai-08 +Ryde-20 +Yong-13 +Venn-12 +Hill-19 +Maas-19 +Cohen-13-LTE +Frebel-10 +Spite-05 +Reddy-03-LTE +Nissen-07-LTE +Nissen-14-NLTE +Bensby-05 +Hansen18 +Bihain-04-LTE +Bensby-14 +Hansen-12 +Frebel-10 +Frebel-14 +Hinkel-14 +Caffau-05 +Caffau-11 +Cayrel-04 +Akerman-04 +Gratton-03 +Ramirez-13-NLTE +Reggiani-17 +Shetrone-13 +Roederer-09 +Jacobson-15-LTE +Gonzalez-06 +Carretta-00-NLTE +Adibekyan-12-LTE +Andrievsky-08-NLTE +Andrievsky-10-NLTE +Andrievsky-07-NLTE +Ishigaki-12&13-LTE +Cohen&Huang-09 +Cohen(CEMP-no)-13-LTE +Bensby&Feltzing-06-LTE +Battistini&Bensby-15-NLTE +Battistini&Bensby-16-LTE +4Be +5B +6C +7N +8O +2 +0 +2 +9F +10Ne +11Na +12Mg +13Al +14Si +2 +0 +2 +[X/Fe] +15P +16S +17Cl +18Ar +19K +20Ca +2 +0 +2 +21Sc +22Ti +23V +24Cr +25Mn +26Fe +6 +4 +2 +0 +2 +0 +2 +27Co +6 +4 +2 +0 +28Ni +6 +4 +2 +0 +29Cu +6 +4 +2 +0 +[Fe/H] +30Zn +6 +4 +2 +0 +31Ga +6 +4 +2 +0 +32Ge +Figure 12. [X/Fe]–[Fe/H] relation abundance plots for the elements up to atomic number 32. To facilitate the navigation of +the table, the atomic number is shown right before the element symbol. The total abundance in the one-zone run is enriched by +BBN (Galli & Palla 2013), SNIa (Iwamoto et al. 1999), AGB/LIMs (Cristallo et al. 2015, with zero initial rotational velocity), +and SNCC (Limongi & Chieffi 2018, set R, with zero initial rotational velocity). The observational data scatter is labeled +according to the legend on top. +are also 31Ga, and 32Ge, enriched by the s-process only. +Fig. 12 shows a summary of the evolution of the [X/Fe]- +[Fe/H] relation for all the elements tracked in the current +paper. +The model is generally in good agreement with +the data, and it is consistent with other literature results +constructed on similar one-zone modeling assumptions. +Therefore we have reached our benchmark. +We stress the fact that the breath of stellar abun- +dance patterns observed in the Milky Way, as well +as in other galaxies, spans a rich breath of composi- +tions and histories that cannot be reproduced by a one- +zone model alone. We also point out at the fact that +the [X/Fe]-[Fe/H] relation is only a proxy for metallic- +ity evolution in the Galactic disk, and that the rela- +tion between iron content and metallicity breaks down +for the metal poor stars found e.g. +in the halo (e.g., +Matteucci 2012; Prantzos et al. 2018). +No one-zone +model is able, without treatments on dynamics or mul- +tiple infall episodes, to reproduce the average patterns +for all elements. +Nonetheless, the one-zone approach +to GCE offers precious and reliable constraints when +multiple abundance patterns are considered simultane- +ously – chiefly a congruent comparison of the different +timescales and rates associated with each enrichment +channel, and how these inform the star formation his- +tory in a galaxy. +We briefly explain the observed patterns in Fig. 12, +and we defer an in-depth analysis for each element to fu- + +GalCEM method presentation +21 +ture works. Intermediate-mass elements are commonly +grouped as follows: the CNO nuclei, the α elements, the +odd-Z elements, and the iron-peak elements. +Among +these, the α elements (C, O, Ne, Mg, Si, S, Ar, Ca) are +the best reproduced in literature, displaying the typical +[α/Fe] plateau at the lowest metallicities, followed by a +mild decrease in the disk metallicity regime (e.g., Chi- +appini et al. 2001; Romano et al. 2010). This is also +the case for GalCEM. We note however that C is often +omitted from the α elements analysis due to its flatter +[Fe/H] dependence compared to the other elements in +this group (Prantzos et al. 2018). +The dominant carbon and oxygen isotopes are 12C +and 16O. They are produced both during the H burning +by the CNO cycle, but they are also the most abun- +dant species produced during the He-burning through +the 3α process (Wallerstein et al. 1997). The debate is +still ongoing on what is the exact breakdown among the +known astrophysical sources of enrichment, but GalCEM +is in agreement with Andrews et al. (2017) and Prant- +zos et al. (2018) on the fact that massive stars produce +a larger quantity of C and O compared to AGB stars. +Nitrogen and fluorine in Fig. 12 behave like secondary +elements. Historically, this has been an issue, particu- +larly for nitrogen, because its observational pattern more +closely resembles a primary element, while yield compu- +tations used to suggest a secondary behavior (like the +one seen in Fig. +12). +The introduction of yields by +low-metallicity, rapidly rotating massive stars (Meynet +& Maeder 2002) solves this issue neatly (Limongi & Chi- +effi 2018) and will be a subject of further investigation +with our code. +The iron-peak elements proper (Cr, Mn, Co, and Ni) +are predominantly synthesized by SNIa. Fig. 13 shows +that, in fact, these are the isotopes where the SNIa +returned mass approaches most closely the returned +masses by the other two enrichment channels. +A list of the observational papers used, and the el- +ement that the observations provide, can be found in +Table 2. The ever-expanding GalCEM library allows the +user to define a list of elements and get a similar printout +of observational references. +Fig. 13 plots the time-dependent returned mass rates +(in units of [M⊙/yr]) for each enrichment channel in- +cluded in the one-zone run presented in this work. They +consist of BBN, SNIa, LIMs, and SNCC as outlined in +Section 2. To improve readability we only show the first +120 isotopes, even though the full simulation computes +451 species. This cut includes every zinc nuclide (atomic +number of 30); i.e., we include all the elements of inter- +est from Fig. 12. Within the first Gyr, all the returned +rates normalize to a given rate, pointing to the fact that +the most variation in chemical evolution models occurs +within the first few hundred million years. +4. DISCUSSION +The present article has presented the features and ra- +tionale behind a new modular publicly available GCE +code that computes using efficient numerical methods +the convolved integrodifferential equations of chemical +evolution. +A GalCEM hallmark is that it computes the enrich- +ment of the full set of individual isotopes from multiple +enrichment channels. It is sufficient to define a list with +the processes one wishes to include in the simulation. +By simply defining flags in the input class, it is possi- +ble to switch yield tabulations from a preprocessed set. +The user may run a script to preprocess custom yields, +or they may choose from a library of popular yields al- +ready analyzed by the GalCEM team. +GalCEM is suitable for studies that involve the simulta- +neous analysis of multiple (or all) isotopes and elements +in given runs, because the code automatically generates +the list of unique isotopes included in the yield tabu- +lations. GalCEM contains a preprocessing interpolation +tool that generates a multidimensional interpolation to +the input yield tables for each isotope. The number of +dimensions in the present work is limited to 3, namely +yield, mass, and metallicity – but the tool can accommo- +date extra dimensions such as initial rotational velocity. +The interpolation tool may also be adapted to handle +stellar lifetime tabulations. +GalCEM allows the user to solve the full one-zone con- +volution integral (Matteucci & Greggio 1986) for multi- +ple channels (AGBs, SNCC, and SNIa on this first re- +lease) without resorting to popular approximations such +as IMF-averaged yields or instantaneous recycling ap- +proximations. +We map the integrand quantities onto +consistent array grids in order to apply the Simpson’s +rule and therefore solve the integrals for each isotope +and each enrichment channel. +The differential equa- +tion is solved with a classic fourth-order Runge-Kutta +method. Parameter dependence and algorithm speeds +will be analyzed in a third paper. In Fig. 12 we com- +pare GalCEM’s abundance patterns with Galactic data of +main-sequence stars. Our results are consistent with the +evolution of all the intermediate elements from carbon +to zinc. A library of observational data will routinely +be updated on GalCEM14. We will provide an analysis +of heavier elements in a second paper, where we will +explore r-process candidate sites. +14 https://github.com/egjergo/GalCEM/tree/main/galcem/input/ +observations/abund + +22 +Gjergo et al. +Table 2. List of observations included in Fig. 12. The element investigated in each paper is marked with an ×. The processed public data is available for download +at https://github.com/egjergo/GalCEM/tree/main/galcem/input/observations/abund +Li +Be +B +C +N +O +F +Ne +Na +Mg +Al +Si +P +S +Cl +Ar +K +Ca +Sc +Ti +V +Cr +Mn +Fe +Co +Ni +Cu +Zn +Adibekyan et al. (2012) +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +× +× +× +× +⃝ +⃝ +⃝ +⃝ +⃝ +× +× +× +× +× +× +× +× +× +⃝ +⃝ +Akerman et al. (2004) +⃝ +⃝ +⃝ +× +⃝ +× +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +× +⃝ +⃝ +⃝ +⃝ +Andrievsky et al. (2007) +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +× +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +× +⃝ +⃝ +⃝ +⃝ +Andrievsky et al. (2008) +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +× +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +× +⃝ +⃝ +⃝ +⃝ +Andrievsky et al. (2010) +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +× +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +× +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +× +⃝ +⃝ +⃝ +⃝ +Battistini & Bensby (2015) +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +× +⃝ +× +⃝ +× +× +× +⃝ +⃝ +⃝ +Bensby & Feltzing (2006) +⃝ +⃝ +⃝ +× +⃝ +× +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +× +⃝ +⃝ +⃝ +⃝ +Bensby et al. (2005) +⃝ +⃝ +⃝ +⃝ +⃝ +× +⃝ +⃝ +× +× +× +× +⃝ +⃝ +⃝ +⃝ +⃝ +× +⃝ +× +⃝ +× +⃝ +× +⃝ +× +⃝ +× +Bensby et al. (2014) +⃝ +⃝ +⃝ +⃝ +⃝ +× +⃝ +⃝ +× +× +× +× +⃝ +⃝ +⃝ +⃝ +⃝ +× +⃝ +× +⃝ +× +⃝ +× +⃝ +× +⃝ +× +Bihain et al. (2004) +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +× +⃝ +⃝ +× +× +Caffau et al. 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Loglog time-dependent returned mass rates (in units of M⊙/yr) for each enrichment channel included in the one- +zone run presented in this work – which include BBN, SNIa, LIMs, and SNCC as outlined in the Methods, Section 2. The +number preceding the atomic symbol represents the atomic mass. To improve readability, we only show the first 120 isotopes, +even though the full simulation computes 451 species. This cut includes every zinc nuclide (atomic number of 30), consistent +with Fig. 12. The full table will be shown in Gjergo et al. (2022b) in prep., with the inclusion of r-process enrichment channels. + +24 +Gjergo et al. +Software: A current version of the code is available at +https://github.com/egjergo/GalCEM/releases. The re- +sults of this article can be reproduced with the GalCEM +1.0.0 package release. Archives of yield processing and re- +sults compilations can be found in the organization page: +https://github.com/GalCEM. +ACKNOWLEDGEMENTS +We are grateful for the thorough feedback provided by +the reviewer, which has helped to significantly improve +the quality of this article. We thank Sergio Cristallo for +clarifications on how to use the F.R.U.I.T.Y. database. +We thank Kai Diethelm for crucial feedback early in the +development of the numerical solutions in GalCEM. We +thank Steve Kuhlmann for providing helpful feedback +on the article. E.G. designed and wrote GalCEM and its +contents in the GitHub repository. E.G. prepared the ar- +ticle, and coordinated the research. A.S. developed the +preprocessing yield interpolation tool, helped with +refactoring GalCEM, and set up the release of GalCEM on +the Python Package Index (PyPI). A.R. provided feed- +back on code design, numerical methods, efficiency, and +convergence tests. A.R. hosts the JupyterHub server at +https://galcem.space/. M.L. guided the early interpre- +tation of the results, clarified relevant issues concerning +stellar evolution, and shared resources on how to han- +dle the yield gap in the 6-13M⊙ mass range. F.M. and +E.S. provided invaluable guidance on GCE theory and +history, as well as on the interpretation of the results. +M.K., T.K. and Y.Y. offered feedback on E.G.’s prelim- +inary tests that matured into GalCEM, and hence influ- +enced the early code design choices. J.L. gathered and +cleaned the observational data on stellar abundances. +X.F. promoted the endeavour to write GalCEM, and was +involved with its development at every stage. X.F. pro- +vided the economic support that made this project pos- +sible. All co-authors helped with interpreting the results +and giving feedback on the article. E.G. and X.F. have +been supported by the National Natural Science Foun- +dation of China under grant (No.11922303) and the Fun- +damental Research Funds for the Central Universities +(No.2042022kf1182). +X.F. is supported by the Hubei +province Natural Science Fund for Distinguished Young +Scholars (No. 2019CFA052). E.S. received funding from +the European Union’s Horizon 2020 research and in- +novation program under SPACE-H2020 grant agree- +ment number 101004214 (EXPLORE project). T.K. is +supported by Grants-in-Aid for Scientific Research of +Japan Society for the Promotion of Science (20K03958, +17K05459). E.G. acknowledges the support of the Na- +tional Natural Science Foundation of China (NSFC) un- +der grants No. 12041305, 12173016. 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O16 ] +( mass , m e t a l l i c i t y ) +train +data +d e s c r i p t i o n +mass +m e t a l l i c i t y +y i e l d +count +86.000000 +86.000000 +86.000000 +mean +3.053488 +0.006049 +0.006087 +std +1.593863 +0.006267 +0.007744 +min +1.300000 +0.000048 +0.000178 +25% +1.500000 +0.000790 +0.000911 +50% +2.500000 +0.003000 +0.002820 +75% +4.000000 +0.010000 +0.008001 +max +6.000000 +0.020000 +0.038573 +train +data +RMSE Abs : +1.83 e−18 +MAE Abs : +1.13 e−18 +Max Abs : +6.94 e−18 +RMSE Rel : +2.78 e−16 +MAE Rel : +2.37 e−16 +Max Rel : +5.46 e−16 +Listing 2. Printout of the interpolant statistics for 16O on the trained LC18 yield data +GalCemInterpolant [ z8 . a16 . irv0 . O16 ] +( mass , m e t a l l i c i t y ) +train +data +d e s c r i p t i o n +mass +m e t a l l i c i t y +y i e l d +count +36.00 +36.000000 +3.600000 e+01 +mean +44.78 +0.005027 +5.726242 e−01 +std +34.27 +0.007688 +8.264590 e−01 +min +13.00 +0.000018 +4.099000 e−07 +25% +20.00 +0.000140 +2.308078 e−03 +50% +30.00 +0.000995 +2.082450 e−01 +75% +60.00 +0.005883 +8.216700 e−01 +max +120.00 +0.018100 +2.702400 e+00 +train +data +metrics +RMSE Abs : +7.43 e−18 +MAE Abs : +2.62 e−18 +Max Abs : +2.78 e−17 +RMSE Rel : +3.01 e−16 +MAE Rel : +1.59 e−16 +Max Rel : +9.61 e−16 + diff --git a/TtE0T4oBgHgl3EQfVAA3/content/tmp_files/load_file.txt b/TtE0T4oBgHgl3EQfVAA3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cb7964efdd65a722327074d021892c73ef5b8517 --- /dev/null +++ b/TtE0T4oBgHgl3EQfVAA3/content/tmp_files/load_file.txt @@ -0,0 +1,2904 @@ 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Japan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2-21-1 Osawa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Mitaka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Tokyo 181-8588,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Japan 14Graduate School of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Hongo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Bunkyo-ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Tokyo 11-0033,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Japan 15School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' and International Research Center for Big-Bang Cosmology and Element Genesis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Beihang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Beijing 100083,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' China (Received January 9, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Revised;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Accepted) Submitted to ApJS ABSTRACT This is the first of a series of papers that will introduce a user-friendly, detailed, and modular GALactic Chemical Evolution Model, GalCEM, that tracks isotope masses as a function of time in a given galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The list of tracked isotopes automatically adapts to the complete set provided by the input yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The present iteration of GalCEM tracks 86 elements broken down in 451 isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The prescription includes massive stars, low-to-intermediate mass stars, and Type Ia supernovae as enrich- ment channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We have developed a preprocessing tool that extracts multi-dimensional interpolation curves from the input yield tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' These interpolation curves improve the computation speeds of the full convolution integrals, which are computed for each isotope and for each enrichment channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We map the integrand quantities onto consistent array grids in order to perform the numerical integration at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The differential equation is solved with a fourth-order Runge-Kutta method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We constrain our analysis to the evolution of all the light and intermediate elements from carbon to zinc, and lithium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Our results are consistent up to the extremely metal poor regime with Galactic abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We provide tools to track the mass rate change of individual isotopes on a typical spiral galaxy with a final baryonic mass of 5 × 1010 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Future iterations of the work will extend to the full periodic table by including the enrichment from neutron-capture channels as well as spatially-dependent treatments of galaxy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' GalCEM is publicly available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='com/egjergo/GalCEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' eda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='gjergo@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='com ∗ Released on MM, DDth, YYYY arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='02257v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='GA] 5 Jan 2023 ID2 Gjergo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Keywords: Publicly available software(1864) — Galaxy chemical evolution(580) — Stellar nucleosyn- thesis(1616) — Chemical enrichment(225) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' INTRODUCTION Galactic Chemical Evolution (GCE) is the field that investigates how the chemical makeup of a galaxy changes with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' To do that, GCE must first con- sider the astrophysical events where each isotope may be synthesized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' These events are point-like sources of enrichment, which occur with varying rates throughout the galaxy and across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' To model how such events are distributed in a variety of galactic morphologies, the field of GCE has developed rate equations which track the abundance of chemical species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' A galactic mor- phology is reconstructed by means of stellar birthrate functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Stellar lifetimes will then determine when the chemical enrichment will contaminate the galactic media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Hereafter we will refer to“astrophysical events” and“enrichment channels” interchangeably (for a recent review, see Matteucci 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The theory of GCE dates its origins to the mid 1950’s with the pioneering works of Salpeter (1955, 1959), and Schmidt (1959), who laid the foundations to the formal- ism best explained in Schmidt (1963) and subsequently in Tinsley (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' A schematic representation of the formalism can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 1 as it applies to one- zone models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', Talbot & Arnett 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Timmes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 1995), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', the treatment of galaxies as homogeneous environments where the gas is mixed instantaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In order to formulate the luminosity function evolu- tion of main-sequence stars, Salpeter (1955) proposed the existence of an “original mass function”, now called initial mass function (IMF) – or, the distribution, with respect to stellar mass M∗, of stars at their birth in a single stellar population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' This IMF distribution is color- coded in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 1 according to main-sequence colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The IMF is one of the two components of the birthrate func- tion B(M∗, t), the other being the star formation rate (SFR, Salpeter 1959;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Schmidt 1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The ansatz is that the birthrate function is separable with respect to stellar mass and time and is represented by the product of SFR and IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' This is a simplistic ansatz, already challenged by numerous works (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2017, 2021), but it nonetheless proved to be valuable in GCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We must also stress the difference between backward and forward approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Cosmological simulations gen- erally follow a forward approach, wherein stellar birth is tracked first, then the chemical enrichment is pro- jected onto future time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' A classic GCE approach, instead, follows a backward approach, namely it re- constructs past stellar distributions by tracking stellar death rates at every time-step (Matteucci 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The interstellar medium (ISM) will generally be en- riched by the nucleosynthesis products (yields) gener- ated in individual astrophysical sites – be them super- novae (SNe), asymptotic giant branch (AGB) winds, or coalescence events, to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' GCE must integrate the occurrence of such events over the lifetime of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' To do that, GCE must couple the birthrate func- tion with stellar lifetimes to extrapolate a “death-rate function” and hence learn when the enrichment events will occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' These, coupled with the yields, define how the abundance of each isotope evolves with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 1, at each time step tn, GCE models reconstruct the birth time and properties of all stars that die within that time step and compute their integrated contribu- tion to the enrichment from each astrophysical site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' A first exhaustive review connecting nuclear physics to as- trophysical sites undergoing nucleosynthesis is the cor- nerstone work by Burbidge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (1957), famously re- ferred to as B2FH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Over the decades, chemical evolution has provided useful insights on galaxy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' For example, Eggen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (1962) first discovered that halo stars are old and metal poor while disk stars are young and metal-rich – hence leading to the understanding that disk stars formed along with gas infall – and that such stars offer a snapshot to the chemical composition of their parent star-forming cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Based on the cornerstone work by Salpeter (1955), Salpeter (1959) and Schmidt (1959) independently de- veloped a very similar GCE formalism that laid the foun- dation for all subsequent works (reviews throughout the years include Tinsley 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Prantzos 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Pagel 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Matteucci 2012, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In particular, Talbot & Arnett (1971) developed a numerical solution to the full con- volution equations in a work that did not require the metallicity to grow monotonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Pagel & Patchett (1975) developed a modified GCE model based on the simple model which included prompt initial enrichment, the early version of metal-enhanced star formation, and inhomogeneous collapse and infall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Chiosi (1980) pro- posed an open model with gas infall and outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Chiosi & Matteucci (1980) and later also Portinari & Chiosi (2000) developed a radially-dependent disk GCE model, which was later improved and expanded in Spitoni & Matteucci (2011) where they found that an inside-out formation scenario (and/or a variable flow) in spiral GalCEM method presentation 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Diagram of the GCE rationale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' At each time step, the SFR, ψ(t), determines how much total gas mass is converted into stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The distribution of stellar masses M∗ follows the given IMF, φ(M∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' At each subsequent time step, the GCE convolution integral determines how many stars die at tn, and therefore how much enrichment these stars return to the gas mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The stellar birth-time t′, the baby blue and magenta grids, and lifetimes τ(M∗, Z∗), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' the curves connecting birth-time with tn, are updated at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Each enrichment channel will cover a mass/lifetime range specific to its process (SNCC – or equivalently for our purposes, SNII – and LIMs are included in the graphic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Onto these enrichment channel grids we compute the integrals, as explained in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' disks is necessary to reproduce Galactic abundance gra- dients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The effect of stellar migration in the Milky way was investigated by Sch¨onrich & Binney (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Spitoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2015) and Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Gas radial flow as well as stellar mixing were considered in a multi- zone GCE model in Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' A Bayesian approach was undertaken in Cˆot´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Rybizki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2017) and Spitoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2020), this last work in particular applied Markov chain Monte Carlo methods to a two-infall Milky Way formation scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Spitoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2017) proposed new analytical solutions to a GCE model that describes SDSS galaxies only as a function of infall timescales, infall masses, and mass loading factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' A handful of GCE models have also been released as public codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2017) presented flexCE, a flexible one-zone chemical evolution code which was recently used in Hasselquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Cˆot´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2017) presented the One-zone Model for the Evolu- tion of GAlaxies (OMEGA) code within the NuGrid Python Chemical Evolution Environment (NuPyCEE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' OMEGA is paired with the Stellar Yields for Galac- tic Modeling Applications (SYGMA, Ritter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2018) which computes the ejecta of single stellar populations (SSP) to reconstruct the enrichment of a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2017) proposed a one-zone model (GalIMF) in which they adopted a variable integrated galactic ini- tial mass function (IGIMF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Rybizki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2017) devel- oped Chempy, which parametrizes open one-zone mod- els within a Bayesian framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' And more recently, Johnson & Weinberg (2020) developed the Versatile In- tegrator for Chemical Evolution (VICE), an efficient and user-friendly code that shortens computation times by SFR(t) (stellar birth) (stellar death) galacticCEM@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content="com IMF (Galaxy age) SNil enrichment (stellar birth-time) t(M) = t - t' (stellar lifetime) LIMs enrichment (stellar birth-time) t(M) = t - t' (stellar lifetime)4 Gjergo et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Symbol Description i index of the tracked isotope (the chemical species) k mass grid integration index n time step index P the label that identifies an astrophysical process that is a channel of chemical enrichment tG final age of the galaxy t age vector of the galaxy t′(M∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Z∗) birth time of a star of mass M∗ τ(M∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Z∗) = t − t′(M∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Z∗) lifetime of a star of mass M∗ and metallicity Z∗ M∗ mass of a single star MP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' MP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='u lower and upper mass limits for stars in the integrals (mass limits are process-specific) Minf(t) baryonic mass of the galaxy as a function of time (determined by the infall rate) Mgas(t) gas mass of the galaxy as a function of time Mi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='gas(t) gas mass of the isotope i in the galaxy as a function of time s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' � i Mi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='gas(t) = Mgas(t) M∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='tot(t) star mass of the galaxy as a function of time Minf(t) = Mgas(t) + M∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='tot(t) ν star formation efficiency ψ(t) star formation rate (SFR) equivalent to ˙M∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='tot(t) φ(M∗) initial mass function (IMF) Z(t) the metallicity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' or the mass fraction of all of the chemical elements with the exclusion of H and He YP (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' M∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Z∗) Yields: tabulations of the mass fractions Mi/M∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' where M∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='R is the total mass returned to the interstellar medium by a star of mass M∗ and of initial metallicity Z∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' for the astrophysical process P Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Glossary of GCE Symbols and Quantities of Interest, as They Have Been Used throughout the Present Article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' applying IMF-averaged yields to the SN enrichment by core collapse SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' There has been a fortunate surge in high precision surveys and instruments — some concluded very re- cently, some ongoing still — that are providing unprece- dented levels of precision to constrain GCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' These in- clude LAMOST (Large Sky Area Multi-Object Fiber Spectroscopic Telescope, Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2012), Gaia-ESO (Gilmore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Randich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2022), Gaia DR3 (Recio-Blanco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2022) whose α-element abundances have already been investigated with the two-infall model (Spitoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2022), RAVE (Kordopatis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Steinmetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2020), APOGEE (Majewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Ahumada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Abdurro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2022) with spectra in the IR, SDSS with photometric optical-nearIR bands, GALAH (De Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Buder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2021) with the goal of obtaining abundances for 30 elements within the Galaxy, Vista Variables in the Via Lactea (VVV Minniti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2010) with a focus on the Galactic Bulge, MUSE (Bacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2010) with a highly resolved integrated field spec- troscopy unit and a broader scope out to high redshifts, 4MOST (de Jong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Walcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2019) which will measure spectroscopic redshifts of X-ray-identified groups and galaxy clusters, and other upcoming instru- ments such as JWST (Gardner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Notably, the Gaia-ESO mission was devised with the purpose of pre- cisely mapping the 3D locations and motions of billions GalCEM method presentation 5 of stars within our galaxy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' the low-resolution LAMOST spectrograph has been able to collect simultaneously up to 4000 spectra over huge volumes of the Galaxy, with both a large aperture and a large field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The anal- ysis of these data which in most cases is coupled with stellar dynamics make it so that this 70-year-old field is mature enough to properly constrain multi-zone sta- tistical analysis and provide further information on the evolutionary history of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Before investigat- ing such pursuits, we lay our foundations by reaching a benchmark with previous studies by means of one-zone modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' With GalCEM we offer a public code that adapts to the complete nuclide list of the chosen yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' GalCEM can solve the GCE integrodifferential equation including infall, outflow, SFR and fully convolved returned ejecta across the whole main-sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' By default, low-mass stars, massive stars, and Type Ia supernova yields are always included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In this work we limit our analysis to these three enrichment channels only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The article is structured as follows: in Section 2 we introduce the GCE formalism and we present our nu- merical solution to our adopted general GCE equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In Section 3 present some preliminary results, as well as the data products available at this stage in GalCEM, and we evaluate aspects of the code performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Finally in Section 4 we summarize the article and we discuss fu- ture aims and scientific goals we expect to achieve with this tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' METHOD In this section we present both the theoretical frame- work on which we base our computation and the nu- merical solution we developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' At present, GalCEM is available as a one-zone code, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', we adopt an instan- taneous mixing approximation and the whole galaxy is treated as a single homogeneous zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The solution has been implemented in the Python code GalCEM, publicly available on GitHub1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We follow the conventional formalism, so the general GCE equation can be succinctly expressed as (Pagel 2009): ˙Mi,gas(t) = I(t) − Xi,gasψ(t) + � P RP,i(t) − O(t), (1) where ˙Mi,gas(t) is the mass rate of change of the gas- phase component of a chemical species or isotope i, I(t) represents the infall term while O(t) is the outflow term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Xi,gasψ(t) is the mass fraction of the i-th isotope 1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='com/egjergo/GalCEM/galcem (Xi,gas = Mi,gas/Mtot,gas) that is subtracted from the to- tal gas component due to star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' ψ(t) refers to the SFR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', of the total gas mass that forms stars at a given time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' RP,i(t) is the rate of the i-th component returned to the gas phase by each enrichment channel P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' At present, such enrichment channels include the fi- nal stages of low-to-intermediate mass stars (LIMs) and their enrichment of the ISM through asymptotic giant branch (AGB) winds, the death of massive stars as core collapse supernovae (SNCC), and Type Ia supernovae (SNIa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In Table 1 is the synopsis all of the main sym- bols involved in the GCE formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The symbols which have not been introduced in the present paragraph will be explained in due course within this section2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The galaxy begins by having no mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The only source of mass growth is provided by the infall term that ac- cretes gas from a primordial (or otherwise very metal poor) intergalactic medium3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We conform our solar metallicity to the value derived in Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2009) of Z⊙ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' An environment, star, or system is de- fined as very metal poor whenever < 10−2Z⊙, and ex- tremely metal poor for < 10−3Z⊙ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', Nomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The presence of gas triggers star formation, described by the SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We take the SFR to be a function of the total galaxy mass and the total gas mass at any given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We assume that the mass distribution of stars is described at every time step by the IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The mass of newly synthesized isotopes at any time will be given by the convolution of the SFR with the IMF and yields, as described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 18, which is the full integrodifferen- tial version of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 1 and the equation being solved by GalCEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The majority of the calibrations in this section are taken from Molero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2021a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In the rest of this Methods section we will provide detailed informa- tion about every necessary GCE ingredient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' General workflow in GalCEM Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2 represents the GalCEM flowchart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Its design fol- lows a simple principle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' namely that GCE considers in- dividual nucleosynthetic enrichment channels and inte- 2 The present article follows the common square bracket notation of the logarithmic abundance: for an element or isotope A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' the abundance relative to another element B will be: [A/B] = log (MA/MB) − log (MA/MB)⊙ = log (µAXA/µBXB) − log � µAXA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='⊙/µBXB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='⊙ � = log (XA/XB) − log (XA/XB)⊙ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2) where µ is the atomic weight of a chemical species and ⊙ marks solar values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 3 Here we follow the gas-phase definition of metallicity Z(t) = (MZ(t) = Mgas(t) − MH(t) − MHe(t))/Mgas(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', where “metals” are all elements aside from H and He 6 Gjergo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' GalCEM flowchart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' A description of the figure is provided in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' grates the occurrence of each event across both time and across the distribution of such events in a galaxy, as con- structed through the combination of birth-rate functions and stellar lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Each event associated with each enrichment channel is represented by individual points for each isotope in yield tabulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' These point sources of enrichment are treated in the yield class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Birth rate and stellar lifetimes are instead treated in the morphol- ogy class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' While GalCEM follows well-established liter- ature prescriptions on the GCE theory, the design and numerical solution to the integrodifferential equation is original to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' GalCEM contains a preprocessing yield tool, and re- quires the input from three classes at setup: inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='py, yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='py, and morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Regarding the prepro- cessing yield tool, it extracts the interpolation surfaces for each yield tabulation selected for a run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='py contains the ingredients that charac- terize the galaxy properties (namely, IMF, SFR, infall, and stellar lifetimes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='py imports the necessary yield properties, including the preprocessed interpola- tions, and provides the tools to combine them within the setup class in Main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='py also contains the solar normalization class, and the class that extracts a combined list of unique isotopes from all of the selected yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The One- Zone class in Main opens and writes the outputs and runs the evolve function, which computes the time- step-dependent GCE quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The global quanti- ties (namely, total gas and total stellar mass) are computed by the total evolution function, while the isotope-dependent quantities are computed by isotopes evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Light-gray font colors repre- sent GalCEM features that will be presented in the near future pre-processing Jets Dynamic Symmetric Dynamic Asymmetric BBN Massive stars LIMs SNla NSM MHDJ Collapsars AGNS Handle Isotopes Solarnormalization Yields Dust Yield classes parameter values options custom functions inputs auxiliary functions Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='hierarchy multi-zones Stellar lifetimes Infall IMF SFR input class Morphology classes Setup OneZone output Maininstantiation Evolve inputsetup total quantities isotopegquantities total evolution Plots integration grid Integration classes isotopes evolution integrals computation observational databaseGalCEM method presentation 7 GalCEM runs on Python3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' After importing the pack- age, an inputs object must be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The inputs object is customizable4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' A minimum working example has the following form5: import galcem as gc inputs = gc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Inputs () oz = gc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' OneZone( inputs , outdir=MYDIR) oz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' main () By defining the oz object, the user initializes the setup, including a series of properties like the time vector, the total gas mass, and the complete list of isotopes gener- ated from the chosen yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The actual run is launched by calling the main function onto the oz object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The parameters of the simulation can either be set within the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='py parameters or they can be edited manually in a script before executing a run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' For ex- ample, the size of the time step, in Gyr, can be easily changed with the following: inputs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' ntime step = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='2 GalCEM has been published as a Python Package In- dex Project, and for pip users can be installed with the terminal or conda command: pip i n s t a l l galcem Alternatively, the interested user who prefers to work on the cloud can request a JupyterHub account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Infall rate and Outflows Infall as well as outflow rates play an essential role in GCE models, in that they can characterize the dynamics of galaxy formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', see Pagel 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Concerning infall rates, declining models are favored because they predict metallicity distributions more closely consistent with those of G-dwarf Milky Way disk stars (Larson 1974, 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Matteucci 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Spitoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2021) recently investigated the assump- tion of a declining infall, as well as the impact of inflow and outflow on the cumulative metallicity distribution of galaxies as a function of their stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In this work we therefore consider a simple single exponential infall rate described by: I(t) = ˙Minf(t) = I0e−t/τinf , (3) where ˙Minf(t) = dMinf(t)/dt is the rate of gas mass from the extragalactic medium falling into the gravita- 4 The full list of input parameters is in https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='com/ egjergo/GalCEM/blob/main/galcem/classes/inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='py 5 The minimum working example script is located in https:// github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='com/egjergo/GalCEM/blob/main/examples/mwe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='py 6 Available at https://galcem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='space/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' tional potential of a galaxy with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We assume that only gas falls into a galaxy’s potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In this prelim- inary work, we ignore the outflow term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' This simplification is justified by the results of Spitoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2009), where they found that the outflow timescales as inferred from the orbit times of clouds subject to the Galactic gravitational potential should be of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='1 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The impact of such outflow timescales on GCE models is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The infall timescale τinf varies depending on the for- mation history of a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Shorter timescales imply more rapid star formation histories and are associated with elliptical galaxies (Pipino & Matteucci 2004), while more relaxed timescales are associated with spiral or dwarf galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In Molero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2021a), the timescale that reproduces the Milky Way thin disk is fine-tuned to 7 Gyr, while 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='2 Gyr is suitable for elliptical galaxies (or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05 Gyr for some dwarf galaxies in Molero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' I0 has units of [M⊙ / Gyr], with M⊙ being the solar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' It is normalized so that: � tG 0 � ˙Minf(t) − ˙Mout(t) � dt = Mtot, (4) where tG and Mtot (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', Minf(tG) = Mtot) represent the present-day age and total baryonic mass of the galaxy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Given that we do not treat dynamics, grav- itational components including dark matter are beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Similarly, in the specific case of no outflow, the total baryonic mass of a galaxy as a function of time is given by: Minf(t) = Mtot(t) = � t 0 ˙Minf(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (5) Infall rate in GalCEM — GalCEM computes the infall mass Minf(t) as well as the total mass Mtot(t) at the setup stage when the one-zone object oz is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We take the final total baryonic mass to be Mtot(tG) = 5 × 1010 M⊙, with tG = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='7 Gyr being the present age of the Galaxy, and the infall timescale τinf = 7 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Outflows —Galactic outflows are an integral component galaxy evolution modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' They have been invoked since the late 70s (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', De Young 1978) as the source of metal-enrichment in the intracluster medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Re- cent Chandra X-ray observations of M82 outflows sug- gest that the hot halo gas is being mass-loaded with cold-phase material (Lopez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' This material can be very metal-rich, like in the case of the edge-on star-forming galaxy Mrk 1485, whose outflows are about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='6 times the ISM metallicity (Cameron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' How these outflows impact the chemical evolution of a galaxy is not yet a settled issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' A number of papers 8 Gjergo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' which do incorporate outflows find that it significantly impacts the evolution of chemical abundances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', An- drews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Weinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Trueman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' However, studies on Galactic fountains triggered by core-collapse SNae associated with the solar annulus Me- lioli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Spitoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2008) find that that out- flow ejecta should fall back into the close proximity to the same Galactocentric regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Furthermore, the typ- ical fall-back delay of such clouds is found to be ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='1 Gyr (Spitoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Despite the dynamical com- plexity of such phenomena, their impact on the chemi- cal evolution appears to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In fact, a number of GCE models which assume no outflow are able to reproduce observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', Minchev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Ro- mano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' But the issue is further complicated by another consideration: if the gas is ejected onto the circumgalactic medium, semi-analytic models find that this gas will become available again for cooling on much longer timescales, of the order of several Gyr (Faerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Outflows in GalCEM —To set a benchmark for com- parison with other works, in this first paper we set the outflow to zero, but the user can easily edit inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='wind efficiency to implement a dimension- less parameter 0 < ω < 1 so that the outflow may be proportional to the SFR, O(t) = ωψ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The proportionality of the outflow to the SFR follows Bradamante et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (1998), where the galactic winds orig- inate whenever the thermal energy of the gas exceeds its binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Alternatively, the user can com- ment out the self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='wind efficiency = 0 override in classes/inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='py to restore the default parameter for the given morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Initial Mass Function (IMF) The initial mass function, IMF, is the number distri- bution of stars in a galaxy as a function of stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Its original single-power-law formulation dates back to Salpeter (1955): φ(M∗) = dN∗/dM∗ = φ0(M∗/M⊙)−x, (6) where M∗ is the stellar mass of individual stars, normal- ized to the solar mass M⊙, and the best fit for the power law index was found to be x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The IMF is one component of the birthrate function, the other being the SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Given that we already express the SFR in units of solar masses over time, we normalize the mass-weighted IMF to 1: � Mu Ml M∗φ(M∗)dM∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' A more appropriate IMF which is representative of the stellar distribution in the Milky Way is the universal IMF (Kroupa 2001): φ(M∗) = � � � � � 2φ0M −α1 ∗ , Ml ≤ M∗/M⊙ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='50 , φ0M −α2 ∗ , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='50 ≤ M∗/M⊙ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00 , φ0M −α3 ∗ , 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00 ≤ M∗/M⊙ < Mu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (7) This IMF is now commonly referred to as canonical IMF, and it is a piecewise power law that accounts for the fact that the low-mass end of the stellar mass distribution is not as steep as the high-mass end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' For a historical overview of the field, see Kroupa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' IMF in GalCEM —For the present paper we adopted the Kroupa (2001) canonical IMF, with α1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='3 and α2 = α3 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='3, according to the canonical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 7 we left the break at M∗/M⊙> 1 because it is representative of GalCEM’s implementation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', the user may readily explore top-light or top-heavy IMFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We chose Ml = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='08 to Mu = 120 M⊙to be the span of the mass limits for the normalization, in accordance with the upper mass limit of our chosen SNCC yields (Limongi & Chieffi 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The Salpeter IMF is also an available op- tion (inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='IMF option = ’Salpeter55’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' GalCEM can take custom IMF laws, as long as they are only de- pendent on the stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Variable IMFs that depend e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' on the metallicity and SFR, such as the integrated galaxy-wide IMF (IGIMF Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Jeˇr´abkov´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2021), will be investigated in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The Star Formation Rate (SFR) One of the prevalent SFR prescriptions, available by default in GalCEM, is: ψ(t) = ˙M∗,tot(t) = ν �Mgas(t) Mtot(t) �κ , (8) is a Kennicutt-Schmidt definition (Kennicutt 1998) of the SFR, where ˙M∗,tot(t) is the rate of total stellar mass formed as a function of time, Mgas(t) is the gas mass at time t, and Mtot(t) is the total baryonic mass fallen within the galaxy by time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Mtot(t) is initially added to the gas component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' This gas mass will be partly com- posed of the pristine infall gas mass, but it will also be enriched over time by the returned ejecta of the astro- physical sites P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' This returned mass is computed by the convolution integrals shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The star formation efficiency (SFE, ν) has units of [Gyr−1] for a SFR defined in terms of surface densities, while κ is a dimensionless parameter that varies depend- ing on the morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' κ normally ranges between 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The best fit found in Kennicutt (1998) is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' This single SFR power law is however a simplifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Kennicutt & De Los Reyes (2021) find breaks in GalCEM method presentation 9 both the power-law index and zero-points of the star for- mation law between starbursting and non-starbursting galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2021) demonstrate that there is significant galaxy-to-galaxy variations in this trend, suggesting that a single relation is not universally appli- cable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Much of the debate on this topic can be traced back to the uncertainties in the CO-to-H2 conversion factor (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', Kennicutt & Evans 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' de los Reyes & Kennicutt (2019) argue that this is still a viable prescription for GCE models, so it is reason- able to use the single power-law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Nonetheless, there is sufficient reason to expect the nuances of the star for- mation law may turn out to be important for chemical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The SFR prescription adopted in GalCEM —We follow the SFR prescription as in Portinari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (1998): ψ(t) = ν �σ(tG) σ(t) �κ−1 Gκ(t), (9) where G(t) = σg(t)/σ(tG) is the normalized surface gas density: ψ(t) = ν �σ(tG) σ(t) �κ−1 � σg(t) σ(tG) �κ = ν [σg(t)]κ [σ(t)]κ−1 σ(tG) = ν [Mgas(t)/V (t)]κ [Mtot(t)/V (t)]κ−1 (Mtot(tG)/V (tG)) , (10) where V (t) is the surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Mtot(tG) = Mfinal is the final baryonic galaxy mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Given that the effective radius of the galaxy does not change with time, V (tG) = V (0) = V (t) = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Let us define the gas fraction fg = σg(t)/σ(t), so that: ψ(t) = ν Mfinal M κ gas(t) M κ−1 tot (t) ≡ ν Mfinal �Mgas(t) Mtot(t) �κ−1 Mgas(t) = νf κ−1 g (t)Mgas(t) Mfinal , (11) this latter expression being the equation implemented in the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Notice that for κ = 1, the SFR is simply proportional to Mgas(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The results in this article are presented assuming that κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Stellar lifetimes Stellar lifetimes is the crucial component that brings together SFR and IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Specifically, lifetimes provide the means of connecting stellar masses with time, and they make the integration component of GCE equations possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 100 101 102 10 3 10 1 101 (M * , Z) 9M 6M 3M Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0004 Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='004 Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='008 Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='02 Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05 100 101 102 Mass 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='8 (X)/ (Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0004) 9M 6M 3M Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='004 Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='008 Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='02 Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 (M * ) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The adopted (Portinari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 1998) metallicity- dependent stellar lifetimes, τ(M∗, Z∗), as a function of stellar mass (top panel) and the ratio of the top 4 metallicity bins normalized by the lowest one (Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0004, bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The 4 next metallicity bins on the bottom panel are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='004, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='008, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='02, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05 (represented in dotted, dashed, and solid blue lines, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The data points of the metal- licity tabulation are color-coded by the actual stellar lifetime as shown in the color bar (where the units are in log10 yr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In both the top and bottom panels, three arbitrary stellar masses are highlighted with red vertical lines (3, 6, and 9 M⊙, solid, dashed, and dotted, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The top-left panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 7 represents this same color-coded grid points in a full 3D graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Stellar lifetimes in GalCEM —In the present version of the code, we adopt the metallicity-dependent lifetimes τ(M∗, Z) from Portinari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The lifetimes are reported on the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 3, where the stellar mass is on the x-axis, the lifetime on the y-axis, and the various curves display varying metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The three red vertical lines aid with tracking 3 reference masses (9 to 3 M⊙) with lifetimes ranging from ∼ 3 × 107 and 3 × 108 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The lifetimes in Portinari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (1998) account for the H- and He-burning timescales and are computed with stellar evolution models in the Padua library (Bres- san et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Fagotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 1994a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The compu- tation of these lifetimes also follow the instantaneous mixing approximation, a common assumption in one- zone GCE models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Other burning stages would anyway be shorter than the resolution of these lifetimes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', Limongi 2017) In the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 3 we show the linear ratio between the three lifetimes computed on larger metallicities divided by the lowest metallicity (Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0004) lifetime, to better highlight trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' For 10 Gjergo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 0 20 40 60 80 100 120 140 160 180 200 Atomic Mass A 0 20 40 60 80 Proton (Atomic) Number Z Tracked isotopes 0 20 40 60 80 100 Isotope abundance % Atomic Mass A 0 50 100 150 200 Atomic Number Z 0 20 40 60 80 isotopic % 0 20 40 60 80 100 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' All the isotopes (451) tracked in the present run, including the ones coming from LIMs (Cristallo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2015), massive stars dying as core-collapse SNe (Limongi & Chieffi 2018), Type Ia SNe (Iwamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 1999), and BBN (Galli & Palla 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The color bar represents the percentage of each isotope that composes each element in terms of solar abundances (Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The subplot is a 3D projection of the figure’s histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' masses ≳ 6 M⊙ and up to solar metallicities, so no life- time varies by more than 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The lifetime at a super-solar metallicity of Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05 does not follow the same trends as the ones at lower metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' This is caused by the assumed increased relative ratio of He abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' While in the other life- times the dependence on metallicity strongly depends on a higher opacity, for super-solar metallicities two states are at play: a lower hydrogen abundance and a higher average molecular weight µ – the latter of which causes a higher luminosity of L ∝ µ7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='4 (Portinari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The combination of these two conditions causes the life- times to fall for super-solar metallicities in the massive star regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In the lower mass regime (∼ 6 M⊙) the lowest metallicity leads to the shortest lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Having noted these trends, we notice that at all stellar masses, the lifetimes never varies by more than a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We will show in future works that GCE models are not particularly sensitive to these variations, so that two-Z- bins approaches like Schaller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (1992) or analytical approaches such as Padovani & Matteucci (1993) are still viable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Yields adopted in GalCEM In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 4 we plot the isotopic solar abundance in the Sun of all the isotopes included in the present run of the simulation, which are 451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We will however limit the discussion to the first 118 isotopes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' from hydrogen to 30Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We will leave the remaining isotopes for the second paper (Gjergo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=') where GCE will be explored thoroughly through the introduction of a multitude of r-process enrichment channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' For LIMs, we submitted a nucleosynthesis query to FUll-Network Repository of Updated Isotopic Tables & Yields (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='Y, Cristallo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2011, 2015)7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We requested the total isotopic yields for the full stellar mass range (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='3 ≤ M∗ ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 M⊙) the full metallicity range (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00002 ≤ Z ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='02), a standard 13C pocket, and all initial rotational velocities, even though in this work we only consider zero initial rotational velocities (IRV = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' yields were calibrated on Lodders (2003) solar metallicities, who reported a present-day photospheric metal abundance of Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In our paper we adopt Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2009) with Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' For consistency, we rescale F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' to the same metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 7 Query submitted to http://fruity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='oa-teramo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='inaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='it/ in July 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' GalCEM method presentation 11 The lowest 4 metallicity bins in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' are re- ported to be Z = 3 × 10−4 to 2 × 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' However, these metallicities are α-enhanced due to a prevalence of early SN core enrichment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Therefore the authors enhance the α isotopes (12C, 16O, 20Ne, 24Mg, 28Si, 32S, 36Ar and 40Ca) by a factor of 3 ([α/Fe]=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5) and that leads to a metallicity for these lower metallicity bins enhanced by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='4 compared to the solar-scaled nominal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In GalCEM we leave the raw input untouched, however we follow the repository’s instructions: wherever [α/Fe] is explicitly indicated, we associate the yields not to the reported metallicity but to a metallicity rescaled by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='8 The SNCC yields are taken from the R set of Limongi & Chieffi (2018, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' )9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Set R is the recom- mended set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' All stars with M∗ > 25 M⊙ fully col- lapse into a black hole, so the ejecta in this range come only from the wind component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The mass range is 13 to 120 M⊙ divided into 9 bins, while there are 4 initial metallicity bins expressed as [Fe/H]=0 to -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Set R is computed assuming mixing and fall-back, and the mass cut is chosen so that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='07 M⊙ of 56Ni is ejected for every supernova event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' To set up a common baseline, also in this case we consider the zero rotational velocity case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We select the table containing the total final explosive yields with stable isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The assumption on the un- stable nuclei is that they fully decay to their closest sta- ble daughter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We however note that theoretical models of massive stars with physically motivated explosion cri- teria do not predict a black hole landscape with a simple mass cutoff (Ertl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2016) as adopted by Limongi & Chieffi (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' This could impact massive star yield ratios based on the mass dependence of explosive yields for different nuclear species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The SNIa yields come from Iwamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (1999)10, specifically, the favored WDD2 model where the mass of synthesized 56Ni is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='69 and the explosion energy to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='40×1051 ergs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' This model works within a deflagra- tion to detonation transition framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' They assume a central density of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='12×109 g cm−3, a slow deflagration speed of vdef/vs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='015 after the thermonuclear run- away, where vdef is the speed of the deflagration wave while vs is the local sound speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 8 ”(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' the case ”0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0001 [α/Fe]=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5” has a metallicity Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00024)”, from the (1) HowTo note in http://fruity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' oa-teramo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='inaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='it/modelli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='pl 9 http://orfeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='iaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='inaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='it/, set R, tab yieldstot iso exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='dec 10 Table 3, formatted in https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='com/egjergo/GalCEM/ tree/main/galcem/input/yields/snia/i99 In GalCEM it is possible to select the flags for the se- lection of the yields or the desired enrichment channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' A constant primordial infall (BBN) is included at setup by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' inputs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' include channel = [ ’SNCC’ , ’LIMs ’ , ’ SNIa ’ ] inputs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' yields LIMs option = ’ c15 ’ inputs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' yields SNCC option = ’ lc18 ’ inputs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' yields SNIa option = ’ i99 ’ inputs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' yields BBN option = ’ gp13 ’ The Karakas (2010) yields have already been pro- cessed in GalCEM as well as all the tables from F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (Cristallo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2011, 2015) and Limongi & Chieffi (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In the near future we plan to explore the yields computed for stars whose initial rotational ve- locity is different from zero, and to further expand the yield library to other popular tabulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Rates Rates return at any given time the newly synthesized isotopes by an enrichment channel in the whole galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In order to carry out the computation, rates reconstruct the occurrence of astrophysical events and pair them with their respective yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In the case of LIMs and SNCC, the occurrence is marked by the death of indi- vidual main-sequence stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' SNIa, on the other hand, are linked to the evolution of a binary system that con- tains at least one white dwarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' All rates are expressed in units of [M⊙/Gyr].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' LIMs and SNCC rates The rates of LIMs and SNCC are given by the follow- ing convolution (a historical definition whose first nu- merical solution dates back to Talbot & Arnett 1971): RP,i(t) = αP � MP,u MP,l ψ (t − τM∗,Z∗) YP,i,M∗,Z∗φ(M∗)dM∗, (12) where ψ and φ are the SFR and IMF respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' αP is the fraction of the stars in the integration limits that undergoes the astrophysical process P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' For the sake of decluttering, hereafter a subscript indicates a variable dependence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', τM∗,Z∗ = τ(M∗, Z∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' YP,i,M∗,Z∗ are the mass- and metallicity-dependent yields for the i-th isotope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The P subscript stands for either LIMs or SNCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Alternative notations as well as reviews can be found in a variety of sources, includ- ing Tinsley (1980);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Prantzos (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Pagel (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Mat- teucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' A classic convolution has the form � a b f(t − x)g(x)dx, while the convolved SFR is a function of time, lifetime, 12 Gjergo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' metallicity, and stellar mass, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' � a b f(t−τ(x, y))g(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The stellar lifetimes, τM∗,Z∗, are treated as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' MP,u and MP,l are the upper and lower mass integration limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The integration is carried with respect to stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' LIMs and SNCC are associ- ated to individual stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Technically, LIMs integrations should not extend above 6 M⊙ and SNCC should not fall below 13 M⊙, as those are the largest and smallest bin, respectively, for the F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Instead of adopting yield computations specific to this mass range, we extrapolate the yields linearly to a mass of 10 M⊙ on both ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Rates for single-star enrichment channels in GalCEM —The integration limits assume that enough time has passed so that stars may die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In GalCEM we impose this condi- tion by ensuring that the birth-time is always positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Birth-time t′ is defined as the difference between galaxy time and stellar lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' This is reflected in the integra- tion limits of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 19 – after a change of variables that switches the independent variable of the integral M∗ to its birth-time t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Birth-time is univocal only to a given stellar population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In the case of LIMs, there will be an overlap with the binary white-dwarf progenitor that will produce SNIa (the very next Section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' By defining αSNIa the frac- tion of binaries that will produce SNIa, the LIMs rate which overlap the SNIa rate will be rescaled by a factor αLIMs = 1 − αSNIa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Type Ia supernova rates There is an extensive body of literature investigating SNIa rates (for a review, see Maoz & Mannucci 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' SNIa have been shown to be responsible for the produc- tion of about 2/3 of the Fe content of a galaxy (Mat- teucci & Chiappini 2005), although we note that this fraction is sensitive to the adopted IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' SNIa may occur either when a white dwarf accretes mass from a close binary companion (single-degenerate scenario, SD) or when two white dwarfs in a binary system coalesce (double-degenerate scenario, DD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Gonz´alez Hern´andez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2012) finds that the SD scenario should not occur in more than 20% of the SNIa events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Studies on solar neighborhood abundances do not show a strong prefer- ence for either a SD or DD scenario (Matteucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' What they do require is a large delay, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', of the total number of SNIa, a fraction not larger that 30% (and preferably smaller than 20%) should have occurred within timescales shorter than 100 Myr (Matteucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2006, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Even larger delay times are predicted in Totani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2008) but in that prescription, a double- degenerate scenario is favored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The necessity for signifi- cant delay times is also apparent through other empirical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' For example, Holoien et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2019) finds that in the ASAS-SN bright supernova catalog, over 10% of the events occurs over 10 kpc away from their host galaxy, suggesting that the binary progenitors migrated consid- erable distances before exploding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' SNIa rates in GalCEM —For SNIa rates we follow the SD scenario proposed by Greggio (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 16, 2005), which is an improvement on the SD models employed in Greggio & Renzini (1983) and Matteucci & Greggio (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In this scenario, the SNIa rate is given by: RSNIa,i(t) = αSNIa YSNIa,i � min(t,τx) τi ψ(t − τ)DTDSNIa(τ)dτ, (13) where αSNIa absorbs kα, the number of stars per unit mass in one stellar generation, which is equivalent to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='55 for the Kroupa (2001) IMF, as well as the realiza- tion probability of the SNIa scenario ASNIa, set to 10−3, according to Greggio (2005), for our adopted canoni- cal IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The yields YSNIa are assumed to not vary as a function of time or mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The delay time distribu- tion of SNIa, DTDSNIa, after being normalized to 1 ( � τx τi DTDSNIa(τ)dτ = 1) is then convolved with the SFR ψ and integrated across delay times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' τi corresponds to the lifetime of a progenitor of ∼ 8M⊙, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', the most massive star capable of producing a WD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In the case of the SD model, τx = min(m2,e) is set by a limit on, the envelope mass m2,e of the mass of the secondary com- ponent of the binary system, m211, which should not fall below m2,e > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='15/ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' ϵ is an efficiency parameter that determines how much of the mass of the secondary companion is accreted onto the WD, and it will appear again at the end of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The DTDSNIa in the SD scenario is proportional to two quantities that depend on m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Specifically, it is proportional to the absolute value of the time derivative of the mass, | ˙m2|, and to the distribution function of the secondary in a progenitor system, n(m2) : DTDSNIa ∝ n(m2)| ˙m2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (14) The mass of the secondary is well approximated by the main-sequence lifetime τMS by Girardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2000) in the mass range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='8 ≲ m2/M⊙ ≲ 8, corresponding to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='04 ≲ τMS/Gyr ≲ 25: log m2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0471 (log τMS)2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='2 log τMS + 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (15) 11 m2 is the companion with a smaller mass – m2 ≤ m1 where m1 is the primary, more massive companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' GalCEM method presentation 13 The distribution function of the secondary mass is given by an integral over the mass of the primary mass, and will depend on the slope of the power law distribu- tion of the binary system, −α, and on the slope of the power law distribution of the ratio between secondary and primary masses, γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The integral simplifies to (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 16 Greggio 2005): n(m2) ∝ m−α 2 � (m2/m1,i)α+γ) − (m2/8)α+γ� , (16) where m1,i is the minimum mass of the primary compan- ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' It is constrained (m1,i = max(m2, m1,n)) so that it is the largest between m2 and the remnant mass of the primary (m1,n = max {2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' + 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (mWD,n − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='6)}), which is constrained by the minimum acceptable mass for the WD, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' mWD,n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='4 + ϵ m2,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In this work we take ϵ = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', the solid curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='2 of Greggio (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The mass of the envelope of the secondary, m2,e = m2 − m2,c, is constrained by its own secondary remnant mass, derived in Nelemans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2001) to be: m2,c = max {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='3 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='1(m2 − 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='15(m2 − 4)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (17) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' General integrodifferential GCE Equation The full GCE equation solved in GalCEM is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' And it expresses the rate of change of the gas mass of isotope i in units of [M⊙/Gyr].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' ˙Mi,gas(t) = Xi,inf ˙Minf(t) − (1 − ω)Xi,gas(t)ψ(t) + � P αP RP,i(t) (18) The first term on the RHS is the infall component, while the second term is the SFR and outflow compo- nent, with ω being the wind efficiency of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The third and last term is the summation of the rate in- tegrals of all the enrichment channels for the isotope i, with the rate expressed in full in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' αP is the frac- tion of the stars in the enrichment channel mass range which will undergo the given astrophysical event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The first and second terms (RHS on the first row of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 18) are rescaled to the respective isotopic gas mass fractions i (Xi,inf = Mi,inf/� i Mi,inf), which in the first term reflects the primordial composition of the infall gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In the second case, by applying the instan- taneous mixing approximation, the fraction X refers to the ith gas mass fraction as a function of time t, Xi,gas(t) = Mi,gas(t)/� i Mi,gas(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In the third term, the summation of the enrichment channel rates, a change of variables has been applied, so that the integral is expressed in terms of birth-time, t′ = t − τM∗,Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The following rate (or a variation thereof) applies to any enrichment channel in which the yield is either mass or metallicity dependent: RP,i(t) = � t−min(t,τ(Mu,ZMu )) t−max(t,τ(Ml,ZMl )) dt′ψ(t′)× � −dM∗ (t − t′, Zt′) dτ φ [M∗(t − t′, Zt′)] YP,i [M∗ (t − t′, Zt′)] � M∗(t−t′,Zt′) , (19) where the integral is computed in the range [t′(Ml,P), t′(Mu,P)], which undergoes the astrophysi- cal process P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' t′(Ml,P) = t − max(t, τ(Ml,ZMl)), and t′(Mu,P) = t − min(t, τ(Mu,ZMu)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' That is to say, the mass limits of integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 12 correspond to the respective birth-time limits t′(Ml,P) and t′(Mu,P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' This prescription is consistent with the Matteucci & Greggio (1986) formalism, later expressed explicitly in Portinari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The integral will be computed when (1) more time has passed than the lifetime of the most massive star, and (2) the birth-time is positive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' there are stars which have died at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The integrand terms within the curly brackets of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 19 are the stellar-mass-dependent convolution pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The terms consist of the IMF, φ(M∗, t − t′, Zt′), the yields YP,i for any given enrich- ment channel P and isotope i, and a chain rule element emerging from the change of variables from M∗ to t′12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The negative sign emerges due to dτ/dt′ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The integral is computed as a function of stellar birth time t′, where t′ = t − τ(M∗, Z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Elsewhere in this article written as τM∗,Z∗, the lifetime is the function described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 that returns the span of existence in Gyr of a star of mass M∗ and metallicity Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Inside the inte- gral, the metallicity of the star Z∗ is reconstructed from the Galaxy metallicity Z(t) via the lifetime of the star of mass M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' So the term dM∗ can be converted to dt′ via the first order derivative of the stellar mass with respect to the stellar lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Integration of the Rates in GalCEM The whole stellar mass range has to be split appropri- ately to isolate only the stars associated with the yield YP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The mass range can be denoted by the stellar lim- its Ml,P and Mu,P for the lower and upper mass limit respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In GalCEM all of the components of the 12 The chain rule reads: dM∗ = dM∗ dt′ dt′ = dM dτ dτ dt′ dt′ = − dM∗(τ∗,Z∗) dτ(M∗,Z∗) dt′, 14 Gjergo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Schematic diagram representing the solution to the integrodifferential GCE equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' A description of the diagram can be found in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='1 integrand are evaluated on a mass grid constructed be- tween t′(Ml,P) and t′(Mu,P) on a grid of size Nk with k = 200 (Portinari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 1998), uniformly distributed in log-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' At each time step and for every process, GalCEM com- putes a grid of quantities starting from the stellar mass, the lifetime, and the birth-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Onto these grid points it evaluates all the functions in the integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' This in- cludes for example the SFR, ψ(t′), which is evaluated with respect to the birth-time: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', the past ψ(t) is in- terpolated in order to reconstruct the SFR at the time of birth of all the stars that die at any given time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 5 is a schematic representation of the grid across which the returned rate integral is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Galaxy age t grows vertically downwards and has a size of n, while stellar mass M∗ is shown horizontally on a grid of length k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' At each time step, there are a series of stored total physical quantities (whose growth is plot- ted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' From right to left there is galaxy age t, total baryonic mass Mtot(t), total gas mass Mgas(t), total stellar mass Mstar(t), SFR(t), metallicity, and fi- nally in dark green is the (Ni, Nn) matrix representing the mass [M⊙] of every isotope as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The boxed line indicates Mi(t) is a matrix instead of a vector like the other quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' On the horizontal axis are the integral components for every enrichment chan- nel, evaluated at every time step and for every isotope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' From bottom to top are the stellar mass M∗, the stellar lifetime, the stellar birth-time, the IMF, the metallicity evaluated at the stellar birth-times, Z(t′), and similarly the SFR, ψ(t′), is also evaluated at the stellar birth- times, and finally the yield interpolations Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' There are two quadrants shaded in blue and pink that represent the SNCC and LIMs enrichment channels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The only independent variables are t and M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' For each time step and for each enrichment channel, an integral must be solved that returns the total yield from all the events that occur at that given time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' On the vertical axis are the time-dependent quantities emerging from the solution of the differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We label a selection of grid points, with a sampling that is not to scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We have also shown plausible lifetimes for the mass grid, evaluated on the metal poor regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' An enrichment channel will activate only when the galaxy time elapsed will be longer than the shortest lifetime of the channel’s stars, and the N = 200 mass grid will be cropped to exclude the lower mass stars which have not died yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The analytical form of the GCE integrands, Fi, have the following form: Fi(t, M∗) = ψ(t − τ(M∗)) × [Yi, φ] (M∗) (20) SNII LIMs V: [Msun] SFR [Msun / yr] SFR Z Z IMF IMF [Gyr] 02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='04 ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='003 ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='07 [Gyr] 60 120 [Msun] M* Mi SFR 003 n 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='8 [Msun [Msun] [Msun][Msun] [Msun][Gyr] /yrl KGalCEM method presentation 15 lifetime_Gyr 3 2 1 0 1 2 metallicity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='25 mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 2 1 0 1 2 lifetime_Gyr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='20 metallicity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='75 mass lifetime_Gyr 0 10 20 30 40 50 60 70 80 metallicity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05 mass 0 20 40 60 80 100 120 10 20 30 40 50 60 70 lifetime_Gyr 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='25 mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='4 d(mass) / d(lifetime_Gyr) 2 1 0 1 2 lifetime_Gyr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='20 metallicity d(mass) / d(lifetime_Gyr) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='34 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='22 mass lifetime_Gyr 0 10 20 30 40 50 60 70 80 metallicity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05 mass 40000 30000 20000 10000 d(mass) / d(lifetime_Gyr) 10 20 30 40 50 60 70 lifetime_Gyr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content="05 metallicity d(mass) / d(lifetime_Gyr) 44800 39200 33600 28000 22400 16800 11200 5600 0 mass MassInterpolant mass by ['lifetime_Gyr', 'metallicity'] Transformed Domain (odd rows) | Original Domain (even rows) Figure 6." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Mass interpolant employed in this work and ex- plained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The even rows represent the origi- nal domain, and the odd rows the transformed domain onto which the interpolation is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The top half of the im- age shows the actual mass interpolant, while the bottom half shows the derivative of the mass with respect to the lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The 3D projections in the first column show the metallicity and lifetime in Gyr for the bottom x-y plane, while mass (or lifetime derivative of the mass for the bottom panels) is shown on the vertical z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The second column displays the respective 2D contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The color-coding of the scatter points is consistent with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' with the change of variables we transform them into: Fi(t, t′) = ψ(t′) × � −dM∗ dτ , Yi, φ � (M∗(t − t′)) , (21) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', the stellar-mass-dependent component is a function dependent on galaxy age, stellar birth-time w(t, t′) = (dM/dτ × φ × Yi) (M∗(t − t′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 metallicity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='25 lifetime_Gyr 2 1 0 1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 mass 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='20 metallicity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='68 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='68 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='24 lifetime_Gyr mass 0 20 40 60 80 100 120 metallicity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05 lifetime_Gyr 0 20 40 60 80 20 40 60 80 100 120 mass 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05 metallicity 0 9 18 27 36 45 54 63 72 81 lifetime_Gyr mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 metallicity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='25 lifetime_Gyr 2 1 0 1 d(lifetime_Gyr) / d(mass) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 mass 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='20 metallicity d(lifetime_Gyr) / d(mass) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='68 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='68 lifetime_Gyr mass 0 20 40 60 80 100 120 metallicity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05 lifetime_Gyr 0 250 500 750 1000 1250 1500 1750 d(lifetime_Gyr) / d(mass) 20 40 60 80 100 120 mass 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content="05 metallicity d(lifetime_Gyr) / d(mass) 0 240 480 720 960 1200 1440 1680 1920 lifetime_Gyr LifetimeInterpolant lifetime_Gyr by ['mass', 'metallicity'] Transformed Domain (odd rows) | Original Domain (even rows) Figure 7." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Lifetime interpolant employed in this work and explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' It follows the same structure as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 6, but for τ(M∗, Z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', the even rows represent the original domain, and the odd rows the transformed domain onto which the interpolation is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The top half of the image shows the lifetime interpolant, while the bottom half shows the derivative of the lifetime with respect to the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The 3D projections in the first column show the metallicity and lifetime in Gyr for the bottom x-y plane, while mass is shown on the vertical z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In the second column are the respective 2D contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The color-coding of the scatter points is consistent with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Having mapped appropriately each item in this inte- grand to its appropriate grid point as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 5, the integrand is simply given by the following product: F (t, t′) = ψ (t′) w(t, t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (22) 16 Gjergo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' metallicity 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='2 yield 6 5 4 3 2 1 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 metallicity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 mass 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='00 yield metallicity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0175 mass 20 40 60 80 100 120 yield 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0175 metallicity 20 40 60 80 100 120 mass 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='4 yield lc18_z8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='a16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='irv0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content="O16 yield by ['metallicity', 'mass'] Transformed Domain (odd rows) | Original Domain (even rows) Figure 8." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Example of interpolating yield tabulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The current plot refers to 16O for the massive yields by Limongi & Chieffi (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The two plots on the bottom show the fit to the raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' On the top is shown the interpolation on the transformed domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' This is the preprocessed interpolation curve which is read inside the SNCC rate integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' An equivalent computation for LIMs is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' So any convenient integration method compatible with the Volterra Equations is capable of dealing with the integrand F (t, t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We apply the Simpson’s rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Solving the Differential Equation in GalCEM —At each time step, the integrals of each isotope and each enrichment channel is summed into a single quantitity to be added to the total gas mass component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' To solve the differential equation we simply solve a classic fourth-order Runge- Kutta method to the total galaxy quantities, namely SFR, stellar mass, and gas mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' At this stage also the metallicity is updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Currently, GalCEM runs on a uniform time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The output of the runs presented in this paper results form a ∆t = 2 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Adaptive time steps will be tested in the future to improve the computation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Summary of the GalCEM framework With the yield selection outlined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='6, GalCEM runs on 451 isotopes i for 86 chemical elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' GalCEM method presentation 17 metallicity 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 mass 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='8 yield 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 metallicity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='8 mass 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='56 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='84 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='60 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='36 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='88 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='40 yield metallicity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0200 mass 2 3 4 5 6 yield 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0200 metallicity 2 3 4 5 6 mass 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0294 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0336 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0378 yield c15_z8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='a16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='irv0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content="O16 yield by ['metallicity', 'mass'] Transformed Domain (odd rows) | Original Domain (even rows) Figure 9." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Example of interpolating yield tabulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The current plot refers to 16O for the LIMs yields by Cristallo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Similarly to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 8, the bottom plot shows the interpolation on the raw data while the top plot shows the interpolation on the transformed domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' This latter curve is the preprocessed interpolation which is read inside the LIMs rate integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The variable t represents the age of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' By solving for Mi,gas(t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 18, the goal is to obtain an (i, t) matrix as output, with each entry representing the mass of every isotope as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 19 is, for each enrichment channel, a system of equa- tions of size i, computed at every time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' ˙Minf(t) is Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 3, fully and uniquely defined by the parameter τinf, chosen at the start of the run, therefore we com- pute both ˙Minf(t), the infall rate, and Minf(t), the total mass of the system at t at the setup stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' ˙M∗,tot(t) is the SFR defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' It depends on ˙Minf(t) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 3) and on Mgas(t) = � i Mi,gas(t) (that we are solving for).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' we save both the one-dimentional vector of length t for ψ(t) = ˙M∗,tot(t), the SFR, and M∗,tot(t), the total stellar mass of the system at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' There is not a single lifetime relation τZ(M∗), in Portinari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (1998) there are 4, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 7, depending on the initial metallicity of the star (Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='02, 8×10−3, 4×10−4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' These functions are inter- 18 Gjergo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 0 5 10 Age [Gyr] 106 107 108 109 1010 1011 Masses [M ] Mstar Mgas Mg, tot, i MH, g MZ, g Mtot Mg + Ms MHe, g Mgal, f 0 5 10 Age [Gyr] 10 3 10 2 10 1 100 101 102 Rates [M /yr] RSNII RSNIa RLIMs Infall SFR 10 3 10 2 10 1 100 101 102 SFRMW CP11 RSNII, MW M05 RSNIa, MW M05 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Evolution of the global quantities in a GCE solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The figure on the left represents mass quantities, while the figure on the right represents rate quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Mtot is the total time-dependent baryonic mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Mstar is the total stellar mass, or M∗,tot in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Mgas is the total gas mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The sum of total stellar and total gas mass, Mgas + M∗,tot as a sanity check coincides with Mtot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Mg,tot,i is the total gas mass as inferred from the isotopic evolution output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' As another sanity check, it should coincide with Mgas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' MH,g, MHe,g, and MZ,g are the time-dependent masses of H, He, and all metals, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Mgal,f represents the final baryonic mass of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' When it comes to the rates, the infall is shown in the black solid line, the SFR is in the dashed orange yellow line, and the dotted dark blue, light blue, and magenta lines represent the SNCC, SNIa and LIMs rate respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The present-day Milky Way estimates of the rates are shown with the vertical segments at time 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='7 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The SFR is taken from CP11, Chomiuk & Povich (2011), while the SNCC and SNIa rates come from M05, Mannucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' polated and extrapolated in accordance with the method described in the following section, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The convolved integral is given by the product of two functions: the SFR as a function of birth time t′, and a product of quantities that depend on the lifetime and metallicity-dependent stellar mass M∗(t − t′ Zt′ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' GalCEM interpolation tool Solving detailed GCE equations requires interpolating at several stages: the yield tabulations must be interpo- lated over mass, metallicity, and occasionally other pa- rameters such as initial rotational velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Inside the integrals, also stellar lifetime, metallicity, and SFR need to be interpolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Lastly, an interpolation is required for the derivative of the stellar mass with respect to stellar lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Yields, mass, and derivatives can be preprocessed with the GalCemInterpolant class in the yield interpolation package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' With the aim of computing the derivative of the stel- lar mass with respect to its lifetime, we derive the fits as they appear in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 6 where lifetimes and metal- licities are interpolated to obtain stellar masses, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 7 where masses and lifetimes are interpolated to obtain lifetimes13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We implement the SciPy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2020) BivariateSpline interpolation which, while not providing as good of a fit like other methods such as LinearNDInterpolator and NearestNDInterpolator, is smooth so it supports taking derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Bivariate splines are ideal methods for the solution of boundary- value problems by finite-element-type methods because they consist of piecewise polynomials triangulated on a polygonal domain (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', N¨urnberger & Zeilfelder 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The even rows of both Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 6 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 7 are shown in the transformed domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The transformations im- plemented by the interpolator are: logarithmic mass, square root metallicity, and logarithmic lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The odd rows depicts the original domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The transforma- tion was necessary to ensure stable fits, especially with the derivatives and specifically the dτ/dM∗ fit which we employ in the returned rate integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 8 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 9 are the interpolation surfaces com- puted for SNCC and LIMs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Both the orig- 13 The interpolation code can be found in https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' com/egjergo/GalCEM/tree/main/yield interpolation/ lifetime mass metallicity/main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='py GalCEM method presentation 19 0 2 4 6 8 10 12 Galaxy Age [Gyr] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 metallicity Z Age linear fit on [M/H] Silva Aguirre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2018) 0 2 4 6 8 10 12 Galaxy Age [Gyr] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 [Fe/H] [Fe/H] Age linear fit on [Fe/H] Silva Aguirre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2018) Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The age-metallicity relation with respect to the metallicity (upper figure) and iron abundance (lower figure) normalized to solar values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The orange lines cross at the solar age and abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The observational scatter comes from the APOKASC sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Stellar age and abundances are taken from Silva Aguirre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2018), while the iron abundance is taken from Pinsonneault et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' inal and the transformed domain are shown in order to display the fit comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The statistics for the good- ness of the fit are reported in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Unlike Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 6 and 7, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 8 and 9 are obtained with a combination of LinearNDInterpolator, NearestNDInterpolator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' These two methods together give better recovery of known val- ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' However, the models are less smooth, especially outside the hull where the derivative is 0 almost ev- erywhere and undefined at points equidistant between neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Bivariate splines do not recover y values at fitted x values while the linear nd interpolator, the near- est neighbor does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' This comes at the cost of lacking smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The linear nd interpolator makes predic- tions for points inside the hull and the nearest neigh- bor outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The errors in Listings 1 and 2 in the Ap- pendix are computed with the data used to fit models (in-sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The original domain plots shows predic- tions from the model fit in the transformed domain i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' the models are not separate, rather they are viewed in different scalings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' RESULTS: GALCEM APPLICATION TO A MILKY WAY-LIKE ONE-ZONE GALAXY We next present our results for a galaxy of final bary- onic mass Mtot = 5×1010M⊙, that forms from an expo- nentially decaying gas infall with an infall timescale of 7 Gyr, no outflow, a Kennicutt (1998) SFR with κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', an invariant canonical IMF (Kroupa 2001), convolved en- richment from AGB winds and core collapse SNae with yields by Cristallo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2011, 2015), and Limongi & Chieffi (2018), respectively, and SNIa enrichment with yields by Iwamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (1999) and a delay-time distri- bution from Greggio (2005), specifically the novel single- degenerate model with ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We first present the results from the global time- dependent quantities in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 10, namely the time- dependent evolution of total baryonic galaxy mass, to- tal gas mass, total stellar mass, metallicity, hydrogen and helium mass on the left-hand side;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' meanwhile in- fall, SFR, and enrichment channel rates (SNCC, SNIa, and LIMs) are on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The evolution is tracked linearly with time expressed in Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The trends of the absolute mass quantities mirrors the slope of the SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' This is owed to the choice of linear SFR law as illustrated in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The mass in metals is a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='05 smaller than the gas mass, which is consistent with super-solar metallicities measured in young stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The total stellar mass to gas mass fraction tends at the present time to a value nearly 4 times larger than the fiducial ≈ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 11 we show the evolution of the metallicity and iron abundance as a linear function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The scatter data are taken from the stellar ages in Silva Aguirre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2018), the ID of the KIC stars is matched to the data from Pinsonneault et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2014) to get the iron abun- dance of said stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The red lines in each plot represents a linear fit to the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The black curve and blue curve are the iron abundance and metallicity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The model is in fairly good agreement with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' It reproduces the solar abundances at the so- lar age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Given that the iron abundance is dominated by SNIa enrichment, we notice that [Fe/H] comes at a delay compared to the remaining metallicity (primarily oxygen) coming from SNCC and LIMs, a delay which is consistent with the SNIa rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 12 is the central result of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We restrict our analysis to an atomic number of 30 with zinc, be- cause heavier elements require r-process enrichment – to be explored in a follow-up paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Visible in the figure 20 Gjergo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='29Cu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='[Fe/H] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='30Zn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='31Ga ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='32Ge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' [X/Fe]–[Fe/H] relation abundance plots for the elements up to atomic number 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' To facilitate the navigation of the table, the atomic number is shown right before the element symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The total abundance in the one-zone run is enriched by BBN (Galli & Palla 2013), SNIa (Iwamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 1999), AGB/LIMs (Cristallo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2015, with zero initial rotational velocity), and SNCC (Limongi & Chieffi 2018, set R, with zero initial rotational velocity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The observational data scatter is labeled according to the legend on top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' are also 31Ga, and 32Ge, enriched by the s-process only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 12 shows a summary of the evolution of the [X/Fe]- [Fe/H] relation for all the elements tracked in the current paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The model is generally in good agreement with the data, and it is consistent with other literature results constructed on similar one-zone modeling assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Therefore we have reached our benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We stress the fact that the breath of stellar abun- dance patterns observed in the Milky Way, as well as in other galaxies, spans a rich breath of composi- tions and histories that cannot be reproduced by a one- zone model alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We also point out at the fact that the [X/Fe]-[Fe/H] relation is only a proxy for metallic- ity evolution in the Galactic disk, and that the rela- tion between iron content and metallicity breaks down for the metal poor stars found e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' in the halo (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', Matteucci 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Prantzos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' No one-zone model is able, without treatments on dynamics or mul- tiple infall episodes, to reproduce the average patterns for all elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Nonetheless, the one-zone approach to GCE offers precious and reliable constraints when multiple abundance patterns are considered simultane- ously – chiefly a congruent comparison of the different timescales and rates associated with each enrichment channel, and how these inform the star formation his- tory in a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We briefly explain the observed patterns in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 12, and we defer an in-depth analysis for each element to fu- GalCEM method presentation 21 ture works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Intermediate-mass elements are commonly grouped as follows: the CNO nuclei, the α elements, the odd-Z elements, and the iron-peak elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Among these, the α elements (C, O, Ne, Mg, Si, S, Ar, Ca) are the best reproduced in literature, displaying the typical [α/Fe] plateau at the lowest metallicities, followed by a mild decrease in the disk metallicity regime (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', Chi- appini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Romano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' This is also the case for GalCEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We note however that C is often omitted from the α elements analysis due to its flatter [Fe/H] dependence compared to the other elements in this group (Prantzos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The dominant carbon and oxygen isotopes are 12C and 16O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' They are produced both during the H burning by the CNO cycle, but they are also the most abun- dant species produced during the He-burning through the 3α process (Wallerstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The debate is still ongoing on what is the exact breakdown among the known astrophysical sources of enrichment, but GalCEM is in agreement with Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2017) and Prant- zos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2018) on the fact that massive stars produce a larger quantity of C and O compared to AGB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Nitrogen and fluorine in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 12 behave like secondary elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Historically, this has been an issue, particu- larly for nitrogen, because its observational pattern more closely resembles a primary element, while yield compu- tations used to suggest a secondary behavior (like the one seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The introduction of yields by low-metallicity, rapidly rotating massive stars (Meynet & Maeder 2002) solves this issue neatly (Limongi & Chi- effi 2018) and will be a subject of further investigation with our code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The iron-peak elements proper (Cr, Mn, Co, and Ni) are predominantly synthesized by SNIa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 13 shows that, in fact, these are the isotopes where the SNIa returned mass approaches most closely the returned masses by the other two enrichment channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' A list of the observational papers used, and the el- ement that the observations provide, can be found in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The ever-expanding GalCEM library allows the user to define a list of elements and get a similar printout of observational references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 13 plots the time-dependent returned mass rates (in units of [M⊙/yr]) for each enrichment channel in- cluded in the one-zone run presented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' They consist of BBN, SNIa, LIMs, and SNCC as outlined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' To improve readability we only show the first 120 isotopes, even though the full simulation computes 451 species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' This cut includes every zinc nuclide (atomic number of 30);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', we include all the elements of inter- est from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Within the first Gyr, all the returned rates normalize to a given rate, pointing to the fact that the most variation in chemical evolution models occurs within the first few hundred million years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' DISCUSSION The present article has presented the features and ra- tionale behind a new modular publicly available GCE code that computes using efficient numerical methods the convolved integrodifferential equations of chemical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' A GalCEM hallmark is that it computes the enrich- ment of the full set of individual isotopes from multiple enrichment channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' It is sufficient to define a list with the processes one wishes to include in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' By simply defining flags in the input class, it is possi- ble to switch yield tabulations from a preprocessed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The user may run a script to preprocess custom yields, or they may choose from a library of popular yields al- ready analyzed by the GalCEM team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' GalCEM is suitable for studies that involve the simulta- neous analysis of multiple (or all) isotopes and elements in given runs, because the code automatically generates the list of unique isotopes included in the yield tabu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' GalCEM contains a preprocessing interpolation tool that generates a multidimensional interpolation to the input yield tables for each isotope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The number of dimensions in the present work is limited to 3, namely yield, mass, and metallicity – but the tool can accommo- date extra dimensions such as initial rotational velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The interpolation tool may also be adapted to handle stellar lifetime tabulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' GalCEM allows the user to solve the full one-zone con- volution integral (Matteucci & Greggio 1986) for multi- ple channels (AGBs, SNCC, and SNIa on this first re- lease) without resorting to popular approximations such as IMF-averaged yields or instantaneous recycling ap- proximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We map the integrand quantities onto consistent array grids in order to apply the Simpson’s rule and therefore solve the integrals for each isotope and each enrichment channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The differential equa- tion is solved with a classic fourth-order Runge-Kutta method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Parameter dependence and algorithm speeds will be analyzed in a third paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 12 we com- pare GalCEM’s abundance patterns with Galactic data of main-sequence stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Our results are consistent with the evolution of all the intermediate elements from carbon to zinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' A library of observational data will routinely be updated on GalCEM14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We will provide an analysis of heavier elements in a second paper, where we will explore r-process candidate sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 14 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='com/egjergo/GalCEM/tree/main/galcem/input/ observations/abund 22 Gjergo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' List of observations included in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The element investigated in each paper is marked with an ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The processed public data is available for download at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='com/egjergo/GalCEM/tree/main/galcem/input/observations/abund Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Adibekyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2012) ⃝ ⃝ ⃝ ⃝ ⃝ ⃝ ⃝ ⃝ × × × × ⃝ ⃝ ⃝ ⃝ ⃝ × × × × × × × × × ⃝ ⃝ Akerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2004) ⃝ ⃝ ⃝ × ⃝ × ⃝ ⃝ ⃝ ⃝ ⃝ ⃝ ⃝ ⃝ ⃝ ⃝ ⃝ ⃝ ⃝ ⃝ ⃝ ⃝ ⃝ × ⃝ ⃝ ⃝ ⃝ Andrievsky et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Loglog time-dependent returned mass rates (in units of M⊙/yr) for each enrichment channel included in the one- zone run presented in this work – which include BBN, SNIa, LIMs, and SNCC as outlined in the Methods, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The number preceding the atomic symbol represents the atomic mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' To improve readability, we only show the first 120 isotopes, even though the full simulation computes 451 species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' This cut includes every zinc nuclide (atomic number of 30), consistent with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The full table will be shown in Gjergo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' (2022b) in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', with the inclusion of r-process enrichment channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 24 Gjergo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Software: A current version of the code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='com/egjergo/GalCEM/releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' The re- sults of this article can be reproduced with the GalCEM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='0 package release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' Archives of yield processing and re- sults compilations can be found in the organization page: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='com/GalCEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We are grateful for the thorough feedback provided by the reviewer, which has helped to significantly improve the quality of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We thank Sergio Cristallo for clarifications on how to use the F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We thank Kai Diethelm for crucial feedback early in the development of the numerical solutions in GalCEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We thank Steve Kuhlmann for providing helpful feedback on the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' designed and wrote GalCEM and its contents in the GitHub repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' prepared the ar- ticle, and coordinated the research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' developed the preprocessing yield interpolation tool, helped with refactoring GalCEM, and set up the release of GalCEM on the Python Package Index (PyPI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' provided feed- back on code design, numerical methods, efficiency, and convergence tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' hosts the JupyterHub server at https://galcem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='space/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' guided the early interpre- tation of the results, clarified relevant issues concerning stellar evolution, and shared resources on how to han- dle the yield gap in the 6-13M⊙ mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' provided invaluable guidance on GCE theory and history, as well as on the interpretation of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' offered feedback on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='G.’s prelim- inary tests that matured into GalCEM, and hence influ- enced the early code design choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' gathered and cleaned the observational data on stellar abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' promoted the endeavour to write GalCEM, and was involved with its development at every stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' pro- vided the economic support that made this project pos- sible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' All co-authors helped with interpreting the results and giving feedback on the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' have been supported by the National Natural Science Foun- dation of China under grant (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='11922303) and the Fun- damental Research Funds for the Central Universities (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='2042022kf1182).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' is supported by the Hubei province Natural Science Fund for Distinguished Young Scholars (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 2019CFA052).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' received funding from the European Union’s Horizon 2020 research and in- novation program under SPACE-H2020 grant agree- ment number 101004214 (EXPLORE project).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' is supported by Grants-in-Aid for Scientific Research of Japan Society for the Promotion of Science (20K03958, 17K05459).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' acknowledges the support of the Na- tional Natural Science Foundation of China (NSFC) un- der grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' 12041305, 12173016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We acknowledge the Program for Innovative Talents, Entrepreneur in Jiangsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf'} +page_content=' We 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Arani†1,2 and Bahram Zonooz†1,2 +1Advanced Research Lab, NavInfo Europe, Eindhoven, The Netherlands +2Department of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands +arnav.varma@navinfo.eu, e.arani@tue.nl, bahram.zonooz@gmail.com +† Contributed equally +Keywords: +Dynamic Neural Networks, Policy Gradients, Lifelong Learning. +Abstract: +Real-world applications often require learning continuously from a stream of data under ever-changing con- +ditions. When trying to learn from such non-stationary data, deep neural networks (DNNs) undergo catas- +trophic forgetting of previously learned information. Among the common approaches to avoid catastrophic +forgetting, rehearsal-based methods have proven effective. However, they are still prone to forgetting due to +task-interference as all parameters respond to all tasks. To counter this, we take inspiration from sparse coding +in the brain and introduce dynamic modularity and sparsity (Dynamos) for rehearsal-based general continual +learning. In this setup, the DNN learns to respond to stimuli by activating relevant subsets of neurons. We +demonstrate the effectiveness of Dynamos on multiple datasets under challenging continual learning evalu- +ation protocols. Finally, we show that our method learns representations that are modular and specialized, +while maintaining reusability by activating subsets of neurons with overlaps corresponding to the similarity of +stimuli. The code is available at https://github.com/NeurAI-Lab/DynamicContinualLearning. +1 +INTRODUCTION +Deep neural networks (DNNs) have achieved human- +level performance in several applications (Greenwald +et al., 2021; Taigman et al., 2014). These networks +are trained on the multiple tasks within an appli- +cation with the data being received under an inde- +pendent and identically distributed (i.i.d.) assump- +tion. +This assumption is satisfied by shuffling the +data from all tasks and balancing and normalizing +the samples from each task in the application (Had- +sell et al., 2020). Consequently, DNNs can achieve +human-level performance on all tasks in these appli- +cations by modeling the joint distribution of the data +as a stationary process. Humans, on the other hand, +can model the world from inherently non-stationary +and sequential observations (French, 1999). Learn- +ing continually from the more realistic sequential and +non-stationary data is crucial for many applications +such as lifelong learning robots (Thrun and Mitchell, +1995) and self-driving cars (Nose et al., 2019). How- +ever, vanilla gradient-based training for such contin- +ual learning setups with a continuous stream of tasks +and data leads to task interference in the DNN’s pa- +rameters, and consequently, catastrophic forgetting on +old tasks (McCloskey and Cohen, 1989; Kirkpatrick +et al., 2017). Therefore, there is a need for methods to +alleviate catastrophic forgetting in continual learning. +Previous works have aimed to address these chal- +lenges in continual learning. These can be broadly +classified into three categories. First, regularization- +based methods (Kirkpatrick et al., 2017; Schwarz +et al., 2018; Zenke et al., 2017) that penalize changes +to the parameters of DNNs to reduce task interfer- +ence. +Second, parameter isolation methods (Adel +et al., 2020) that assign distinct subsets of parame- +ters to different tasks. Finally, rehearsal-based meth- +ods (Chaudhry et al., 2019) that co-train on cur- +rent and stored previous samples. +Among these, +regularization-based and parameter isolation-based +methods often require additional information (such as +task-identity at test time and task-boundaries during +training), or unconstrained growth of networks. These +requirements fail to meet general continual learning +(GCL) desiderata (Delange et al., 2021; Farquhar +and Gal, 2018), making these methods unsuitable for +GCL. +Although rehearsal-based methods improve over +other categories and meet GCL desiderata, they still +suffer from catastrophic forgetting through task in- +terference in the DNN parameters, as all parameters +respond to all examples and tasks. This could be re- +arXiv:2301.00620v1 [cs.CV] 2 Jan 2023 + +solved by inculcating task or example specific param- +eter isolation in the rehearsal-based methods. How- +ever, it is worth noting that unlike parameter isolation +methods, modularity and sparsity in the brain is not +static. There is evidence that the brain responds to +stimuli in a dynamic and sparse manner, with differ- +ent modules or subsets of neurons responding ”dy- +namically” to different stimuli (Graham and Field, +2006). The advantages of a dynamic and sparse re- +sponse to stimuli have been explored in deep learn- +ing in stationary settings through mechanisms such as +gating of modules (Veit and Belongie, 2018), early- +exiting (Li et al., 2017; Hu et al., 2020), and dy- +namic routing (Wang et al., 2018), along with train- +ing losses that incentivize sparsity of neural activa- +tions (Wu et al., 2018). These studies observed that +DNNs trained to predict dynamically also learn to +respond differently to different inputs. Furthermore, +the learned DNNs demonstrate clustering of parame- +ters in terms of tasks such as similarity, difficulty, and +resolution of inputs (Wang et al., 2018; Veit and Be- +longie, 2018), indicating dynamic modularity. Hence, +we hypothesize that combining rehearsal-based meth- +ods with dynamic sparsity and modularity could help +further mitigate catastrophic forgetting in a more bi- +ologically plausible fashion while adhering to GCL +desiderata. +To this end, we propose Dynamic Modularity and +Sparsity (Dynamos), a general continual learning al- +gorithm that combines rehearsal-based methods with +dynamic modularity and sparsity. Concretely, we seek +to achieve three objectives: dynamic and sparse re- +sponse to inputs with specialized modules, compe- +tent performance, and reducing catastrophic forget- +ting. +To achieve dynamic and sparse responses to +inputs, we define multiple agents in our DNN, each +responsible for dynamically zeroing out filter acti- +vations of a convolutional layer based on the input +to that layer. The agents are rewarded for choosing +actions that remove activations (sparse responses) if +the network predictions are accurate, but are penal- +ized heavily for choosing actions that lead to inaccu- +rate predictions. Agents also rely on prototype losses +to learn specialized features. +To reduce forgetting +and achieve competent performance, we maintain a +constant-size memory buffer in which we store pre- +viously seen examples. The network is retrained on +previous examples alongside current examples to both +maintain performance on current and previous tasks, +as well as to enforce consistency between current +and previous responses to stimuli. Dynamos demon- +strates competent performance on multiple continual +learning datasets under multiple evaluation protocols, +including general continual learning. +Additionally, +our method demonstrates similar and overlapping re- +sponses for similar inputs and disparate responses for +dissimilar inputs. Finally, we demonstrate that our +method can simulate the trial-to-trial variability ob- +served in humans (Faisal et al., 2008; Werner and +Mountcastle, 1963). +2 +RELATED WORK +Research in deep learning has approached the dy- +namic compositionality and sparsity observed in +the human brain through dynamic neural networks, +where different subsets of neurons or different sub- +networks are activated for different stimuli (Bengio +et al., 2015; Bolukbasi et al., 2017). +This can be +achieved through early exiting (Hu et al., 2020), dy- +namic routing through mixtures of experts or multi- +ple branches (Collier et al., 2020; Wang et al., 2022), +and through gating of modules (Wang et al., 2018). +Early-exiting might force the DNN to learn specific +features in its earlier layers and consequently hurt +performance (Wu et al., 2018) as the earlier layers +of DNNs are known to learn general purpose fea- +tures (Yosinski et al., 2014). Dynamic routing, on +the other hand, would require the growth of new ex- +perts in response to new tasks that risk unconstrained +growth, or the initialization of a larger DNN with +branches corresponding to the expected number of +tasks (Chen et al., 2020). Dynamic networks with +gating mechanisms, meanwhile, have been shown +to achieve competent performance in i.i.d. training +with standard DNNs embedded with small gating net- +works (Veit and Belongie, 2018; Wu et al., 2018; +Wang et al., 2018). These gating networks emit a dis- +crete keep/drop decision for each module, depending +on the input to the module or the DNN. As this opera- +tion is non-differentiable, a Gumbel Softmax approx- +imation (Veit and Belongie, 2018; Wang et al., 2018), +or an agent trained with policy gradients (Wu et al., +2018; Sutton and Barto, 2018) is commonly used in +each module to enable backpropagation. However, +unlike the latter, the Gumbel-Softmax approximation +induces an asymmetry between the forward pass acti- +vations at inference and training (Wang et al., 2018). +Furthermore, these methods are not applicable to con- +tinual learning. +Recent works have attempted to build dynamic +networks for continual learning setups (Chen et al., +2020; Abati et al., 2020), where data arrive in a more +realistic sequential manner. InstAParam (Chen et al., +2020), Random Path Selection (RPS) (Rajasegaran +et al., 2019), and MoE (Collier et al., 2020) start +with multiple parallel blocks at each layer, finding + +input-specific or task-specific paths within this large +network. +Nevertheless, this requires knowledge of +the number of tasks to be learned ahead of training. +More importantly, initializing a large network might +be unnecessary as indicated by the competent perfor- +mance of dynamic networks with gating mechanisms +in i.i.d training. +In contrast to this, MNTDP (Ve- +niat et al., 2021), LMC (Ostapenko et al., 2021), and +CCGN (Abati et al., 2020) start with a standard archi- +tecture and grow units to respond to new data or tasks. +Of these, MNTDP and LMC develop task-specific +networks where all inputs from the same task elicit +the same response and therefore do not show a truly +dynamic response to stimuli. CCGN, however, com- +poses convolutional filters dynamically to respond to +stimuli, using a task-specific vector for every convo- +lutional filter, and task boundaries to freeze frequently +active filters. +However, this leads to unrestrained +growth and fails in the absence of task-boundaries, +which makes it unsuitable for general continual learn- +ing. +Therefore, we propose a general continual learn- +ing method with dynamic modularity and sparsity +(Dynamos) induced through reinforcement learning +agents trained with policy gradients. +3 +METHODOLOGY +Humans learn continually from inherently non- +stationary and sequential observations of the world +without catastrophic forgetting, even without super- +vision about tasks to be performed or the arrival of +new tasks, maintaining a bounded memory through- +out. This involves, among other things, making multi- +scale associations between current and previous ob- +servations (Goyal and Bengio, 2020) and respond- +ing sparsely and dynamically to stimuli (Graham and +Field, 2006). The former concerns consolidation of +previous experiences and ensuring that learned ex- +periences evoke a similar response. The latter con- +cern dynamically composing a subset of the special- +ized neural modules available to respond to stimuli, +reusing only the relevant previously learned informa- +tion. This also avoids erasure of information irrele- +vant to current stimuli but relevant to previous experi- +ences. We now formulate an approach for dynamic +sparse and modular general continual learning that +mimics these procedures with DNNs. +3.1 +Dynamic, Modular, and Sparse +response to stimuli +To achieve a dynamic, modular, and sparse response +to inputs, we use a DNN F with a policy to compose +a subset of the available modules in each layer to re- +spond to the input to that layer. More specifically, we +use a CNN which is incentivized to drop some chan- +nels in its activations adaptively using policy gradi- +ents (Sutton and Barto, 2018; Williams, 1992). +Let us consider the lth convolutional layer with +cl output channels ∀l ∈ {1,2,...L}, where L is the +total number of convolutional layers in the network. +The input to the convolutional layer is processed us- +ing an agent module with actions al ∈ {0,1}cl as out- +put, where each action represents the decision to drop +(action = 0) or keep (action = 1) the corresponding +channel of the output of the convolutional layer. The +agent module uses a self-attention network to obtain +a channel-wise attention vector vl of dimension cl, +which is converted into ”action probabilities” using +a probability layer. The policy for choosing actions is +then sampled from a cl-dimensional Bernoulli distri- +bution; +pl = σ(vl) +πl(al) = +cl +∏ +i=1 +p +al,i +l,i (1− pl,i)(1−al,i), +(1) +where pl ∈ (0,1)cl is the output of the probability +layer σ, and πl is the policy function. The final output +of the convolutional layer is the channel-wise product +of the actions with the output of the convolution. This +policy formulation is used at each convolutional layer +in the CNN, leading to L agents in total. The over- +all structure of an agent for a convolutional layer is +shown in Figure 1. +These agents are rewarded for dropping channels +while making accurate predictions through a reward +function. For an input to the DNN X applied to clas- +sification with label Y: +Z,V = F(X), V = [v1|v2,...|vL] +ˆY = argmaxZ, +(2) +where Z refers to the logits. Now, the ratio of activa- +tions or channels that were retained in the layer l is +determined by 1 +cl ∑cl +i=1 al,i. So, for a target activation +retention rate per layer or ”keep ratio” kr, the reward +function is as follows: +Rl(X,Y) = +� +−(kr − 1 +cl ∑cl +i=1 al,i)2, +if ˆY = Y +−λ(kr − 1 +cl ∑cl +i=1 al,i)2, +otherwise. +(3) + +Bernoulli +Sampling +Probability +Layer +Convolutional Layer with cl filters +Action +Probabilities +Actions: +Channel-wise +Attention Vector +Input +Batch +Normalization +Global +Average +Pooling +Point-wise +Convolution +FC +FC +Self-Attention Network +0 +1 +1 +10 +1 +Output with channels removed +Figure 1: An overview of Dynamos’ dynamic and sparse response mechanism at the lth convolutional layer. Blacked acti- +vations are removed. The agent (bottom path) self-attention network uses a pointwise convolution to match output channels +and global average pooling to get a channel-length flattened vector. This is sent through an MLP with one hidden layer +and Sigmoid activation, and multiplied with the original channel-length representation to get the channel-wise self-attention +vector. +Therefore, when the DNN’s predictions are correct, +each agent is rewarded for dropping enough activa- +tions to match the ”keep ratio” from its correspond- +ing convolutional layer. However, when the predic- +tion is incorrect, each agent is penalized for the same, +scaled by a constant penalty factor λ. The global na- +ture of the reward function, achieved through depen- +dence on the correctness of the prediction, also en- +forces coordination between agents. Following RE- +INFORCE (Williams, 1992), the loss from all agents +t = 1,2,...L is: +LR(X,Y) = ElEπ[−Rl(X,Y)logπl(al)] += ElEπ[−Rl(X,Y)log +cl +∏ +i=1 +pl,ial,i ++(1− pl,i)(1−al,i)] += ElEπ[−Rl(X,Y) +cl +∑ +i=1 +log[pl,ial,i ++(1− pl,i)(1−al,i)]]. +(4) +Although the agents along with this loss ensure +sparse and dynamic responses from the DNN, they +do not explicitly impose any specialization of com- +positional neural modules seen in humans. As the +channel-wise ”modules” activated in the DNN are di- +rectly dependent on the channel-wise attention vec- +tors, we finally apply a specialization loss that we call +prototype loss to them. Concretely, for classification, +in any batch of inputs, we pull the vectors belonging +to the same class together while pushing those from +different classes away. +This would cause different +subsets of channel-wise modules to be used for in- +puts of different classes. When combined with a suf- +ficiently high ”keep ratio”, this will encourage overlap +and therefore, reuse of relevant previously learned in- +formation (for example, reusing channels correspond- +ing to a learned class for a newly observed class) and, +consequently, learning of general-purpose features by +the modules. For an input batch X with correspond- +ing labels Y, and the corresponding batch of concate- +nated channel-wise attention vectors V (Equation 2), +the prototype loss is given by: +LP(X,Y) = +1+Σ(V1,V2)∈V 2:Y1=Y2MSE(V1,V2) +1+Σ(V1,V2)∈V 2,Y1̸=Y2MSE(V1,V2), +(5) +where MSE refers to the Mean Squared Error estima- +tor. Note that we only apply this loss to samples for +which the predictions were correct. +3.2 +Multi-Scale associations +As discussed earlier, one of the mechanisms em- +ployed by humans to mitigate forgetting is multi- +scale associations between current and previous ex- +periences. +With this goal in mind, we follow recent rehearsal- +based approaches (Buzzega et al., 2020; Riemer et al., +2019) that comply with GCL and use a memory buffer +during training to store previously seen examples and +responses. The buffer is updated using reservoir sam- +pling (Vitter, 1985), which helps to approximate the +distribution of the samples seen so far (Isele and Cos- +gun, 2018). However, we only consider the subset +of batch samples on which the prediction was made + +correctly for addition to the memory buffer. These +buffer samples are replayed through the DNN along- +side new samples with losses that associate the current +response with the stored previous response, resulting +in consistent responses over time. +Let M denote the memory buffer and DT de- +note the current task stream, from which we sample +batches (XM,YM,ZM,VM) and (Xt,Yt), respectively. +Here, ZM and VM are the saved logits and channel- +wise attention vectors corresponding to XM when it +was initially observed. The consistency losses associ- +ated with current and previous responses are obtained +during the task T as follows: +Z +′ +M,V +′ +M = F(XM) +LC(ZM,Z +′ +M) = EXM[∥ZM −Z +′ +M∥2 +2] +LC(VM,V +′ +M) = EXM[∥VM −V +′ +M∥2 +2]. +(6) +In addition to consistency losses, we also enforce +accuracy, and dynamic sparsity and modularity on the +memory samples. +Therefore, we have four sets of +losses: +• Task performance loss on current and memory +samples to ensure correctness on current and pre- +vious tasks. +For classification, we use cross- +entropy loss (LCE). +• Reward losses (Equation 4) on current and mem- +ory samples to ensure dynamic modularity and +sparsity on current and previous tasks. +• Prototype losses (Equation 5) on current and +memory samples to ensure the specialization of +modules on current and previous tasks. +• Consistency losses (Equation 6) for multi-scale +associations between current and previous sam- +ples. +Putting everything together, the total loss becomes: +Ltotal = LCE(XB,YB)+γLr(XB) ++β[LCE(XM,YM)+γLr(XM)] ++αLC(ZM,Z +′ +M)+αpLC(VM,V +′ +M) ++wp[LP(XB,YB)+LP(XM,YM)]. +(7) +The weights given to the losses - α, αp, β, wp, +and γ, and the penalty for misclassification (λ) and +keep ratio (kr) in Equation 3, are hyperparameters. +Note that we employ a warm-up stage at the begin- +ning of training, where neither the memory buffer nor +the agents are employed. This is equivalent to train- +ing using only the cross-entropy loss for this period, +while the agents are kept frozen. This gives agents +a better search space when they start searching for a +solution. We call our method as described above Dy- +namic modularity and sparsity - Dynamos. +4 +EXPERIMENT DETAILS +Datasets. +We show results on sequential variants of +MNIST (LeCun et al., 1998) and SVHN: Seq-MNIST +and Seq-SVHN (Netzer et al., 2011), respectively. +Seq-MNIST and Seq-SVHN divide their respective +datasets into 5 tasks, with 2 classes per task. Fur- +thermore, to test the applicability of Dynamos under +general continual learning, we also use the MNIST- +360 dataset (Buzzega et al., 2020). +Architecture. +We use a network based on the +ResNet-18 (He et al., 2016) structure by removing the +later two of its four blocks and reducing the number +of filters per convolutional layer from 64 to 32. The +initial convolution is reduced to 3 × 3 to work with +smaller image sizes. For the baseline experiments, +we did not use any agents. For our method, while +agents can be used for all convolutional layers, we +only use agents in the second block. We make this +choice based on recent studies that observe that ear- +lier layers undergo minimal forgetting (Davari et al., +2022), are highly transferrable (Yosinski et al., 2014), +and are used for most examples even when learned +with dynamic modularity (Abati et al., 2020). We use +a sigmoid with a temperature layer as the probabil- +ity layer in the agents and a probability of 0.5 as a +threshold for picking actions, i.e., channels during in- +ference. The temperature serves the purpose of tuning +the range of outputs of the self-attention layers, en- +suring that the probabilities being sampled to choose +the actions are not too small and that enough activa- +tions are chosen to enable learning. The exact net- +work structure used for each experiment, including +the self-attention networks of the agents, can be found +in Appendix, in Table 3 and Table 4. +Settings. +All methods are implemented in the Mam- +moth repository1 in PyTorch 1.6 and were trained +on Nvidia V100 GPUs. +The hyperparameters cor- +responding to each experiment can be found in Ap- +pendix, Table 5. +We always maintain a keep ra- +tio higher than 1/Num tasks to allow the learning +of overlapping, reusable, and general-purpose mod- +ules. The temperature of the Sigmoid activation of +the probability layers is kept at 0.15 unless mentioned +otherwise. +1https://github.com/aimagelab/mammoth/ + +500 +1000 +2000 +60 +64 +68 +72 +76 +80 +84 +88 +Accuracy +Seq-SVHN +500 +1000 +2000 +Buffer size +90 +92 +94 +96 +98 +100 +Accuracy +Seq-MNIST +Models +CCGN +Dynamos +Figure 2: +Quantitative results under Class-Incremental +Learning protocol. Results are averaged across three seeds. +CCGN values taken from the original paper. The precise +accuracies can be found in Table 2. +5 +RESULTS +We will evaluate Dynamos under two standard eval- +uation protocols that adhere to the core desiderata of +GCL. +5.1 +Class-Incremental Learning (CIL) +Class-incremental learning (CIL) refers to the eval- +uation protocol in which mutually exclusive sets of +classes are presented sequentially to the network, and +the identity of the task is not provided at the test +time, which meets the core desiderata of GCL (Far- +quhar and Gal, 2018). We compare against Condi- +tional Convolutional Gated Network (CCGN) (Abati +et al., 2020), which also dynamically composes con- +volutional filters for continual learning. We observe +in Figure 2 that Dynamos shows higher accuracies +on both the Seq-MNIST and Seq-SVHN datasets un- +der all buffer sizes. +However, CCGN requires a +separate task vector for every task per convolutional +layer, resulting in unrestricted growth during train- +ing, whereas we maintain a bounded memory through +training. Furthermore, unlike CCGN, we do not lever- +age the task boundaries or the validation set during +training. Therefore, Dynamos outperforms the previ- +ous state-of-the-art for dynamic compositional con- +tinual learning in class-incremental learning, while +showing bounded memory consumption during train- +ing. +5.2 +General Continual Learning (GCL) +So far, we have observed Dynamos under the CIL +protocol. Unlike CIL, real-world data streams with- +out clear task boundaries, where the same data may +reappear under different distributions (e.g. different +poses). Following (Buzzega et al., 2020), we approxi- +mate this setting using MNIST-360, where tasks over- +lap in digits (i.e. classes), reappear under different ro- +tations (i.e. distributions), and each example is seen +exactly once during training. This serves as a verifica- +tion of the adherence to the GCL desiderata (Farquhar +and Gal, 2018; Delange et al., 2021). We study the +impact of both dynamic modularity as well as multi- +scale associations by removing them incrementally +from Dynamos. When neither is used, the learning +is done using vanilla gradient-based training, with no +strategy to counter forgetting. When dynamic mod- +ularity is removed, the learning strategy forms our +baseline, where no agents are used, simplifying the +total training loss from Equation 7 to: +Lbase = LCE(XB,YB)+βLCE(XM,YM)+ +αLC(ZM,Z +′ +M). +(8) +Table 1 shows that Dynamos outperforms the baseline +in all buffer sizes, proving that dynamic modularity is +advantageous in GCL. Furthermore, when multi-scale +associations are also removed, no buffer is used, and +the DNN undergoes catastrophic forgetting. +Thus, +Dynamos is applicable to general continual learning, +with dynamic modularity improving over the base- +line. We hypothesize that dynamic modularity makes +dealing with the blurred task boundaries of GCL eas- +ier by adaptively reusing relevant previously learned +information, which in this case corresponds to learned +filters. +6 +MODEL CHARACTERISTICS +We now analyze some of the characteristics and ad- +vantages of Dynamos. For all experiments in this sec- +tion, we use our model trained on Sequential-MNIST +with buffer size 500. +6.1 +Dynamic Modularity and +Compositionality +Humans show modular and specialized responses to +stimuli(Meunier et al., 2010) with dynamic and sparse + +Table 1: General continual learning results for multiple buffer sizes. All results are averaged across five seeds. +Multi-Scale +Associations +Dynamic +Modularity +Buffer Size +100 +200 +500 + + +64.418±4.095 +79.638±2.853 +90.519±0.737 + + +61.192±3.072 +75.364±1.259 +88.150±0.888 + + +18.712±0.690 +Filters +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Class ID +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(a) Classwise +Filters +0 +1 +2 +3 +4 +Task ID +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(b) Taskwise +Figure 3: Filter activation rates on the test set for each filter with respect to tasks and classes. For ease of visualization, we +only look at the last 40 filters. Full visualizations can be found in Appendix (Figure 7). +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Class ID +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Class ID +1.33 +0.13 +1.00 +0.65 +0.71 +0.46 +0.31 +0.67 +0.20 +0.45 +0.11 +1.17 +0.08 +0.50 +0.31 +0.18 +1.87 +0.27 +1.07 +0.50 +0.31 +1.30 +0.04 +0.93 +0.64 +0.59 +1.12 +1.81 +0.17 +1.90 +0.23 +0.94 +0.48 +0.31 +0.05 +1.84 +0.72 +0.32 +0.51 +0.32 +0.25 +0.62 +1.06 +0.24 +1.11 +Figure 4: Jensen-Shanon Divergences (×100) of the activa- +tion rates of class pairs on the test set. +response to inputs (Graham and Field, 2006) - a capa- +bility that we instilled in our DNN while learning a +sequence of tasks by dynamically removing channel +activations of convolutional layers. Therefore, we ex- +amine the task- and class-wise tendencies of the firing +rates of each neuron (filter) in Figure 3. +It can be seen that Dynamos learns a soft separa- +tion of both tasks and classes, as evidenced by the per- +task and per-class firing rates, respectively, of each +filter. +This is in contrast to static methods, where +all filters react to all examples. +Figure 3a further +shows that this allows learning of similar activation +patterns for similar examples. For example, MNIST +digit pairs 1 and 7, and 6 and 8, which share some +shape similarities, also share similarities in their acti- +vation patterns/rates. This could be attributed to be- +ing able to reuse and drop learned filters dynamically, +which causes the DNN to react similarly to similar +inputs, partitioning its responses based on example +similarities. Additionally, the ability to dynamically +reuse filters allows DNNs to learn overlapping acti- +vation patterns for dissimilar examples and classes, +instead of using completely disparate activation pat- +terns. This also facilitates the learning of sequences +of tasks without having to grow the DNN capacity or +having a larger capacity at initialization, as opposed +to the static parameter isolation methods for contin- +ual learning. +Following (Abbasi et al., 2022), we quantify the +overlap between the activation rates for each class pair +in the final layer using the Jensen-Shanon divergence +(JSD) between them in Figure 4. Lower JSDs sig- +nify higher overlap. The JSD is lowest for the class +pair (1,7) (both digits look like vertical lines), and is +∼ +1 +15th the average JSD across class pairs, and ∼ +1 +42th +that of the least overlapping class pair (1,8) (1 is a +line, 8 is formed of loops). Now, as per Equation 1, +filters in the layer are activated based on the channel- +wise attention vector vL (see Equation 2), which are +pushed together for examples of the same classes, and +pushed away from each other for examples of differ- +ent classes using prototype loss (Equation 5). We vi- +sualize the t-SNEs of these vLs on the test set in Fig- + +75 +50 +25 +0 +25 +50 +75 +75 +50 +25 +0 +25 +50 +75 +100 +Class IDs +0.0 +1.0 +2.0 +3.0 +4.0 +5.0 +6.0 +7.0 +8.0 +9.0 +Figure 5: t-SNEs on the test set of class prototypes learned +from channel-wise self-attention vectors for all classes. +ure 5 and observe that the samples belonging to the +same classes are clustered, confirming the effective- +ness of our prototype loss. Moreover, the clusters of +visually similar classes are close together, which is +concomitant with the JSDs and class-wise activation +rates seen earlier. Class similarities are also reflected +through multiple clusters for the digit 9, indicating its +similarity with the digits 6 (loop) and 1 (line) in one +cluster, but also with 7 (line) and 4 (line + loop) in +another cluster. Finally, we observe that there are ex- +amples that are scattered away from their class clus- +ters and overlap with other clusters, probably indicat- +ing that these particular examples are visually closer +to other digits. Note, however, that these similar ex- +amples and classes are distributed across tasks, which +explains the lower similarities in activation patterns +between task pairs in Figure 3b compared to the class +pairs in Figure 3a. +Therefore, Dynamos is capable of learning modu- +lar and specialized units that result in input-adaptive +dynamic separation and overlap of activations, based +on the extent of similarities with previously learned +examples. We also contend that the overlapping acti- +vations for digits of similar shape suggest the learning +of general-purpose features. +6.2 +Trial-to-trial variability +The brain is known to show variability in response +across trials (Faisal et al., 2008; Werner and Mount- +castle, 1963). +For the same stimulus, the precise +neuronal response could differ between trials, a be- +havior absent in most conventional DNNs. +In our +method, this aspect of brains can be mimicked by us- +ing Bernoulli sampling instead of thresholding to pick +keep/drop decisions at each convolutional layer. In +Figure 6, we plot the response variability in the last +convolutional layer of our DNN with the same exam- +ple in four trials. We only pick responses for which +Filters +1 +2 +3 +4 +Trial +0 +1 +Figure 6: Trial-to-trial variability of responses to same input +in Dynamos. +the predictions were correct. It can be seen that each +trial evoked a different response from the DNN. Fur- +thermore, despite the differences, there are also some +similarities in the response. There are some filters that +are repeatedly left unused, as well as some filters that +are used in every trial. This demonstrates that Dy- +namos can additionally simulate the trial-to-trial vari- +ability observed in brains. +7 +CONCLUSION AND FUTURE +WORK +We propose Dynamos, a method for general contin- +ual learning, that simulates the dynamically modu- +lar and sparse response to stimuli observed in the +brain. Dynamos rewards the input-adaptive removal +of channel activations of convolutional layers using +policy gradients for dynamic and sparse responses. To +further induce modularity, channel-wise self-attention +vectors corresponding to each convolutional layer +are pulled together for examples from same classes, +and are pushed apart for examples from different +classes; these vectors are then used to sample the +keep/drop decision for the corresponding channel. +Using a memory buffer, we enforce multi-scale con- +sistency between previous and current responses to +prevent forgetting. +Dynamos outperforms previous +baselines on multiple datasets when evaluated us- +ing class-incremental learning (CIL) and general con- +tinual learning (GCL) protocols. Dynamos exhibits +similar and overlapping responses for similar inputs, +yet distinct responses to dissimilar inputs by utiliz- +ing subsets of learned filters in an adaptive manner. +We quantified the extent of class-wise overlaps and +showed that the semantic similarity of classes (dig- +its in MNIST, e.g. 1 and 7) are reflected in higher +representation overlaps. 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PMLR. +APPENDIX +Table 2: Class-Incremental learning accuracies for CCGN +and Dynamos corresponding to Figure 2. +Buffer +Size +Model +Seq-SVHN +Seq-MNIST +500 +CCGN +67.45 +94.01 +Dynamos +84.815 +97.19 +1000 +CCGN +73.99 +95.94 +Dynamos +87.38 +97.51 +2000 +CCGN +81.02 +95.94 +Dynamos +88.54 +97.57 +Table 3: Agent architecture with input features of shape +B×Cin ×H ×W, output features of shape B×Cout ×1×1, +where B is the batch size, Cin is the number of channels in +the input, and Cout is the number of channels expected in +the output, which is same as the number of keep/drop ac- +tions required for the corresponding convolutional layer. +Operations +Input size +Output size +Pointwise Conv +B×Cin ×h×w +B×Cout ×h×w +Average Pooling +B×Cout ×h×w +B×Cout ×1×1 +Reshape +B×Cout ×1×1 +B×Cout +Linear +B×Cout +B×Cout/16 +ReLU +B×Cout/16 +B×Cout/16 +Linear +B×Cout/16 +B×Cout +Reshape +B×Cout +B×Cout ×1×1 +Sigmoidτ +B×Cout ×1×1 +B×Cout ×1×1 + +Table 4: Architectures used in our experiments. For baseline experiments without dynamic compositionality, we do not use +the ”Agent” branch. Conv(k, n, s, p) refers to convolutional layer with kernel size k, number of filters n, stride s, and padding +p. BN refers to Batch Normalization. Linear(M, N) refers to a linear layer with M-dimensional input and N-dimensional +output. Agent(Cin, Cout, τ) refers to the Agent subnetwork with Cin input channels, Cout output channels, and τ temperature of +the sigmoid in the probability layer (See Figure 1, Section 4). The elementwise multiplication of the actions from the agents +with the output of a convolutional layer is done after the application of batch normalization, if present, but before the ReLU +activation function, if present. For complete description of Agent architecture, refer to Table 3. num classes refers to the +number of classes to be predicted. +Component +Main Branch +Residual branch +Agent branch +Conv1 +Conv(3,32,1,1),BN,ReLU +− +− +Block1 +� +Conv(3,32,1,1),BN,ReLU +Conv(3,32,1,1),BN,ReLU +� +� +Identity +Identity +� +� +Agent(64,64,0.15) +Agent(64,64,0.15) +� +Block2 +� +Conv(3,32,1,1),BN,ReLU +Conv(3,32,1,1),BN,ReLU +� +� +Conv(1,64,2,0),BN +Conv(1,64,2,0),BN +� +� +Agent(64,64,0.15) +Agent(64,64,0.15) +� +Classifier +Linear(64,num classes) +− +− +Table 5: Hyperparameters for all the datasets for Dynamos. +Dataset +Buffer +Size +lr +#Epochs +#Warmup +Ep./Itr. +Batch +Size +Memory +Batch +Size +α +β +αP +λ +γ +wP +kr +Seq-MNIST +500 +0.07 +1 +10 it +10 +10 +0.2 +2.0 +0.2 +500 +0.5 +0.3 +0.7 +1000 +0.07 +1 +10 it +10 +10 +0.1 +2.5 +0.2 +200 +0.7 +0.5 +0.7 +2000 +0.07 +1 +10 it +10 +10 +0.5 +3.0 +0.2 +200 +0.5 +0.5 +0.7 +SVHN +500 +0.07 +70 +10 ep +16 +16 +2.0 +3.0 +1.0 +500 +1.0 +0.5 +0.7 +1000 +0.07 +70 +10 ep +16 +16 +2.5 +2.0 +0.2 +500 +1.0 +0.5 +0.7 +2000 +0.07 +70 +10 ep +16 +16 +2.5 +2.0 +0.2 +500 +1.0 +0.5 +0.7 +MNIST-360 +100 +0.07 +1 +10 it +16 +16 +0.2 +1.0 +0.1 +200 +0.5 +0.5 +0.7 +200 +0.07 +1 +10 it +16 +16 +0.2 +1.5 +0.1 +200 +1.0 +0.5 +0.7 +500 +0.07 +1 +10 it +16 +16 +0.1 +1.5 +0.1 +200 +0.3 +0.3 +0.7 + +Filters +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Class ID +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(a) Classwise activations for Layer 1 +Filters +0 +1 +2 +3 +4 +Task ID +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(b) Taskwise activations for Layer 1 +Filters +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Class ID +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(c) Classwise activations for Layer 2 +Filters +0 +1 +2 +3 +4 +Task ID +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(d) Taskwise activations for Layer 2 +Filters +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Class ID +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(e) Classwise activations for Layer 3 +Filters +0 +1 +2 +3 +4 +Task ID +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(f) Taskwise activations for Layer 3 +Filters +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Class ID +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(g) Classwise activations for Layer 4 +Filters +0 +1 +2 +3 +4 +Task ID +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(h) Taskwise activations for Layer 4 +Figure 7: Filter activation rates for each filter in each convolutional layer of Block 2 with respect to MNIST tasks and classes. +Overlapping activations of tasks and classes indicative of similarities between them can still be observed. For e.g. 1 and 7 +still show very similar responses. + diff --git a/VNAyT4oBgHgl3EQfuvm-/content/tmp_files/load_file.txt b/VNAyT4oBgHgl3EQfuvm-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e44053b9223e14e558ffd25014379c994bb66419 --- /dev/null +++ b/VNAyT4oBgHgl3EQfuvm-/content/tmp_files/load_file.txt @@ -0,0 +1,929 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf,len=928 +page_content='Dynamically Modular and Sparse General Continual Learning Arnav Varma1, Elahe Arani†1,2 and Bahram Zonooz†1,2 1Advanced Research Lab, NavInfo Europe, Eindhoven, The Netherlands 2Department of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands arnav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='varma@navinfo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='eu, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='arani@tue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='nl, bahram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='zonooz@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='com † Contributed equally Keywords: Dynamic Neural Networks, Policy Gradients, Lifelong Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Abstract: Real-world applications often require learning continuously from a stream of data under ever-changing con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' When trying to learn from such non-stationary data, deep neural networks (DNNs) undergo catas- trophic forgetting of previously learned information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Among the common approaches to avoid catastrophic forgetting, rehearsal-based methods have proven effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' However, they are still prone to forgetting due to task-interference as all parameters respond to all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' To counter this, we take inspiration from sparse coding in the brain and introduce dynamic modularity and sparsity (Dynamos) for rehearsal-based general continual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' In this setup, the DNN learns to respond to stimuli by activating relevant subsets of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' We demonstrate the effectiveness of Dynamos on multiple datasets under challenging continual learning evalu- ation protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Finally, we show that our method learns representations that are modular and specialized, while maintaining reusability by activating subsets of neurons with overlaps corresponding to the similarity of stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='com/NeurAI-Lab/DynamicContinualLearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' 1 INTRODUCTION Deep neural networks (DNNs) have achieved human- level performance in several applications (Greenwald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Taigman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' These networks are trained on the multiple tasks within an appli- cation with the data being received under an inde- pendent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=') assump- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' This assumption is satisfied by shuffling the data from all tasks and balancing and normalizing the samples from each task in the application (Had- sell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Consequently, DNNs can achieve human-level performance on all tasks in these appli- cations by modeling the joint distribution of the data as a stationary process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Humans, on the other hand, can model the world from inherently non-stationary and sequential observations (French, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Learn- ing continually from the more realistic sequential and non-stationary data is crucial for many applications such as lifelong learning robots (Thrun and Mitchell, 1995) and self-driving cars (Nose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' How- ever, vanilla gradient-based training for such contin- ual learning setups with a continuous stream of tasks and data leads to task interference in the DNN’s pa- rameters, and consequently, catastrophic forgetting on old tasks (McCloskey and Cohen, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Therefore, there is a need for methods to alleviate catastrophic forgetting in continual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Previous works have aimed to address these chal- lenges in continual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' These can be broadly classified into three categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' First, regularization- based methods (Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Schwarz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Zenke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2017) that penalize changes to the parameters of DNNs to reduce task interfer- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Second, parameter isolation methods (Adel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2020) that assign distinct subsets of parame- ters to different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Finally, rehearsal-based meth- ods (Chaudhry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2019) that co-train on cur- rent and stored previous samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Among these, regularization-based and parameter isolation-based methods often require additional information (such as task-identity at test time and task-boundaries during training), or unconstrained growth of networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' These requirements fail to meet general continual learning (GCL) desiderata (Delange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Farquhar and Gal, 2018), making these methods unsuitable for GCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Although rehearsal-based methods improve over other categories and meet GCL desiderata, they still suffer from catastrophic forgetting through task in- terference in the DNN parameters, as all parameters respond to all examples and tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' This could be re- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='00620v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='CV] 2 Jan 2023 solved by inculcating task or example specific param- eter isolation in the rehearsal-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' How- ever, it is worth noting that unlike parameter isolation methods, modularity and sparsity in the brain is not static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' There is evidence that the brain responds to stimuli in a dynamic and sparse manner, with differ- ent modules or subsets of neurons responding ”dy- namically” to different stimuli (Graham and Field, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The advantages of a dynamic and sparse re- sponse to stimuli have been explored in deep learn- ing in stationary settings through mechanisms such as gating of modules (Veit and Belongie, 2018), early- exiting (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2020), and dy- namic routing (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2018), along with train- ing losses that incentivize sparsity of neural activa- tions (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' These studies observed that DNNs trained to predict dynamically also learn to respond differently to different inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Furthermore, the learned DNNs demonstrate clustering of parame- ters in terms of tasks such as similarity, difficulty, and resolution of inputs (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Veit and Be- longie, 2018), indicating dynamic modularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Hence, we hypothesize that combining rehearsal-based meth- ods with dynamic sparsity and modularity could help further mitigate catastrophic forgetting in a more bi- ologically plausible fashion while adhering to GCL desiderata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' To this end, we propose Dynamic Modularity and Sparsity (Dynamos), a general continual learning al- gorithm that combines rehearsal-based methods with dynamic modularity and sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Concretely, we seek to achieve three objectives: dynamic and sparse re- sponse to inputs with specialized modules, compe- tent performance, and reducing catastrophic forget- ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' To achieve dynamic and sparse responses to inputs, we define multiple agents in our DNN, each responsible for dynamically zeroing out filter acti- vations of a convolutional layer based on the input to that layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The agents are rewarded for choosing actions that remove activations (sparse responses) if the network predictions are accurate, but are penal- ized heavily for choosing actions that lead to inaccu- rate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Agents also rely on prototype losses to learn specialized features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' To reduce forgetting and achieve competent performance, we maintain a constant-size memory buffer in which we store pre- viously seen examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The network is retrained on previous examples alongside current examples to both maintain performance on current and previous tasks, as well as to enforce consistency between current and previous responses to stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Dynamos demon- strates competent performance on multiple continual learning datasets under multiple evaluation protocols, including general continual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Additionally, our method demonstrates similar and overlapping re- sponses for similar inputs and disparate responses for dissimilar inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Finally, we demonstrate that our method can simulate the trial-to-trial variability ob- served in humans (Faisal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Werner and Mountcastle, 1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' 2 RELATED WORK Research in deep learning has approached the dy- namic compositionality and sparsity observed in the human brain through dynamic neural networks, where different subsets of neurons or different sub- networks are activated for different stimuli (Bengio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Bolukbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' This can be achieved through early exiting (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2020), dy- namic routing through mixtures of experts or multi- ple branches (Collier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2022), and through gating of modules (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Early-exiting might force the DNN to learn specific features in its earlier layers and consequently hurt performance (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2018) as the earlier layers of DNNs are known to learn general purpose fea- tures (Yosinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Dynamic routing, on the other hand, would require the growth of new ex- perts in response to new tasks that risk unconstrained growth, or the initialization of a larger DNN with branches corresponding to the expected number of tasks (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Dynamic networks with gating mechanisms, meanwhile, have been shown to achieve competent performance in i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' training with standard DNNs embedded with small gating net- works (Veit and Belongie, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' These gating networks emit a dis- crete keep/drop decision for each module, depending on the input to the module or the DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' As this opera- tion is non-differentiable, a Gumbel Softmax approx- imation (Veit and Belongie, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2018), or an agent trained with policy gradients (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Sutton and Barto, 2018) is commonly used in each module to enable backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' However, unlike the latter, the Gumbel-Softmax approximation induces an asymmetry between the forward pass acti- vations at inference and training (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Furthermore, these methods are not applicable to con- tinual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Recent works have attempted to build dynamic networks for continual learning setups (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Abati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2020), where data arrive in a more realistic sequential manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' InstAParam (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2020), Random Path Selection (RPS) (Rajasegaran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2019), and MoE (Collier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2020) start with multiple parallel blocks at each layer, finding input-specific or task-specific paths within this large network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Nevertheless, this requires knowledge of the number of tasks to be learned ahead of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' More importantly, initializing a large network might be unnecessary as indicated by the competent perfor- mance of dynamic networks with gating mechanisms in i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='d training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' In contrast to this, MNTDP (Ve- niat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2021), LMC (Ostapenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2021), and CCGN (Abati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2020) start with a standard archi- tecture and grow units to respond to new data or tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Of these, MNTDP and LMC develop task-specific networks where all inputs from the same task elicit the same response and therefore do not show a truly dynamic response to stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' CCGN, however, com- poses convolutional filters dynamically to respond to stimuli, using a task-specific vector for every convo- lutional filter, and task boundaries to freeze frequently active filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' However, this leads to unrestrained growth and fails in the absence of task-boundaries, which makes it unsuitable for general continual learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Therefore, we propose a general continual learn- ing method with dynamic modularity and sparsity (Dynamos) induced through reinforcement learning agents trained with policy gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' 3 METHODOLOGY Humans learn continually from inherently non- stationary and sequential observations of the world without catastrophic forgetting, even without super- vision about tasks to be performed or the arrival of new tasks, maintaining a bounded memory through- out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' This involves, among other things, making multi- scale associations between current and previous ob- servations (Goyal and Bengio, 2020) and respond- ing sparsely and dynamically to stimuli (Graham and Field, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The former concerns consolidation of previous experiences and ensuring that learned ex- periences evoke a similar response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The latter con- cern dynamically composing a subset of the special- ized neural modules available to respond to stimuli, reusing only the relevant previously learned informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' This also avoids erasure of information irrele- vant to current stimuli but relevant to previous experi- ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' We now formulate an approach for dynamic sparse and modular general continual learning that mimics these procedures with DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='1 Dynamic, Modular, and Sparse response to stimuli To achieve a dynamic, modular, and sparse response to inputs, we use a DNN F with a policy to compose a subset of the available modules in each layer to re- spond to the input to that layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' More specifically, we use a CNN which is incentivized to drop some chan- nels in its activations adaptively using policy gradi- ents (Sutton and Barto, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Williams, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Let us consider the lth convolutional layer with cl output channels ∀l ∈ {1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='L}, where L is the total number of convolutional layers in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The input to the convolutional layer is processed us- ing an agent module with actions al ∈ {0,1}cl as out- put, where each action represents the decision to drop (action = 0) or keep (action = 1) the corresponding channel of the output of the convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The agent module uses a self-attention network to obtain a channel-wise attention vector vl of dimension cl, which is converted into ”action probabilities” using a probability layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The policy for choosing actions is then sampled from a cl-dimensional Bernoulli distri- bution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' pl = σ(vl) πl(al) = cl ∏ i=1 p al,i l,i (1− pl,i)(1−al,i), (1) where pl ∈ (0,1)cl is the output of the probability layer σ, and πl is the policy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The final output of the convolutional layer is the channel-wise product of the actions with the output of the convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' This policy formulation is used at each convolutional layer in the CNN, leading to L agents in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The over- all structure of an agent for a convolutional layer is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' These agents are rewarded for dropping channels while making accurate predictions through a reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' For an input to the DNN X applied to clas- sification with label Y: Z,V = F(X), V = [v1|v2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='|vL] ˆY = argmaxZ, (2) where Z refers to the logits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Now, the ratio of activa- tions or channels that were retained in the layer l is determined by 1 cl ∑cl i=1 al,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' So, for a target activation retention rate per layer or ”keep ratio” kr, the reward function is as follows: Rl(X,Y) = � −(kr − 1 cl ∑cl i=1 al,i)2, if ˆY = Y −λ(kr − 1 cl ∑cl i=1 al,i)2, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' (3) Bernoulli Sampling Probability Layer Convolutional Layer with cl filters Action Probabilities Actions: Channel-wise Attention Vector Input Batch Normalization Global Average Pooling Point-wise Convolution FC FC Self-Attention Network 0 1 1 10 1 Output with channels removed Figure 1: An overview of Dynamos’ dynamic and sparse response mechanism at the lth convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Blacked acti- vations are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The agent (bottom path) self-attention network uses a pointwise convolution to match output channels and global average pooling to get a channel-length flattened vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' This is sent through an MLP with one hidden layer and Sigmoid activation, and multiplied with the original channel-length representation to get the channel-wise self-attention vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Therefore, when the DNN’s predictions are correct, each agent is rewarded for dropping enough activa- tions to match the ”keep ratio” from its correspond- ing convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' However, when the predic- tion is incorrect, each agent is penalized for the same, scaled by a constant penalty factor λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The global na- ture of the reward function, achieved through depen- dence on the correctness of the prediction, also en- forces coordination between agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Following RE- INFORCE (Williams, 1992), the loss from all agents t = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='L is: LR(X,Y) = ElEπ[−Rl(X,Y)logπl(al)] = ElEπ[−Rl(X,Y)log cl ∏ i=1 pl,ial,i +(1− pl,i)(1−al,i)] = ElEπ[−Rl(X,Y) cl ∑ i=1 log[pl,ial,i +(1− pl,i)(1−al,i)]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' (4) Although the agents along with this loss ensure sparse and dynamic responses from the DNN, they do not explicitly impose any specialization of com- positional neural modules seen in humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' As the channel-wise ”modules” activated in the DNN are di- rectly dependent on the channel-wise attention vec- tors, we finally apply a specialization loss that we call prototype loss to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Concretely, for classification, in any batch of inputs, we pull the vectors belonging to the same class together while pushing those from different classes away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' This would cause different subsets of channel-wise modules to be used for in- puts of different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' When combined with a suf- ficiently high ”keep ratio”, this will encourage overlap and therefore, reuse of relevant previously learned in- formation (for example, reusing channels correspond- ing to a learned class for a newly observed class) and, consequently, learning of general-purpose features by the modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' For an input batch X with correspond- ing labels Y, and the corresponding batch of concate- nated channel-wise attention vectors V (Equation 2), the prototype loss is given by: LP(X,Y) = 1+Σ(V1,V2)∈V 2:Y1=Y2MSE(V1,V2) 1+Σ(V1,V2)∈V 2,Y1̸=Y2MSE(V1,V2), (5) where MSE refers to the Mean Squared Error estima- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Note that we only apply this loss to samples for which the predictions were correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='2 Multi-Scale associations As discussed earlier, one of the mechanisms em- ployed by humans to mitigate forgetting is multi- scale associations between current and previous ex- periences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' With this goal in mind, we follow recent rehearsal- based approaches (Buzzega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Riemer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2019) that comply with GCL and use a memory buffer during training to store previously seen examples and responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The buffer is updated using reservoir sam- pling (Vitter, 1985), which helps to approximate the distribution of the samples seen so far (Isele and Cos- gun, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' However, we only consider the subset of batch samples on which the prediction was made correctly for addition to the memory buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' These buffer samples are replayed through the DNN along- side new samples with losses that associate the current response with the stored previous response, resulting in consistent responses over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Let M denote the memory buffer and DT de- note the current task stream, from which we sample batches (XM,YM,ZM,VM) and (Xt,Yt), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Here, ZM and VM are the saved logits and channel- wise attention vectors corresponding to XM when it was initially observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The consistency losses associ- ated with current and previous responses are obtained during the task T as follows: Z ′ M,V ′ M = F(XM) LC(ZM,Z ′ M) = EXM[∥ZM −Z ′ M∥2 2] LC(VM,V ′ M) = EXM[∥VM −V ′ M∥2 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' (6) In addition to consistency losses, we also enforce accuracy, and dynamic sparsity and modularity on the memory samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Therefore, we have four sets of losses: Task performance loss on current and memory samples to ensure correctness on current and pre- vious tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' For classification, we use cross- entropy loss (LCE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Reward losses (Equation 4) on current and mem- ory samples to ensure dynamic modularity and sparsity on current and previous tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Prototype losses (Equation 5) on current and memory samples to ensure the specialization of modules on current and previous tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Consistency losses (Equation 6) for multi-scale associations between current and previous sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Putting everything together, the total loss becomes: Ltotal = LCE(XB,YB)+γLr(XB) +β[LCE(XM,YM)+γLr(XM)] +αLC(ZM,Z ′ M)+αpLC(VM,V ′ M) +wp[LP(XB,YB)+LP(XM,YM)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' (7) The weights given to the losses - α, αp, β, wp, and γ, and the penalty for misclassification (λ) and keep ratio (kr) in Equation 3, are hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Note that we employ a warm-up stage at the begin- ning of training, where neither the memory buffer nor the agents are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' This is equivalent to train- ing using only the cross-entropy loss for this period, while the agents are kept frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' This gives agents a better search space when they start searching for a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' We call our method as described above Dy- namic modularity and sparsity - Dynamos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' 4 EXPERIMENT DETAILS Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' We show results on sequential variants of MNIST (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 1998) and SVHN: Seq-MNIST and Seq-SVHN (Netzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2011), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Seq-MNIST and Seq-SVHN divide their respective datasets into 5 tasks, with 2 classes per task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Fur- thermore, to test the applicability of Dynamos under general continual learning, we also use the MNIST- 360 dataset (Buzzega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' We use a network based on the ResNet-18 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2016) structure by removing the later two of its four blocks and reducing the number of filters per convolutional layer from 64 to 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The initial convolution is reduced to 3 × 3 to work with smaller image sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' For the baseline experiments, we did not use any agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' For our method, while agents can be used for all convolutional layers, we only use agents in the second block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' We make this choice based on recent studies that observe that ear- lier layers undergo minimal forgetting (Davari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2022), are highly transferrable (Yosinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2014), and are used for most examples even when learned with dynamic modularity (Abati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' We use a sigmoid with a temperature layer as the probabil- ity layer in the agents and a probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='5 as a threshold for picking actions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', channels during in- ference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The temperature serves the purpose of tuning the range of outputs of the self-attention layers, en- suring that the probabilities being sampled to choose the actions are not too small and that enough activa- tions are chosen to enable learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The exact net- work structure used for each experiment, including the self-attention networks of the agents, can be found in Appendix, in Table 3 and Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' All methods are implemented in the Mam- moth repository1 in PyTorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='6 and were trained on Nvidia V100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The hyperparameters cor- responding to each experiment can be found in Ap- pendix, Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' We always maintain a keep ra- tio higher than 1/Num tasks to allow the learning of overlapping, reusable, and general-purpose mod- ules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The temperature of the Sigmoid activation of the probability layers is kept at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='15 unless mentioned otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='com/aimagelab/mammoth/ 500 1000 2000 60 64 68 72 76 80 84 88 Accuracy Seq-SVHN 500 1000 2000 Buffer size 90 92 94 96 98 100 Accuracy Seq-MNIST Models CCGN Dynamos Figure 2: Quantitative results under Class-Incremental Learning protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Results are averaged across three seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' CCGN values taken from the original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The precise accuracies can be found in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' 5 RESULTS We will evaluate Dynamos under two standard eval- uation protocols that adhere to the core desiderata of GCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='1 Class-Incremental Learning (CIL) Class-incremental learning (CIL) refers to the eval- uation protocol in which mutually exclusive sets of classes are presented sequentially to the network, and the identity of the task is not provided at the test time, which meets the core desiderata of GCL (Far- quhar and Gal, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' We compare against Condi- tional Convolutional Gated Network (CCGN) (Abati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2020), which also dynamically composes con- volutional filters for continual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' We observe in Figure 2 that Dynamos shows higher accuracies on both the Seq-MNIST and Seq-SVHN datasets un- der all buffer sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' However, CCGN requires a separate task vector for every task per convolutional layer, resulting in unrestricted growth during train- ing, whereas we maintain a bounded memory through training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Furthermore, unlike CCGN, we do not lever- age the task boundaries or the validation set during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Therefore, Dynamos outperforms the previ- ous state-of-the-art for dynamic compositional con- tinual learning in class-incremental learning, while showing bounded memory consumption during train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='2 General Continual Learning (GCL) So far, we have observed Dynamos under the CIL protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Unlike CIL, real-world data streams with- out clear task boundaries, where the same data may reappear under different distributions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' different poses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Following (Buzzega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2020), we approxi- mate this setting using MNIST-360, where tasks over- lap in digits (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' classes), reappear under different ro- tations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' distributions), and each example is seen exactly once during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' This serves as a verifica- tion of the adherence to the GCL desiderata (Farquhar and Gal, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Delange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' We study the impact of both dynamic modularity as well as multi- scale associations by removing them incrementally from Dynamos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' When neither is used, the learning is done using vanilla gradient-based training, with no strategy to counter forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' When dynamic mod- ularity is removed, the learning strategy forms our baseline, where no agents are used, simplifying the total training loss from Equation 7 to: Lbase = LCE(XB,YB)+βLCE(XM,YM)+ αLC(ZM,Z ′ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' (8) Table 1 shows that Dynamos outperforms the baseline in all buffer sizes, proving that dynamic modularity is advantageous in GCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Furthermore, when multi-scale associations are also removed, no buffer is used, and the DNN undergoes catastrophic forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Thus, Dynamos is applicable to general continual learning, with dynamic modularity improving over the base- line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' We hypothesize that dynamic modularity makes dealing with the blurred task boundaries of GCL eas- ier by adaptively reusing relevant previously learned information, which in this case corresponds to learned filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' 6 MODEL CHARACTERISTICS We now analyze some of the characteristics and ad- vantages of Dynamos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' For all experiments in this sec- tion, we use our model trained on Sequential-MNIST with buffer size 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='1 Dynamic Modularity and Compositionality Humans show modular and specialized responses to stimuli(Meunier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2010) with dynamic and sparse Table 1: General continual learning results for multiple buffer sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' All results are averaged across five seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Multi-Scale Associations Dynamic Modularity Buffer Size 100 200 500 \x13 \x13 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='418±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='095 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='638±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='853 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='519±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='737 \x13 \x17 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='192±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='072 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='364±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='259 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='150±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='888 \x17 \x17 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='712±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='690 Filters 0 1 2 3 4 5 6 7 8 9 Class ID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='0 (a) Classwise Filters 0 1 2 3 4 Task ID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='0 (b) Taskwise Figure 3: Filter activation rates on the test set for each filter with respect to tasks and classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' For ease of visualization, we only look at the last 40 filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Full visualizations can be found in Appendix (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' 0 1 2 3 4 5 6 7 8 9 Class ID 0 1 2 3 4 5 6 7 8 9 Class ID 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='11 Figure 4: Jensen-Shanon Divergences (×100) of the activa- tion rates of class pairs on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' response to inputs (Graham and Field, 2006) - a capa- bility that we instilled in our DNN while learning a sequence of tasks by dynamically removing channel activations of convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Therefore, we ex- amine the task- and class-wise tendencies of the firing rates of each neuron (filter) in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' It can be seen that Dynamos learns a soft separa- tion of both tasks and classes, as evidenced by the per- task and per-class firing rates, respectively, of each filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' This is in contrast to static methods, where all filters react to all examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Figure 3a further shows that this allows learning of similar activation patterns for similar examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' For example, MNIST digit pairs 1 and 7, and 6 and 8, which share some shape similarities, also share similarities in their acti- vation patterns/rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' This could be attributed to be- ing able to reuse and drop learned filters dynamically, which causes the DNN to react similarly to similar inputs, partitioning its responses based on example similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Additionally, the ability to dynamically reuse filters allows DNNs to learn overlapping acti- vation patterns for dissimilar examples and classes, instead of using completely disparate activation pat- terns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' This also facilitates the learning of sequences of tasks without having to grow the DNN capacity or having a larger capacity at initialization, as opposed to the static parameter isolation methods for contin- ual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Following (Abbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2022), we quantify the overlap between the activation rates for each class pair in the final layer using the Jensen-Shanon divergence (JSD) between them in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Lower JSDs sig- nify higher overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The JSD is lowest for the class pair (1,7) (both digits look like vertical lines), and is ∼ 1 15th the average JSD across class pairs, and ∼ 1 42th that of the least overlapping class pair (1,8) (1 is a line, 8 is formed of loops).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Now, as per Equation 1, filters in the layer are activated based on the channel- wise attention vector vL (see Equation 2), which are pushed together for examples of the same classes, and pushed away from each other for examples of differ- ent classes using prototype loss (Equation 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' We vi- sualize the t-SNEs of these vLs on the test set in Fig- 75 50 25 0 25 50 75 75 50 25 0 25 50 75 100 Class IDs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='0 Figure 5: t-SNEs on the test set of class prototypes learned from channel-wise self-attention vectors for all classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' ure 5 and observe that the samples belonging to the same classes are clustered, confirming the effective- ness of our prototype loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Moreover, the clusters of visually similar classes are close together, which is concomitant with the JSDs and class-wise activation rates seen earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Class similarities are also reflected through multiple clusters for the digit 9, indicating its similarity with the digits 6 (loop) and 1 (line) in one cluster, but also with 7 (line) and 4 (line + loop) in another cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Finally, we observe that there are ex- amples that are scattered away from their class clus- ters and overlap with other clusters, probably indicat- ing that these particular examples are visually closer to other digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Note, however, that these similar ex- amples and classes are distributed across tasks, which explains the lower similarities in activation patterns between task pairs in Figure 3b compared to the class pairs in Figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Therefore, Dynamos is capable of learning modu- lar and specialized units that result in input-adaptive dynamic separation and overlap of activations, based on the extent of similarities with previously learned examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' We also contend that the overlapping acti- vations for digits of similar shape suggest the learning of general-purpose features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='2 Trial-to-trial variability The brain is known to show variability in response across trials (Faisal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Werner and Mount- castle, 1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' For the same stimulus, the precise neuronal response could differ between trials, a be- havior absent in most conventional DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' In our method, this aspect of brains can be mimicked by us- ing Bernoulli sampling instead of thresholding to pick keep/drop decisions at each convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' In Figure 6, we plot the response variability in the last convolutional layer of our DNN with the same exam- ple in four trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' We only pick responses for which Filters 1 2 3 4 Trial 0 1 Figure 6: Trial-to-trial variability of responses to same input in Dynamos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' the predictions were correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' It can be seen that each trial evoked a different response from the DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Fur- thermore, despite the differences, there are also some similarities in the response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' There are some filters that are repeatedly left unused, as well as some filters that are used in every trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' This demonstrates that Dy- namos can additionally simulate the trial-to-trial vari- ability observed in brains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' 7 CONCLUSION AND FUTURE WORK We propose Dynamos, a method for general contin- ual learning, that simulates the dynamically modu- lar and sparse response to stimuli observed in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Dynamos rewards the input-adaptive removal of channel activations of convolutional layers using policy gradients for dynamic and sparse responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' To further induce modularity, channel-wise self-attention vectors corresponding to each convolutional layer are pulled together for examples from same classes, and are pushed apart for examples from different classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' these vectors are then used to sample the keep/drop decision for the corresponding channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Using a memory buffer, we enforce multi-scale con- sistency between previous and current responses to prevent forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Dynamos outperforms previous baselines on multiple datasets when evaluated us- ing class-incremental learning (CIL) and general con- tinual learning (GCL) protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Dynamos exhibits similar and overlapping responses for similar inputs, yet distinct responses to dissimilar inputs by utiliz- ing subsets of learned filters in an adaptive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' We quantified the extent of class-wise overlaps and showed that the semantic similarity of classes (dig- its in MNIST, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' 1 and 7) are reflected in higher representation overlaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' We additionally visualized the channel-wise attention vectors and observed that they are clustered by the classes and the clusters of se- mantically similar classes lie together or overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Fi- nally, we also demonstrated the ability of our method to mimic the trial-to-trial variability seen in the brain, where same inputs achieve same outputs through dif- ferent “responses”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Thus, we con- sider our work as a step toward achieving dynamically modular and general-purpose continual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' REFERENCES Abati, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', Tomczak, J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' In Pro- ceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 8817–8826.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Yosinski, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', Clune, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', Bengio, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', and Lipson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' How transferable are features in deep neural net- works?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Advances in neural information processing systems, 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Zenke, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', Poole, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=', and Ganguli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Continual learning through synaptic intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' In Interna- tional Conference on Machine Learning, pages 3987– 3995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' APPENDIX Table 2: Class-Incremental learning accuracies for CCGN and Dynamos corresponding to Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Buffer Size Model Seq-SVHN Seq-MNIST 500 CCGN 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='45 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='01 Dynamos 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='815 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='19 1000 CCGN 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='99 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='94 Dynamos 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='38 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='51 2000 CCGN 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='02 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='94 Dynamos 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='54 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='57 Table 3: Agent architecture with input features of shape B×Cin ×H ×W, output features of shape B×Cout ×1×1, where B is the batch size, Cin is the number of channels in the input, and Cout is the number of channels expected in the output, which is same as the number of keep/drop ac- tions required for the corresponding convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Operations Input size Output size Pointwise Conv B×Cin ×h×w B×Cout ×h×w Average Pooling B×Cout ×h×w B×Cout ×1×1 Reshape B×Cout ×1×1 B×Cout Linear B×Cout B×Cout/16 ReLU B×Cout/16 B×Cout/16 Linear B×Cout/16 B×Cout Reshape B×Cout B×Cout ×1×1 Sigmoidτ B×Cout ×1×1 B×Cout ×1×1 Table 4: Architectures used in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' For baseline experiments without dynamic compositionality, we do not use the ”Agent” branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Conv(k, n, s, p) refers to convolutional layer with kernel size k, number of filters n, stride s, and padding p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' BN refers to Batch Normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Linear(M, N) refers to a linear layer with M-dimensional input and N-dimensional output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Agent(Cin, Cout, τ) refers to the Agent subnetwork with Cin input channels, Cout output channels, and τ temperature of the sigmoid in the probability layer (See Figure 1, Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' The elementwise multiplication of the actions from the agents with the output of a convolutional layer is done after the application of batch normalization, if present, but before the ReLU activation function, if present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' For complete description of Agent architecture, refer to Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' num classes refers to the number of classes to be predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Component Main Branch Residual branch Agent branch Conv1 Conv(3,32,1,1),BN,ReLU − − Block1 � Conv(3,32,1,1),BN,ReLU Conv(3,32,1,1),BN,ReLU � � Identity Identity � � Agent(64,64,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='15) Agent(64,64,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='15) � Block2 � Conv(3,32,1,1),BN,ReLU Conv(3,32,1,1),BN,ReLU � � Conv(1,64,2,0),BN Conv(1,64,2,0),BN � � Agent(64,64,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='15) Agent(64,64,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='15) � Classifier Linear(64,num classes) − − Table 5: Hyperparameters for all the datasets for Dynamos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' Dataset Buffer Size lr #Epochs #Warmup Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='/Itr.' metadata={'source': 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and classes indicative of similarities between them can still be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' For e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} +page_content=' 1 and 7 still show very similar responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfuvm-/content/2301.00620v1.pdf'} diff --git a/WdE1T4oBgHgl3EQfbgRN/content/tmp_files/2301.03173v1.pdf.txt b/WdE1T4oBgHgl3EQfbgRN/content/tmp_files/2301.03173v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..70c3331e13a2d50b712099c178a9d7eddddb7673 --- /dev/null +++ b/WdE1T4oBgHgl3EQfbgRN/content/tmp_files/2301.03173v1.pdf.txt @@ -0,0 +1,1964 @@ +APCTP Pre2022 - 030 +Popcorn transitions and approach to conformality in homogeneous holographic +nuclear matter +Jes´us Cruz Rojas,1 Tuna Demircik,2 and Matti J¨arvinen1, 3 +1Asia Pacific Center for Theoretical Physics, Pohang 37673, Republic of Korea +2Institute for Theoretical Physics, Wroc�law University of Science and Technology, 50-370 Wroc�law, Poland +3Department of Physics, Pohang University of Science and Technology Pohang 37673, Republic of Korea +We study cold and dense nuclear matter by using the gauge/gravity duality. +To this end we +use the Witten-Sakai-Sugimoto model and the V-QCD models with an approach where the nuclear +matter is taken to be spatially homogeneous. We focus on the “popcorn” transitions, which are +phase transitions in the nuclear matter phases induced by changes in the layer structure of the +configuration on the gravity side. We demonstrate that the equation of state for the homogeneous +nuclear matter becomes approximately conformal at high densities, and compare our results to other +approaches. +I. +INTRODUCTION +Recent observations of neutron star mergers by the +LIGO/Virgo collaboration have opened a new window +for studying dense matter in quantum chromodynam- +ics (QCD). In particular, the gravitational and electro- +magnetic waves observed from the GW170817 merger +event [1] already set highly nontrivial constrains for the +QCD equation of state [2] at low temperatures and high +densities. This progress has boosted interest in theoret- +ical studies of dense QCD, which is a challenging topic +as standard theoretical and computational tools do not +work in extensive regions of the phase diagram (see the +overview in [3]). These include the region of dense nuclear +matter, i.e., a nucleon liquid at baryon number densities +well above the nuclear saturation density ρs ≈ 0.16 fm−3. +The difficulty of solving the properties of dense mat- +ter calls for new methods. +A possibility is to use the +gauge/gravity duality. +Indeed, applications of holo- +graphic QCD models to dense matter have received a +lot of interest recently. There has been progress in de- +veloping models both for quark matter [4–9], for nuclear +matter [10–14], and for other phases (such as color super- +conducting or quarkyonic phases) [15–18]. See also the +reviews [19, 20]. +The natural starting point for describing nuclear mat- +ter is to study the holographic duals for nucleons. The +standard approach [21] boils down to describing them as +solitonic “instanton” solutions of bulk gauge fields, i.e., +the gauge fields living in the higher dimensional grav- +ity theory. These solitons are localized both in the spa- +tial directions and in the holographic direction (but not +in time). +Solitons that are duals of isolated nucleons +have been solved in various holographic models [22–31]. +Constructing more complicated solutions, and eventu- +ally the holographic dual of dense nucleon matter out +of these bulk solitons, is however challenging. Some re- +sults, which use instanton gases without interactions, are +available [13, 32–36] and also with two-body interactions +included [10]. Moreover, at large Nc the nuclear matter +is a crystal rather than a liquid of nucleons [37]. Such +crystals have been studied by using different toy models +and approximations [38–43]. +In this article, we focus on a simpler approach, which +treats dense nuclear matter as a homogeneous configura- +tion of the non-Abelian gauge fields in the bulk. This ap- +proach was applied to the Witten-Sakai-Sugimoto (WSS) +model [44–46], in [47], and argued to be a reasonable ap- +proximation at high density.1 It was further developed +in [49, 50] and applied to other models in [11, 14]. In- +terestingly, dense (and cold) homogeneous holographic +nuclear matter was seen to have a high speed of sound, +clearly above the value c2 +s = 1/3 of conformal theories [11] +(see also [51, 52]). That is, the equation of state is “stiff”. +This is important as it helps to construct models which +pass observational bounds [53–55]. +Changes in the structure of dense nuclear matter may +give rise to transitions within the nuclear matter phase. +At large Nc, nuclear matter is a crystal of Skyrmions, +solitons of the low energy chiral effective theory [56]. As +the density increases, the Skyrmion crystal is expected to +undergo a transition into half-solitons, where each node +of the crystal carries baryon number of one half [57– +60]. Similar structures have been studied by using the +gauge/gravity duality in [39]. +This topology changing transition has been studied +extensively by using an effective field theory approach, +which introduces the σ meson of QCD as a pseudo- +Nambu-Goldstone mode of broken scale invariance, and +vector mesons through the hidden local symmetry ap- +proach [61, 62]. +This approach is supported by the +analysis of the nucleon axial coupling gA for heavy nu- +clei [63]. +Above the transition density, it was found +that the speed of sound rapidly approaches the confor- +mal value c2 +s = 1/3 [64]. At the same time, the poly- +tropic index γ = d log p/d log ϵ takes small values [65, 66] +as compared to what is usually found in nuclear theory +models [67, 68]. The transition has also been argued [69] +to be indicative of quark-hadron continuity [70], which +1 An even simpler approach is to treat the baryons as point-like +sources in the bulk, which may be a better approximation at low +density [47, 48]. +arXiv:2301.03173v1 [hep-th] 9 Jan 2023 + +2 +states that there is no phase transition between nuclear +and quark matter. Whether the continuity is a feasible +possibility is a matter of ongoing debate, see [71–74] (see +also [15] for a holographic discussion). +A closely related transition realised in holographic se- +tups is the transition from a single-layer configuration +into a double layer configuration: +In the low density +limit, the location of each soliton is found by individ- +ually minimizing its energy, so that the solitons form a +single layer at a specific value of the holographic coor- +dinate. For dense configurations, however, the repulsive +interactions between solitons will eventually force them +out of this layer, which leads to a double layer or a more +complicated configuration. +This transition was coined +the “popcorn” transition in [40]. If interactions between +the solitons are attractive at large distances (as is the +case for real QCD) the picture is more complicated as +the solitons clump together even at low densities, but +the transition may still be present. Various phases ap- +pear as the density increases further [41, 42] in setups +motivated by the WSS model. The simplest case of the +transition is however the separation of a single layer into +two layers. +This kind of transition was also found to +take place in the WSS model in various approximations: +when the instantons were approximated as point-like ob- +jects [17], when including finite widths [36], and when +using a homogeneous approach [50]. Indications of such +a transition were also seen when using a homogeneous +Ansatz for nuclear matter in the hard wall model of [14], +where it was interpreted as a transition to a quarkyonic +phase [75]. +In this article, we study the popcorn transitions within +cold homogeneous holographic nuclear matter by using +two different models, the top-down WSS model and the +bottom-up V-QCD model [19, 76]. +These two are ar- +guably the most developed holographic top-down and +bottom-up models for QCD at finite temperature and +density. For the WSS model a similar analysis was car- +ried out in [50]. This reference used an approach which +is slightly different from ours: In their case, a zero cur- +vature condition for the non-Abelian gauge fields in the +Lagrangian density is imposed before approximating the +density to be homogeneous. We use a somewhat simpler +approach where the fields are assumed to be homoge- +neous to start with. In our case, as we will discuss in de- +tail below, a discontinuity of the gauge fields as a function +of the holographic coordinate is required to have nonzero +baryon density [47]. This may appear to be a weakness +of the simpler approach, but we remark that the disconti- +nuity is actually well motivated, as it can be seen to arise +from non-analyticity of the instanton solutions at their +centers after smearing over the spatial dimensions [19]. +The main goal in this article is to analyze the soften- +ing of the equation of state at the phase transition. The +main indicators for this are the speed of sound and the +polytropic index γ. We compute these quantities in both +holographic models and compare to results in other se- +tups. In particular, we find interesting similarities with +the effective theory approach for the topology changing +transition [62, 64]. +The rest of the article is organized as follows. In sec- +tion II we review the setup with homogeneous nuclear +matter for the WSS model, and in section III we do the +same for the V-QCD model. In section IV we discuss the +numerical results for the solutions, the phase transitions, +and the equation of state. Finally, we discuss our findings +in section V. +II. +HOMOGENEOUS NUCLEAR MATTER IN +THE WITTEN-SAKAI-SUGIMOTO MODEL +The phase diagram of QCD has been studied by using +several holographic “top-down” models, i.e., models di- +rectly based on string theory, such as the Witten-Sakai +Sugimoto model [45, 46]. In this model, Witten’s non- +supersymmetric model for low-energy QCD [44] has been +successfully applied to study the spectra and the proper- +ties of mesons and baryons. +In the WSS model, the pure glue physics of the QFT is +described by the dual gravitational background and it is +sourced by Nc D4-branes in type-IIA superstring theory. +Fundamental degrees of freedom are included by adding +Nf pairs of D8 and D8-branes, such that the strings +connecting D4 − D8 and D4 − D8 branes are associated +with left and right-handed fermions. +Witten’s model includes a phase transition involving +a topologically nontrivial change in geometry from a +low temperature “cigar” geometry to a high temperature +black hole geometry [77]. We focus here in the low tem- +perature geometry which we will give explicitly below. +In the low temperature geometry the D8 and D8-branes +join at the tip of the cigar (see figure 1), which locks +together the flavor transformations on the branes, indi- +cating chiral symmetry breaking. As shown in the figure +we assume the simplest case where the D8 and D8-branes +are antipodal, i.e., at located at exactly opposite curves +on the cigar. A chemical potential for the baryon num- +ber can be turned on by adding a nonzero source for the +temporal component of the Abelian gauge field on the +D8-branes. +A. +Expanding the Dirac-Born-Infeld action +The 10-dimensional metric of the confined low tem- +perature geometry in the Witten model can be written +as [78, 79] +ds2 = +�U +R +�3/2 � +dxµdxµ + f(U)dx2 +4 +� ++ +�R +U +�3/2 � dU 2 +f(U) + U 2dΩ2 +4 +� +(1) + +3 +x4 +UKK +∞ +U +z =+∞ +z =0 +z =-∞ +z =-zc +x4 +UKK +∞ +U +z =-∞ +z =+∞ +z =+zc +FIG. 1. +Setup in the WSS model. The coordinate z runs between z = −∞ and z = ∞ between the two boundaries of the D8 +brane embedding as indicated in the figure. The blobs show the locations of the discontinuities for the single layer configuration +(left) and for the double layer configuration (right). +where R is the curvature radius, +f(U) = 1 − U 3 +KK +U 3 +(2) +with UKK denoting the end of space, and dΩ2 +4 is the +metric of S4. For the Minkowski metric dxµdxµ we use +mostly plus conventions, and the x4 coordinate is com- +pactified on a circle. The dilaton is given by +eφ = gs +�U +R +�3/4 +, +(3) +where gs is the string coupling. +In the (x4, U)-coordinates this geometry takes the form +of a cigar and the regularity at the tip of the cigar links +the radius of compactification R4 of the x4 coordinate to +the Kaluza-Klein scale characterized by UKK, as R4 = +(4π)R3/2 +3√UKK . The simplest D8 brane embedding within the +cigar geometry is the antipodal one, given (for example) +by x4 = 0 and x4 = πR4. By changing the coordinates +to U = UKK(1 + z2)1/3 the induced metric on the brane +can be written as +ds2 +ind = +�UKK +R +�3/2� +1 + z2 dxµdxµ ++ +� +R +UKK +�3/2 +4dz2 +9 (1 + z2)5/6 ++ R3/2� +UKK +6� +1 + z2dΩ2 +4 . +(4) +Here the coordinate z takes both positive and negative +values on different branches of the brane. The boundary +is at z = ±∞ and the tip of the cigar at z = 0. See +Figure 1 for illustration. +We work in units where UKK = 1 and R3 = 9/4. We +also start from the Dirac-Born-Infeld action +SDBI = −τ8 +� +d9x e−φtr +� +− det(g + F) , +(5) +where the trace is over flavor indices the brane tension is +given by +τ8 = +1 +(2π)8l9s += +λ9/2 +157464 +√ +2π8 . +(6) +Here ls is the string length, λ is the ’t Hooft coupling, F +is the field strength tensor of the gauge field A, and we +used the relations R3 = πgsNcl3 +s and 2πlsgsNc = λ. +We use a similar expansion as in the case of V-QCD be- +low [11] so that the non-Abelian component of the gauge +fields are treated as small but the Abelian terms are kept +unexpanded. To do so, we separate the gauge field into +non-Abelian and Abelian components: A = A+ ˆA where +A is non-Abelian and ˆA is Abelian, i.e., proportional to +the unit matrix in flavor space (and similarly F = F + ˆF +for the field strengths). We take only the temporal com- +ponent of the Abelian gauge field to be nonzero, assume +that it depends only on the holographic coordinate z, and +assume no dependence on the angular coordinates of Ω4 +for all fields, so that these coordinates can be integrated +out. + +4 +Then the five-dimensional effective action for the gauge fields, to leading order in the non-Abelian F 2, is given as +S = S(0) +DBI + S(1) +DBI + SCS +(7) +where the terms arising from the Dirac-Born-Infeld action read +S(0) +DBI = −λ3NcNf +19683π5 +� +d5x +3� +1 + z2 +� +(1 + z2)2/3 − (1 + z2) Φ′(z)2 +(8) +and +S(1) +DBI = − λNc +216π5 +� +d5x tr +� +− +F 2 +tz +� +1 + z2� +� +1 − +3√ +1 + z2Φ′(z)2�3/2 − +F 2 +ti +3√ +1 + z2 +� +1 − +3√ +1 + z2Φ′(z)2 ++ +F 2 +ij +� +1 − +3√ +1 + z2Φ′(z)2 +2 +3√ +1 + z2 ++ +F 2 +zi +� +1 + z2� +� +1 − +3√ +1 + z2Φ′(z)2 +� +. +(9) +Notice that the general Dirac-Born-Infeld action is am- +biguous for non-Abelian fields, but up to second order in +the expansion the action is non-ambiguous. The Chern- +Simons term is +SCS = +Nc +24π2 +� � +ω5 + d +� +ˆA ∧ tr +� +2A ∧ F + i +2A3 +�� ++3 ˆA ∧ tr (F ∧ F) +� +(10) +with the Abelian gauge field normalized as Φ += +2λ ˆAt/( +√ +729π). Here +ω5 = tr +� +A ∧ F 2 + i +2A3 ∧ F − 1 +10A5 +� +(11) +gives the standard Chern-Simons term for the brane. We +used conventions where F = dA − iA ∧ A. Notice that +in (10) the Abelian field couples to the instanton density +in the bulk as expected (see the last term). Indeed, notice +that S(0) +DBI and S(1) +DBI depend on the Abelian gauge field +only through its z-derivative, and only SCS contains non- +derivative dependence on this field. +The total baryon +charge density is defined as +ρ0 = − +� +δS +δ ˆA′ +t +� +bdry += +� +dz δS +δ ˆAt +, +(12) +according to holographic dictionary, where only the +Chern-Simons term contributes to the last expression. +Therefore the baryon charge is given by the coupling of +the non-Abelian field to ˆAt in SCS. In other words, the +Chern-Simons term determines how the solitons source +baryonic charge. +We also remark that the construction of the precisely +consistent Chern-Simons term is actually rather involved, +in general [80], but in the simple case considered here +complications do not arise. +B. +The homogeneous Ansatz +Then as the next step, we set Nf = 2 and insert the +homogeneous Ansatz +Ai = h(z)σi +(13) +where h(z) is a scalar function and σi are the Pauli ma- +trices. The non-Abelian At and Az components are set +to zero. We then find that +Fzi = h′(z)σi , +Fij = 2h(z)2ϵijkσk +(14) +while other components of the field strength are zero. We +obtain +S(1) +DBI = − λNc +36π5 +� +d5x +�4h(z)4 +� +1 − +3√ +1 + z2Φ′(z)2 +3√ +1 + z2 ++ +(h′(z))2 � +1 + z2� +� +1 − +3√ +1 + z2Φ′(z)2 +� +(15) +and the Chern-Simons action contributes as +SCS = 3Nc +π2 +� +h(z)2h′(z) ˆA ∧ dz ∧ dx1 ∧ dx2 ∧ dx3 (16) +as well as a boundary term +SCS,bdry = 3Nc +4π2 +� +bdry +h(±∞)3 ˆA ∧ dx1 ∧ dx2 ∧ dx3 (17) +which however will vanish when it is evaluated on the +solution in our case, because the solution for h(z) will +vanish on the boundary. +C. +The single layer solution +In order to have explicit parity invariance, we assume +that h(z) = −h(−z). +Following [47], we assume that + +5 +the field h has a discontinuity at z = 0, denoted by the +blob in figure 1 (left), and approaches different constant +values as z → 0 either from above or from below. As we +mentioned above, the discontinuity is required to have a +non-vanishing baryon density. Defining the bulk charge +density as +ρ(z, xµ) = − +δS +δ ∂z ˆAt(z, xµ) +(18) +the equation of motion for ˆA implies +ρ′(z) = 3Nc +π2 h(z)2h′(z) . +(19) +The continuous and symmetric solution is given by +ρ(z) = +� +ρ0 + Nc +π2 h(z)3 , +(z < 0) +−ρ0 + Nc +π2 h(z)3 , +(z > 0) +(20) +where +ρ0 = Nc +π2 +lim +z→0+ h(z)3 = −Nc +π2 +lim +z→0− h(z)3 +(21) +is the boundary charge density. Notice that as expected, +it is sourced by the discontinuity of h. +This solution +is identified as the single layer solution. To finalize the +construction, we require that h satisfies the equation of +motion arising from minimizing the action, except at z = +0 where the discontinuity is located. +D. +The double layer solution +A slightly more general solution than the single layer +solution however exists: it is possible that the disconti- +nuity of the h field does not take place at the tip but +at a generic value of the holographic coordinate. +The +simplest of such solutions, which still respects the sym- +metry h(z) = −h(−z), is where h(z) vanishes when +−zc < z < zc so that the discontinuity is located at +z = ±zc, see figure 1 (right). Similar solutions were con- +sidered in [17] for point-like instantons. In this case, the +solution for the bulk charge density is given by +ρ(z) = +� +� +� +ρ0 + Nc +π2 h(z)3 , +(z < −zc) +−ρ0 + Nc +π2 h(z)3 , +(z > zc) +0 , +(−zc < z < zc) +(22) +where +ρ0 = Nc +π2 +lim +z→zc+ h(z)3 = −Nc +π2 +lim +z→(−zc)− h(z)3 . +(23) +This solution is identified as the double layer solution. +E. +Legendre transform to canonical ensemble +To determine the phase diagram, one needs to deter- +mine the free energy, the energy densities, and the grand +potential for the different phases. Thus, we first need to +compute the free energy of the baryonic phase. For this +purpose we minimize the action for h to determine the +location of the discontinuity. +It is convenient to work at fixed baryonic charge rather +than chemical potential. To this end, we perform a Leg- +endre transformation for the action (7): +�S = S + +� +d +dz +� +ˆAt ρ +� +dz +(24) +For convenience we rescale ρ as ρ → +4λ4 +531441π6 ρ ≡ ˆρ. Ex- +panding to first nontrivial order in h(z) and h′(z), and +using equation (18), we can solve for Φ′(z): +Φ′(z) = − +ˆρ +R(z, ˆρ)(1 + z2)Nc +− +2187ˆρ +� +−4h(z)4(1 + z2) +2 +3 + h′(z)2(1 + z2)2R(z, ˆρ)2� +8λ2Nc(1 + z2) +8 +3 R(z, ˆρ)3 +, +(25) +where we define +R(z, ˆρ) = +� +1 + +ˆρ2 +(1 + z2)5/3 N 2c +. +(26) +Then the Legendre transformed action is +�S = − Nc +� +d5x +� +2λ3(1 + z2) +2 +3 R(z, ˆρ) +19683π5 ++ +λ +� +4h(z)4(1 + z2) +2 +3 + h′(z)2 � +(1 + z2)2R(z, ˆρ)2�� +36π5 ((1 + z2)R(z, ˆρ)) +� +. +(27) +Now we can find the equation of motion for h(z) and solve it. For this purpose, we need to find the appropriate +asymptotics of the field h at the boundary: +h(z) ≃ h1 +z , +(28) + +6 +with h1 remaining as a free parameter. +III. +HOMOGENEOUS NUCLEAR MATTER IN +THE V-QCD MODEL +V-QCD is bottom-up holographic model which con- +tains both glue and flavor sectors. +The glue sector +is given by the improved holographic QCD framework +[81, 82] in which a dilaton field and the potential de- +pending on it are used to implement the essential fea- +tures of the related QCD sector, i.e. asymptotic freedom, +and confinement to deconfinement phase transition. The +flavour sector arises from a pair of dynamical space filling +flavor branes [83, 84]. In V-QCD, the full back-reaction of +the flavor branes is taken into account via the Veneziano +limit [85] in which both Nc and Nf are large but their +ratio is kept O(1) as it is in real QCD [76]. In the V-QCD +flavor sector, a tachyon field is used to realize the break- +ing/restoration of the chiral symmetry. In both sectors, +the model parameters are also fixed by considering per- +turbative QCD results (running of coupling constant and +quark mass) at weak coupling [76, 81, 82], by requiring +qualitative agreement with QCD (e.g. confinement and +discrete spectrum) at strong coupling [86], and by fitting +to QCD data (e.g. meson and glueball masses and the +equation of state at finite temperature) [6, 31, 87–89]. +For the more complete review about the construction of +the V-QCD model, the fit to fix the potentials and com- +parison with the data, we refer the reader to [19]. In this +article, we use one of the models defined in [6] (potentials +7a). The parameter b appearing in the Chern-Simons ac- +tion [11] is set to b = 10. +There are two possible geometries in V-QCD: a +horizon-less geometry ending in a “good” kind of singu- +larity [90] (dual to a chirally broken confined phase) and +a geometry of charged “planar” black hole [91, 92] (dual +to a chirally symmetric deconfined phase). In this arti- +cle, we focus on the former geometry which is the relevant +geometry for cold and dense nuclear matter. This phase +also includes chiral symmetry breaking which is induced +by the condensate os a scalar field τ (the “tachyon”) in +the bulk. +In order to discuss nuclear matter, we will employ here +an approach which is essentially the same as the homoge- +neous approach introduced for the WSS model above [11]. +This approach has been improved by combining the +predictions of V-QCD with other models [93–96]. The re- +sultant equation of states have been widely investigated. +It was shown that the resultant equations of state are fea- +sible in the sense of being consistent with neutron star +observations [93, 94, 96–100]. +They were also used in +phenomenological applications such as modeling spinning +neutrons stars [98] and neutron star merger simulations +[93, 99, 100]. +In the first two subsections below, we outline the imple- +mentation of homogeneous Ansatz in V-QCD and discuss +the single layer solution. For more details we refer to [19]. +In the last two subsections, we present the generalization +to double layer solution and investigate the possibility +of a transition from the single layer to a double layer +configuration. +A. +The homogeneous Ansatz +For V-QCD, we use the action with finite baryon den- +sity which can be written as +SV −QCD = Sglue + SDBI + SCS. +(29) +The explicit expression for the action can be found in +[11]. The renormalization group flow of QCD is modeled +through a nontrivial evolution of the geometry between +the weak coupling (ultraviolet, UV) and strong coupling +(infrared, IR) regions. We will be using here the confor- +mal coordinate r in the holographic direction [81, 82] for +which the UV boundary is located at r = 0 while the IR +singularity is at r = ∞. As in the case of the WSS model +above, we separate the gauge field into non-Abelian and +Abelian components: +AL/R = AL/R + ˆAL/R . +(30) +Here the left and right handed fields arise from D4 and +D4 branes, respectively [83, 84]. Similarly as in the case +of the WSS model above, we turn on temporal component +of the vectorial Abelian gauge field +ˆAL = ˆAR = INf ×Nf Φ(r)dt . +(31) +Then on top this background, the non-Abelian baryonic +terms are treated as a perturbation. We expand the DBI +action up to a first nontrivial order in the non-Abelian +fields (quadratic in the field strengths F(L/R)). +After the expansion, +we insert the homogeneous +Ansatz for non-Abelian gauge field, i.e. +Ai +L = −Ai +R = h(r)σi +(32) +where h(r) is a smooth function and σi are Pauli matri- +ces introducing non-trivial flavor dependence, SU(2). As +result, the action for the flavour sector is written as +Sh = S(0) +DBI + S(1) +DBI + SCS +(33) +where S(0) +DBI is the DBI action in the absence of solitons, +S(1) +DBI is the expansion of the DBI action with homoge- +neous Ansatz at the second order, SCS is Chern-Simons +term with the homogeneous Ansatz (the explicit expres- +sions are given in [11]). + +7 +B. +The single layer solution +The solution for the bulk charge density is found by +considering the Φ equation of motion [11] +ρ′ = − d +dr +δSh +δΦ′ = −δSh +δΦ = 2Nc +π2 +d +dr +� +e−bτ 2h3(1 − 2bτ 2) +� +, +(34) +where b is a parameter in the Chern-Simons term, ρ is the +bulk charge density, and τ is the tachyon field. However, +the solution for ρ implied by this equation vanishes both +in the UV and in the IR. That is to say, diverging tachyon +in the IR set the solution to zero via exponential factor +and boundary condition for h in UV requires it to vanish +(since there is no external baryon source). Therefore, as +was the case in with the WSS model above, the baryon +density is zero, unless we impose an abrupt discontinuity +in the field h. +Motivated by these considerations, we write the “single +layer” solution for V-QCD as [11] +ρ = +� +ρ0 + 2Nc +π2 e−bτ 2h3(1 − 2bτ 2), +(r < rc) +2Nc +π2 e−bτ 2h3(1 − 2bτ 2), +(r > rc) +(35) +where ρ0 is boundary baryon charge density (the physical +density) and rc is the location of the discontinuity. The +explicit expression for ρ0 is +ρ0 = 2Nc +π2 e−bτ(rc)2(1 − 2bτ 2(rc))Disc(h3(rc)), +(36) +where we use the notation Disc(g(rc)) ≡ limϵ→0+(g(r + +ϵ) − g(r − ϵ)). +For future convenience, we briefly discuss the asymp- +totics of the field h. In the UV, h has the asymptotics +typical for gauge fields: +h ≃ h1 + h2r2 . +(37) +We require that non-Abelian sources vanish and there- +fore h1 = 0, but h2 remains as a free parameter (which +also determines rc for given ρ0, see appendix A 2). Fol- +lowing [11], we set h(r) = 0 for r > rc. +C. +The double layer solution +In this subsection, we generalize the single layer solu- +tion for baryon field h to have two discontinuities, i.e. +ρ = +� +� +� +ρ01 + 2Nc +π2 e−bτ 2h3(1 − 2bτ 2), +(r < rc1) +ρ02 + 2Nc +π2 e−bτ 2h3(1 − 2bτ 2), +(rc1 < r < rc2) +2Nc +π2 e−bτ 2h3(1 − 2bτ 2), +(r > rc2) +(38) +which we will be calling the double layer solution. There +is also a continuity condition on ρ that is to be satisfied, +which is given as +ρ02 = −2Nc +π2 (1 − 2bτ 2)e−bτ 2Disc h3|r=rc2 , +ρ01 − ρ02 = −2Nc +π2 (1 − 2bτ 2)e−bτ 2Disc h3|r=rc1 . +(39) +Therefore, summing the equalities above, we identify the +boundary baryon charge density ρ0 as ρ01: +ρ0 =ρ(r = 0) +=ρ01 = −2Nc +π2 +2 +� +i=1 +(1 − 2bτ 2)e−bτ 2Disc h3|r=rci . (40) +We stress however that even though we call this solu- +tion by the same name as the double layer solution for the +WSS model, these solutions are quite different. In partic- +ular, the double layer V-QCD solution has discontinuities +at two values of the holographic coordinate whereas the +WSS solution only has discontinuities at a single value. +Actually the single layer solution of the V-QCD model +is closer to the double layer solution of the WSS model +than the double layer solution of the V-QCD model. We +will discuss this difference in more detail below. +The double layer solution depends on four parameters +at fixed ρ0: there is one additional parameter from the +location of the extra discontinuity with respect to the +single layer solution, and as the solution for h in the sec- +ond interval rc1 < r < rc2 is independent of the solution +in the first interval, there are two additional integration +constants from the solution of h. Finally, the generaliza- +tion to triple layer or even to a solution with a higher +number of flavors is straightforward. One only needs to +modify the piecewise solution for the charge density ρ +with addition of new intervals. This will introduce three +new parameters for each interval. +D. +Legendre transform to canonical ensemble +As in the analysis of the WSS model above, it is con- +venient to work in the canonical ensemble. The Legendre +transformed action for V-QCD becomes [11] +�Sh = − +� +d5xVρG +� +1 + +ρ2 +(Vρwe−2A)2 +× +� +1 + 6w2e−4Ah4 + 6κτ 2e−2Ah2 +1 + ρ2(Vρwe−2A)−2 ++ 3w2e−4Af(h′)2 +2G2 +� +. +(41) +IV. +RESULTS +A. +Second order transition in the +Witten-Sakai-Sugimoto model +We start by analyzing the configurations in the WSS +model. We set λ = 16.63 [46] and analyse the solutions +numerically (see appendix A). As a function of the chem- +ical potential, we find three phases: + +8 +ρ0 = 0.0002 +ρ0 = 0.0003 +ρ0 = 0.0005 +ρ0 = 0.0010 +1 +2 +3 +4 +5 +z +-0.15 +-0.10 +-0.05 +h +1 +2 +3 +4 +5 +z +0.2 +0.4 +0.6 +0.8 +1.0 +ρ/ρ0 +FIG. 2. The profile of the gauge field h(z) (left) and the bulk charge density ρ(z) (right) for the single layer (solid curves) and +double layer (dashed curves) configurations for various values of the charge density. The vertical dashed lines in the left hand +plot denote the discontinuities of the double layer solutions at z = zc. +1. Vacuum for µ < µc with µc ≃ 0.205. +2. Single layer phase for µc < µ < µl with µl ≃ 0.342 +3. Double layer phase for µ > µl. +The phase transition at µ = µc (µ = µl) is of first (sec- +ond) order. +Here the second order transition (at the +higher value of the chemical potential, µ = µl), is iden- +tified as the popcorn transition. Notice that in the ap- +proach of [50], which used a different variation of the +homogeneous approach, both the vacuum to nuclear and +popcorn transitions were of first order. Even though we +are not attempting a serious comparison to QCD data, we +note that setting MKK = 949 MeV as determined by the +mass of the ρ meson [46], we have (for the quark chemical +potential) µc ≃ 195 MeV and µl ≃ 325 MeV, i.e., num- +bers in the correct ballpark. We note that µl/µc ≃ 1.67. +Denoting the density of the single layer configuration at +µ = µc as ρc (i.e., the analogue of the saturation den- +sity), the density ρl at the second order transition satis- +fies ρl/ρc ≃ 3.4. +Here we are mostly interested in the second order tran- +sition from the single to double layer phase. We show the +relevant configurations in figure 2 for a choice of densi- +ties ρ0 around the critical value ρl ≃ 2.52 × 10−4. Recall +that the single layer configuration is unique for fixed ρ0, +whereas the double layer configuration also depends on +zc. We show here the double layer profiles which mini- +mize the free energy. They are separate from the single +layer configuration only for ρ0 > ρl (the three highest +values in the figure), where they have lower free energies +than the single layer solutions. Interestingly, the single +and double layer solutions at the same ρ0 are close: The +functions h(z) deviate by at most a few percent in the +region z > zc. The deviations for ρ(z) are slightly higher, +and the single layer solution can be viewed as a smoothed +out version of the double layer solution. That is, even if +we were not considering the double layer solutions explic- +itly, their presence could be guessed from the single layer +solutions. In both cases, deviation is largest close to zc. +We also remark that the single layer profiles h(z) appear +to be qualitatively similar to the solutions found in the +approach of [50] (see figure 4 in this reference), up to a +shift by a constant. +B. +Analysis of configurations in V-QCD +We construct the double layer and single layer solution +by the procedure which is outlined in appendix A 2. The +essence of the procedure is the minimization of the free +energy density at fixed ρ0 depending on the free parame- +ters. In the case of the single layer, there is only one pa- +rameter: rc or equivalently h2, and it is straightforward +to solve the equation of state in this case. For the dou- +ble layer solution, there are four parameters which would +make the numerical minimization procedure challenging +in contrast to single layer solution. Therefore, while we +perform minimization of single layer solution for large +domain of ρ0 values, we investigate presence of lower the +free energy density of the double layer solution only for +solutions obtained by gluing together single layer solu- +tions for some representative values of ρ0 changing from +0.8 to 2.5. +Denoting ∆hi = Disc h(rci), we investigate three qual- +itatively different configurations: we consider ∆h1 < 0 +and ∆h1 > 0 for double layer solution and ∆h1 > 0 +, ∆h2 > 0 for a triple layer solution. +For bound- +ary baryon number charge we consider the values of +ρ0 = 0.5, ρ0 = 0.8 and ρ0 = 2.5, which will correspond to + +9 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.0 +0.5 +1.0 +1.5 +r +h +rc1 +rc2rc +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.0 +0.2 +0.4 +0.6 +0.8 +r +ρ +rc1 +rc2rc +ρ0 =ρ01 +ρ02 +0.46 +0.52 +0.5 +0.8 +ρ02 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.0 +0.5 +1.0 +1.5 +r +h +rc1 rc2 rc +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.0 +0.2 +0.4 +0.6 +0.8 +r +ρ +rc1 rc2 rc +ρ0 =ρ01 +ρ02 +0.46 +0.52 +0.5 +0.8 +ρ02 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.0 +0.5 +1.0 +1.5 +r +h +rc1rc2 rc3rc +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.0 +0.2 +0.4 +0.6 +0.8 +r +ρ +rc1 rc2rc3 rc +ρ0 =ρ01 +ρ02 +ρ03 +0.46 +0.52 +0.5 +0.8 +ρ02 +ρ03 +FIG. 3. The profile of the gauge field h(r) (left) and the bulk charge density ρ(r) (right) for double layer with ∆h1 < 0 (first +row), double layer with ∆h1 > 0 (second row), triple layer solution with ∆h1 > 0 and ∆h2 > 0 (third row) configurations. The +single layer configuration with same boundary charge density ρ0 = 0.8 is showed with the gray dashed curve in each plots. The +parameters rci and ρ0i that characterize the multilayer configurations are shown by blobs. The values of rci, h2i, ρ0i and f are +given in table I. +µ/µc = 1.65, µ/µc = 2.04 and µ/µc = 3.57 for the ther- +modynamics determined by the single layer solution, re- +spectively. While the first choice roughly corresponds to +chemical potential values in which double layer solutions +is WSS become dominant (as it is seen from figure 3), +the other two choices are even much larger than that. +In figure 3, the results for the three representative case +are shown. The baryon field profile h(r) and correspond- +ing baryon number densities ρ0(r) in the bulk are shown +in the first and second column respectively. In each plot, +the single layer solution minimizing the free energy is +shown with gray dashed curves whose parameters are + +10 +rc +h2 +ρ0 +f +0.570 +2.991 +0.80 +1.745 +{0.483, 0.539} +{3.90, 3.10} +{0.80, 0.59} +2.003 +{0.487, 0.525} +{2.90, 4.10} +{0.80, 0.72} +1.626 +{0.476, 0.498, 0.533} {2.90, 3.20, 3.90} {0.80, 0.74, 0.64} 1.642 +TABLE I. The values of {rc, h2, ρ0, f} for the single layer configuration (first row) and {rci, h2i, ρ0i, f} for the multi layer +configurations (second-forth rows) that is shown in figure 3. +given in the first row of table I. The red, blue and green +solid curves show ∆h1 < 0 and ∆h1 > 0 double layer and +(∆h1 > 0, ∆h2 > 0) triple layer solutions. The parame- +ters rci, h2i, ρ0i, where h2i are the asymptotic constants +h2 for the single layer solutions that were glued together +to obtain the multilayer solutions, and the corresponding +free energy densities f are shown in table I. The locations +of the discontinuities and ρ0i are also shown in the figures +with the blobs. +We were able to find double layer solutions which have +lower free energy than the single layer solution for the +cases of ∆h1 > 0 i.e., the second row of figure 3. However, +we were not able to find double solutions with ∆h1 < 0 +that would have lower free energy than the single layer +solution (configurations in the first row of the figure). +Notice that having solutions with ∆h1 > 0 means that +contributions to the total charge from the two discontinu- +ities have opposite signs. This means that in the instan- +ton picture, the discontinuities must arise from smear- +ing instantons with opposite charges. This suggests that +proton-antiproton pairs are created, which should be for- +bidden due to the large energy required for such a pair +creation. Therefore the configuration of the first row is +not physically sound. We suspect that it appears because +the homogeneous approximation works poorly with con- +figurations with discontinuities at several values of the +holographic coordinate. We also show the example of a +triple layer configuration with ∆h1 > 0 and ∆h2 > 0 on +the third row of the plot. +C. +Speed of sound and polytropic index +We now study the physical implications of the phase +transition. To this end, we plot the speed of sound and +the polytropic index γ = d log p/d log ϵ for the WSS and +V-QCD models in figure 4. In these plots, the chemical +potential was normalized using the value at the vacuum +to nuclear matter transition. +In both models, the speed of sound is below the value +c2 +s = 1/3 of conformal theories right above the transi- +tion to nuclear matter. When µ increases, however, the +speed of sound crosses this value and reaches values well +above it [11]. The speed of sound has a maximum in both +model. Even though the location of the maximum is dif- +ferent between the models, the maximal values are rather +close: the maximum of c2 +s is 0.463 (at µ/µc = 1.355) for +the WSS model and 0.504 (at µ/µc = 2.246) for V-QCD. +Eventually at higher densities, the speed of sounds de- +creases to values closer to the conformal value. This is +clearer in the WSS than in the V-QCD model. In the +WSS model, where the popcorn transition from a sin- +gle to a double layer configuration is found, the speed +of sound drops to a roughly constant value which closely +agrees with the conformal value in the double layer phase: +the speed of sound squared is about one per cent higher +than the conformal value 1/3. +Similar results are found for the polytropic index γ in +the right hand plot of figure 4. In both models, γ de- +creases with µ in the (single layer) nuclear matter phase. +This decrease is fast in the sense that γ drops below +the value of γ = 1.75, which was used as a criterion +to separate nuclear matter from quark matter in [67, 68], +where equations of state obtained as interpolations be- +tween known results from nuclear theory at low density +and perturbation theory at high density were considered. +For µ/µc > 1.5 the results from both model are below this +value. At the popcorn transition of the WSS model, γ +drops to a value close to one. +Our findings indicate that the homogeneous holo- +graphic nuclear matter behaves approximately confor- +mally at high densities, i.e., at densities well above the +nuclear saturation density (see also [101]). This is partic- +ularly clear for the WSS model, which becomes approxi- +mately conformal at the popcorn transition. These find- +ings are consistent with earlier studies of homogeneous +nuclear matter in the WSS (see, e.g., [102]) and the V- +QCD (see, e.g., [94]) models. They also agree with the +results found in the effective theory approach of [62, 64]. +This agreement in strikingly good for the WSS model, +where the results both for the speed of sound (see [64]) +and for the polytropic index (see [65]) have been com- +puted. For example, our results for the maximal value of +the speed of sound (our value is cs,max ≈ 0.68) and the +density at the popcorn transition (we found nl/nc ≈ 3.4) +agree rather well with those of these references – our value +for the speed of sound (transition density) is a bit below +(above) the values of the effective theory approach. +We also remark that the non-monotonic behavior for +the speed of sound in the WSS model qualitatively agrees +with that found in the point-like instanton gas approach +in [15], albeit with a different embedding for the D8 +branes. The maximal value found in this reference is also +close to the maximal value obtained here. This agree- + +11 +1.0 +1.5 +2.0 +2.5 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +μ/μc +cs2 +V-QCD s.l. +WSS s.l. +WSS d.l. +1.0 +1.5 +2.0 +2.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +μ/μc +γ +V-QCD s.l. +WSS s.l. +WSS d.l. +FIG. 4. +The speed of sound (left) and the polytropic index γ = d log p/d log ϵ (right). +The solid red, dashed green, and +dot-dashed blue curves are the results for the single layer configuration in the WSS model, double layer configuration in the +WSS model, and the V-QCD model, respectively. +ment is interesting as it obtained in a completely different +approach, which is expected to be reliable at lower den- +sities. Moreover we compare our results to the different +approach of homogeneous nuclear matter derived in [50] +in appendix B, and mostly find qualitative agreement. +V. +CONCLUSIONS +In this article, we analyzed nuclear matter using a ho- +mogeneous approach in two different holographic mod- +els: in the top-down WSS model and in the bottom-up +V-QCD model. We focused on two topics: the popcorn +transitions, where the layer structure of the nuclear mat- +ter changes in the bulk, and approach to conformal be- +havior at high densities. We found a second order pop- +corn transition in the WSS model, and signs of approach +to conformality in both holographic setups. +We have several remarks about our results. +Firstly, +the results in the WSS and V-QCD models appeared to +be quite different: in particular, the popcorn transition +was only found to take place in the WSS model. This is +however not surprising at all and can be seen to follow +from the differences in the geometry and the realisation +of chiral symmetry breaking between the models as we +now explain. Recall that in the WSS model, the geome- +try ends at the tip of the cigar in the confined phase as +shown in figure 1, and chiral symmetry breaking is re- +alised by the joining of the two branches of flavor branes +at the tip. In the V-QCD picture there is no cigar struc- +ture, and chiral symmetry breaking arises from a con- +densate of a bulk scalar field. In the WSS model, nuclear +matter at low densities is seen to arise from instantons +located at the tip, and it is not possible to assign such +instantons to be left or right handed. In V-QCD, how- +ever, nuclear matter is stabilized at a nontrivial value of +the holographic coordinate due to interaction with the +bulk scalar field [11], and by definition always contains +left and right handed components. Therefore, in V-QCD +separate configurations analogous the single and double +layer configurations of the WSS in figure 1 do not exist. +The configurations of this figure map to the same con- +figuration in V-QCD, which is what we called the single +layer configuration. The double layer configuration in V- +QCD defined in (38) would map to a more complicated +configuration in the WSS model where discontinuities of +the h field are found at two distinct values of z. +We found that the results for the equation of state near +the popcorn transition of the WSS model closely resem- +ble those obtained by the framework of [62, 64], where +effective theory was used to analyse the transition of the +Skyrmion crystal to a crystal of half-Skyrmions. +This +suggests that the transition in the holographic model +should be identified with the topology changing transi- +tion where half-Skyrmions appear.2 +It is however dif- +ficult to say anything definite about this because the +holographic approach which we used does not contain +individual instantons. +Moreover, in [50] it was argued +that the topology changing transition should not be iden- +tified as the transition between the single and double +layer solutions, but should take place between solutions +of qualitatively different behavior within the single layer +solution. Another point is that chiral symmetry should +be restored globally at the topology changing transition +(meaning that the averages of the condensate over large +regions should vanish). This however will not happen for +any of the configurations in the WSS approach because +the D8 brane action is treated in the probe approxima- +tion, and the embedding of the brane is independent of +the density. Nevertheless we remark that, as seen from +the expressions for the single and double layer configu- +rations in (20) and in (22), the bulk charge density has +2 We thank N. Kovensky and A. Schmitt for correspondence on +this question. + +12 +support near the tip of the cigar only for the single layer +configuration, where the flavor branes join, breaking chi- +ral symmetry. Therefore the double layer configuration +can also exist in chirally symmetric backgrounds. Exam- +ples of such chirally symmetric double layer configura- +tions were indeed found in [17] (the chirally symmetric +quarkyonic matter phase of this reference). +Finally, we demonstrated that the homogeneous nu- +clear matter becomes approximately conformal at high +densities, i.e., above few times the nuclear saturation +density. That is, the values of the speed of sound lay +close to the value c2 +s = 1/3 of conformal theories, and +similarly γ lay values close to the value γ = 1. In par- +ticular, the polytropic index reached values well below +the value γ = 1.75 both in the V-QCD model and in +the WSS model, which has been used to classify equa- +tions of state for nuclear and quark matter in the ap- +proach of [67, 68]. That is, the part of the single layer +phase and all of the double layer phase would be clas- +sified as quark matter in this approach. +This appears +consistent with the interpretation that the double layer +phase is smoothly connected to quark matter [69]. +In +the V-QCD setup, however, there is a separate strong +first order phase transition from nuclear to quark mat- +ter at higher densities [11, 94]. In the WSS model there +is a separate quark matter phase also, but in this case +the transition is weak and even continuity between the +phases is a possibility [10]. +ACKNOWLEDGMENTS +We thank Mannque Rho for the invitation to con- +tribute to the special issue “Symmetries and Ultra +Dense Matter in Compact Stars” in Symmetry. +We +also thank Elias Kiritsis, Nicolas Kovensky, Yong-Liang +Ma, and Andreas Schmitt for discussions and correspon- +dence. This work benefited from discussions during the +APCTP focus program “QCD and gauge/gravity dual- +ity”. J. C. R. and M. J. have been supported by an ap- +pointment to the JRG Program at the APCTP through +the Science and Technology Promotion Fund and Lottery +Fund of the Korean Government. J. C. R. and M. J. have +also been supported by the Korean Local Governments +– Gyeongsangbuk-do Province and Pohang City – and +by the National Research Foundation of Korea (NRF) +funded by the Korean government (MSIT) (grant number +2021R1A2C1010834). T.D. acknowledges the support of +the Narodowe Centrum Nauki (NCN) Sonata Bis Grant +No. 2019/34/E/ST3/00405. +Appendix A: Numerical details +1. +Constructing the solution in the +Witten-Sakai-Sugimoto setup +Here we summarize the basic steps we follow to find +the free energy and the equation of state for the case of +the simple profile for the charge density (20) +1. We derive from the action (27) the equation of mo- +tion for h(z). +After plugging the baryon charge +density ρ and fixing Nc → 3 and λ → 16.63, the +only free parameter is the boundary charge density +ρ0. Then we can simply solve the equation for h(z) +for fixed ρ0 from the UV boundary (we still need +to fix h1). +2. We fix the value of h1 by solving for h for a given +fixed ρ0 and chose a value of h1 such that ρ(h) = 0 +at z = 0 , we can determine bulk charge density ρ +profile by considering (20). +3. The free energy density is given by explicit integra- +tion of (27) from zero to a large cut-off. At this +step, we (re)normalize the free energy by subtract- +ing �S in the absence of baryons from the original +�S. +4. From the tabulated data {ρ0, F}, we can construct +F(ρ0) and find at which value of ρ the transition to +nuclear matter happens. The corresponding chem- +ical potential and grand potential can be obtained +via µ = dF/dρ0 and Ω = F − ρ0µ = −p. +For the case of the more general solution (22), we need +to find the value zc where the charge density vanishes but +then the procedure to find the energy as a function of ρ0 +is analogous to the single layer solution above. However +one difference with respect to the previous single layer +solution is that the value of h1 that minimizes the energy +changes for densities larger than a critical value. +From the comparison of the free energy we can see that +there is a second order phase transition at this critical +density ρc from the single layer solution to the double +layer solution. +2. +Constructing the solution in the V-QCD setup +In this subsection, we summarize and outline the calcu- +lation of free energy density and minimization procedure: +1. We work in the probe limit. +We first construct +the thermal gas background solution for the geom- +etry [76] in the absence of the baryons. +2. Then, from (41) we derive equations of motion for +h. +After plugging background fields and baryon +charge density ρ, the only free parameter is the +boundary charge density ρ0. So we can simply solve + +13 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +μ/μc +cs2 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +μ/μc +γ +FIG. 5. The speed of sound (left) and the polytropic index γ = d log p/d log ϵ (right) for single layer (cyan curves) and double +layer (magenta curves) solutions in the approach of [50]. The gray curves show the WSS and V-QCD results that are presented +in figure 4. +the equation of motion for h for fixed ρ0 by from +UV boundary. +3. After solving for h for given fixed ρ0 and chosen h2, +we can determine bulk charge density ρ profile by +considering (35). Note that the vanishing point of +bulk density profile gives the location of the soliton, +i.e ρ(rc) = 0. +4. The free energy density is given by explicit inte- +gration of (41) from boundary to the location of +the discontinuity. +At this step, we also subtract +�Sh in the absence of baryons from original �Sh to +(re)normalize the free energy. +5. Now, we can return to our main purpose of min- +imizing free energy at fixed ρ0 depending on free +parameter rc or equivalently h2. +We can simply +perform above mention procedure with a loop over +h2 values. +6. From the tabulated data, we can construct F(h2) +and minimize it. The corresponding chemical po- +tential and grand potential can be obtained via +µ = dF/dρ0 and Ω = F − ρ0µ = −p. +For the case of the multi-layer configurations, the +number of parameters which should be used in the +minimization procedure increase and this makes the +similar analysis numerically expensive. This is beyond +the scope of this project. +Therefore, we decide to +analyze the situation by considering some representative +situations (the details of them are given in the main +text in subsection IV B) and searching for solutions with +lower f than that of single layer configurations. +Appendix B: Comparison to a different +homogeneous approach +In this appendix we compare our results to those ob- +tained by employing the homogeneous approach of [50], +where one uses a zero curvature condition before taking +the system to be homogeneous in the WSS model. We +set the parameter Λ = 8λ/(27π) to the value Λ ≈ 1.568 +for consistent comparison with our approach. The results +are shown in figure 5. We see that the maximal value of +the speed of sound is higher in the approach of [50] than +in the other approaches, and the value of µ at the pop- +corn transition is likewise higher than in the approach we +used here. 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Schmitt, SciPost Phys. 11, 029 +(2021), arXiv:2105.03218 [hep-ph]. + diff --git a/WdE1T4oBgHgl3EQfbgRN/content/tmp_files/load_file.txt b/WdE1T4oBgHgl3EQfbgRN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b319b452f0c2712cdfe6c69cde320cc50eb32d63 --- /dev/null +++ b/WdE1T4oBgHgl3EQfbgRN/content/tmp_files/load_file.txt @@ -0,0 +1,1377 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf,len=1376 +page_content='APCTP Pre2022 - 030 Popcorn transitions and approach to conformality in homogeneous holographic nuclear matter Jes´us Cruz Rojas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='1 Tuna Demircik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='2 and Matti J¨arvinen1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 3 1Asia Pacific Center for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Pohang 37673,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Republic of Korea 2Institute for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Wroc�law University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 50-370 Wroc�law,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Poland 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Pohang University of Science and Technology Pohang 37673,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Republic of Korea We study cold and dense nuclear matter by using the gauge/gravity duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' To this end we use the Witten-Sakai-Sugimoto model and the V-QCD models with an approach where the nuclear matter is taken to be spatially homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We focus on the “popcorn” transitions, which are phase transitions in the nuclear matter phases induced by changes in the layer structure of the configuration on the gravity side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We demonstrate that the equation of state for the homogeneous nuclear matter becomes approximately conformal at high densities, and compare our results to other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' INTRODUCTION Recent observations of neutron star mergers by the LIGO/Virgo collaboration have opened a new window for studying dense matter in quantum chromodynam- ics (QCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In particular, the gravitational and electro- magnetic waves observed from the GW170817 merger event [1] already set highly nontrivial constrains for the QCD equation of state [2] at low temperatures and high densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This progress has boosted interest in theoret- ical studies of dense QCD, which is a challenging topic as standard theoretical and computational tools do not work in extensive regions of the phase diagram (see the overview in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' These include the region of dense nuclear matter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=', a nucleon liquid at baryon number densities well above the nuclear saturation density ρs ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='16 fm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The difficulty of solving the properties of dense mat- ter calls for new methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' A possibility is to use the gauge/gravity duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Indeed, applications of holo- graphic QCD models to dense matter have received a lot of interest recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' There has been progress in de- veloping models both for quark matter [4–9], for nuclear matter [10–14], and for other phases (such as color super- conducting or quarkyonic phases) [15–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' See also the reviews [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The natural starting point for describing nuclear mat- ter is to study the holographic duals for nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The standard approach [21] boils down to describing them as solitonic “instanton” solutions of bulk gauge fields, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=', the gauge fields living in the higher dimensional grav- ity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' These solitons are localized both in the spa- tial directions and in the holographic direction (but not in time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Solitons that are duals of isolated nucleons have been solved in various holographic models [22–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Constructing more complicated solutions, and eventu- ally the holographic dual of dense nucleon matter out of these bulk solitons, is however challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Some re- sults, which use instanton gases without interactions, are available [13, 32–36] and also with two-body interactions included [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Moreover, at large Nc the nuclear matter is a crystal rather than a liquid of nucleons [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Such crystals have been studied by using different toy models and approximations [38–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In this article, we focus on a simpler approach, which treats dense nuclear matter as a homogeneous configura- tion of the non-Abelian gauge fields in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This ap- proach was applied to the Witten-Sakai-Sugimoto (WSS) model [44–46], in [47], and argued to be a reasonable ap- proximation at high density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='1 It was further developed in [49, 50] and applied to other models in [11, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In- terestingly, dense (and cold) homogeneous holographic nuclear matter was seen to have a high speed of sound, clearly above the value c2 s = 1/3 of conformal theories [11] (see also [51, 52]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' That is, the equation of state is “stiff”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This is important as it helps to construct models which pass observational bounds [53–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Changes in the structure of dense nuclear matter may give rise to transitions within the nuclear matter phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' At large Nc, nuclear matter is a crystal of Skyrmions, solitons of the low energy chiral effective theory [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' As the density increases, the Skyrmion crystal is expected to undergo a transition into half-solitons, where each node of the crystal carries baryon number of one half [57– 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Similar structures have been studied by using the gauge/gravity duality in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This topology changing transition has been studied extensively by using an effective field theory approach, which introduces the σ meson of QCD as a pseudo- Nambu-Goldstone mode of broken scale invariance, and vector mesons through the hidden local symmetry ap- proach [61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This approach is supported by the analysis of the nucleon axial coupling gA for heavy nu- clei [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Above the transition density, it was found that the speed of sound rapidly approaches the confor- mal value c2 s = 1/3 [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' At the same time, the poly- tropic index γ = d log p/d log ϵ takes small values [65, 66] as compared to what is usually found in nuclear theory models [67, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The transition has also been argued [69] to be indicative of quark-hadron continuity [70], which 1 An even simpler approach is to treat the baryons as point-like sources in the bulk, which may be a better approximation at low density [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='03173v1 [hep-th] 9 Jan 2023 2 states that there is no phase transition between nuclear and quark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Whether the continuity is a feasible possibility is a matter of ongoing debate, see [71–74] (see also [15] for a holographic discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' A closely related transition realised in holographic se- tups is the transition from a single-layer configuration into a double layer configuration: In the low density limit, the location of each soliton is found by individ- ually minimizing its energy, so that the solitons form a single layer at a specific value of the holographic coor- dinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' For dense configurations, however, the repulsive interactions between solitons will eventually force them out of this layer, which leads to a double layer or a more complicated configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This transition was coined the “popcorn” transition in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' If interactions between the solitons are attractive at large distances (as is the case for real QCD) the picture is more complicated as the solitons clump together even at low densities, but the transition may still be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Various phases ap- pear as the density increases further [41, 42] in setups motivated by the WSS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The simplest case of the transition is however the separation of a single layer into two layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This kind of transition was also found to take place in the WSS model in various approximations: when the instantons were approximated as point-like ob- jects [17], when including finite widths [36], and when using a homogeneous approach [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Indications of such a transition were also seen when using a homogeneous Ansatz for nuclear matter in the hard wall model of [14], where it was interpreted as a transition to a quarkyonic phase [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In this article, we study the popcorn transitions within cold homogeneous holographic nuclear matter by using two different models, the top-down WSS model and the bottom-up V-QCD model [19, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' These two are ar- guably the most developed holographic top-down and bottom-up models for QCD at finite temperature and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' For the WSS model a similar analysis was car- ried out in [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This reference used an approach which is slightly different from ours: In their case, a zero cur- vature condition for the non-Abelian gauge fields in the Lagrangian density is imposed before approximating the density to be homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We use a somewhat simpler approach where the fields are assumed to be homoge- neous to start with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In our case, as we will discuss in de- tail below, a discontinuity of the gauge fields as a function of the holographic coordinate is required to have nonzero baryon density [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This may appear to be a weakness of the simpler approach, but we remark that the disconti- nuity is actually well motivated, as it can be seen to arise from non-analyticity of the instanton solutions at their centers after smearing over the spatial dimensions [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The main goal in this article is to analyze the soften- ing of the equation of state at the phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The main indicators for this are the speed of sound and the polytropic index γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We compute these quantities in both holographic models and compare to results in other se- tups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In particular, we find interesting similarities with the effective theory approach for the topology changing transition [62, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The rest of the article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In sec- tion II we review the setup with homogeneous nuclear matter for the WSS model, and in section III we do the same for the V-QCD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In section IV we discuss the numerical results for the solutions, the phase transitions, and the equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Finally, we discuss our findings in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' HOMOGENEOUS NUCLEAR MATTER IN THE WITTEN-SAKAI-SUGIMOTO MODEL The phase diagram of QCD has been studied by using several holographic “top-down” models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=', models di- rectly based on string theory, such as the Witten-Sakai Sugimoto model [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In this model, Witten’s non- supersymmetric model for low-energy QCD [44] has been successfully applied to study the spectra and the proper- ties of mesons and baryons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In the WSS model, the pure glue physics of the QFT is described by the dual gravitational background and it is sourced by Nc D4-branes in type-IIA superstring theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Fundamental degrees of freedom are included by adding Nf pairs of D8 and D8-branes, such that the strings connecting D4 − D8 and D4 − D8 branes are associated with left and right-handed fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Witten’s model includes a phase transition involving a topologically nontrivial change in geometry from a low temperature “cigar” geometry to a high temperature black hole geometry [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We focus here in the low tem- perature geometry which we will give explicitly below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In the low temperature geometry the D8 and D8-branes join at the tip of the cigar (see figure 1), which locks together the flavor transformations on the branes, indi- cating chiral symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' As shown in the figure we assume the simplest case where the D8 and D8-branes are antipodal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=', at located at exactly opposite curves on the cigar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' A chemical potential for the baryon num- ber can be turned on by adding a nonzero source for the temporal component of the Abelian gauge field on the D8-branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Expanding the Dirac-Born-Infeld action The 10-dimensional metric of the confined low tem- perature geometry in the Witten model can be written as [78, 79] ds2 = �U R �3/2 � dxµdxµ + f(U)dx2 4 � + �R U �3/2 � dU 2 f(U) + U 2dΩ2 4 � (1) 3 x4 UKK ∞ U z =+∞ z =0 z =-∞ z =-zc x4 UKK ∞ U z =-∞ z =+∞ z =+zc FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Setup in the WSS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The coordinate z runs between z = −∞ and z = ∞ between the two boundaries of the D8 brane embedding as indicated in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The blobs show the locations of the discontinuities for the single layer configuration (left) and for the double layer configuration (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' where R is the curvature radius, f(U) = 1 − U 3 KK U 3 (2) with UKK denoting the end of space, and dΩ2 4 is the metric of S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' For the Minkowski metric dxµdxµ we use mostly plus conventions, and the x4 coordinate is com- pactified on a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The dilaton is given by eφ = gs �U R �3/4 , (3) where gs is the string coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In the (x4, U)-coordinates this geometry takes the form of a cigar and the regularity at the tip of the cigar links the radius of compactification R4 of the x4 coordinate to the Kaluza-Klein scale characterized by UKK, as R4 = (4π)R3/2 3√UKK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The simplest D8 brane embedding within the cigar geometry is the antipodal one, given (for example) by x4 = 0 and x4 = πR4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' By changing the coordinates to U = UKK(1 + z2)1/3 the induced metric on the brane can be written as ds2 ind = �UKK R �3/2� 1 + z2 dxµdxµ + � R UKK �3/2 4dz2 9 (1 + z2)5/6 + R3/2� UKK 6� 1 + z2dΩ2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' (4) Here the coordinate z takes both positive and negative values on different branches of the brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The boundary is at z = ±∞ and the tip of the cigar at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' See Figure 1 for illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We work in units where UKK = 1 and R3 = 9/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We also start from the Dirac-Born-Infeld action SDBI = −τ8 � d9x e−φtr � − det(g + F) , (5) where the trace is over flavor indices the brane tension is given by τ8 = 1 (2π)8l9s = λ9/2 157464 √ 2π8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' (6) Here ls is the string length, λ is the ’t Hooft coupling, F is the field strength tensor of the gauge field A, and we used the relations R3 = πgsNcl3 s and 2πlsgsNc = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We use a similar expansion as in the case of V-QCD be- low [11] so that the non-Abelian component of the gauge fields are treated as small but the Abelian terms are kept unexpanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' To do so, we separate the gauge field into non-Abelian and Abelian components: A = A+ ˆA where A is non-Abelian and ˆA is Abelian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=', proportional to the unit matrix in flavor space (and similarly F = F + ˆF for the field strengths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We take only the temporal com- ponent of the Abelian gauge field to be nonzero, assume that it depends only on the holographic coordinate z, and assume no dependence on the angular coordinates of Ω4 for all fields, so that these coordinates can be integrated out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 4 Then the five-dimensional effective action for the gauge fields,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' to leading order in the non-Abelian F 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' is given as S = S(0) DBI + S(1) DBI + SCS (7) where the terms arising from the Dirac-Born-Infeld action read S(0) DBI = −λ3NcNf 19683π5 � d5x 3� 1 + z2 � (1 + z2)2/3 − (1 + z2) Φ′(z)2 (8) and S(1) DBI = − λNc 216π5 � d5x tr � − F 2 tz � 1 + z2� � 1 − 3√ 1 + z2Φ′(z)2�3/2 − F 2 ti 3√ 1 + z2 � 1 − 3√ 1 + z2Φ′(z)2 + F 2 ij � 1 − 3√ 1 + z2Φ′(z)2 2 3√ 1 + z2 + F 2 zi � 1 + z2� � 1 − 3√ 1 + z2Φ′(z)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' (9) Notice that the general Dirac-Born-Infeld action is am- biguous for non-Abelian fields, but up to second order in the expansion the action is non-ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The Chern- Simons term is SCS = Nc 24π2 � � ω5 + d � ˆA ∧ tr � 2A ∧ F + i 2A3 �� +3 ˆA ∧ tr (F ∧ F) � (10) with the Abelian gauge field normalized as Φ = 2λ ˆAt/( √ 729π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Here ω5 = tr � A ∧ F 2 + i 2A3 ∧ F − 1 10A5 � (11) gives the standard Chern-Simons term for the brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We used conventions where F = dA − iA ∧ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Notice that in (10) the Abelian field couples to the instanton density in the bulk as expected (see the last term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Indeed, notice that S(0) DBI and S(1) DBI depend on the Abelian gauge field only through its z-derivative, and only SCS contains non- derivative dependence on this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The total baryon charge density is defined as ρ0 = − � δS δ ˆA′ t � bdry = � dz δS δ ˆAt , (12) according to holographic dictionary, where only the Chern-Simons term contributes to the last expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Therefore the baryon charge is given by the coupling of the non-Abelian field to ˆAt in SCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In other words, the Chern-Simons term determines how the solitons source baryonic charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We also remark that the construction of the precisely consistent Chern-Simons term is actually rather involved, in general [80], but in the simple case considered here complications do not arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The homogeneous Ansatz Then as the next step, we set Nf = 2 and insert the homogeneous Ansatz Ai = h(z)σi (13) where h(z) is a scalar function and σi are the Pauli ma- trices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The non-Abelian At and Az components are set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We then find that Fzi = h′(z)σi , Fij = 2h(z)2ϵijkσk (14) while other components of the field strength are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We obtain S(1) DBI = − λNc 36π5 � d5x �4h(z)4 � 1 − 3√ 1 + z2Φ′(z)2 3√ 1 + z2 + (h′(z))2 � 1 + z2� � 1 − 3√ 1 + z2Φ′(z)2 � (15) and the Chern-Simons action contributes as SCS = 3Nc π2 � h(z)2h′(z) ˆA ∧ dz ∧ dx1 ∧ dx2 ∧ dx3 (16) as well as a boundary term SCS,bdry = 3Nc 4π2 � bdry h(±∞)3 ˆA ∧ dx1 ∧ dx2 ∧ dx3 (17) which however will vanish when it is evaluated on the solution in our case, because the solution for h(z) will vanish on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The single layer solution In order to have explicit parity invariance, we assume that h(z) = −h(−z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Following [47], we assume that 5 the field h has a discontinuity at z = 0, denoted by the blob in figure 1 (left), and approaches different constant values as z → 0 either from above or from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' As we mentioned above, the discontinuity is required to have a non-vanishing baryon density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Defining the bulk charge density as ρ(z, xµ) = − δS δ ∂z ˆAt(z, xµ) (18) the equation of motion for ˆA implies ρ′(z) = 3Nc π2 h(z)2h′(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' (19) The continuous and symmetric solution is given by ρ(z) = � ρ0 + Nc π2 h(z)3 , (z < 0) −ρ0 + Nc π2 h(z)3 , (z > 0) (20) where ρ0 = Nc π2 lim z→0+ h(z)3 = −Nc π2 lim z→0− h(z)3 (21) is the boundary charge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Notice that as expected, it is sourced by the discontinuity of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This solution is identified as the single layer solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' To finalize the construction, we require that h satisfies the equation of motion arising from minimizing the action, except at z = 0 where the discontinuity is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The double layer solution A slightly more general solution than the single layer solution however exists: it is possible that the disconti- nuity of the h field does not take place at the tip but at a generic value of the holographic coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The simplest of such solutions, which still respects the sym- metry h(z) = −h(−z), is where h(z) vanishes when −zc < z < zc so that the discontinuity is located at z = ±zc, see figure 1 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Similar solutions were con- sidered in [17] for point-like instantons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In this case, the solution for the bulk charge density is given by ρ(z) = � � � ρ0 + Nc π2 h(z)3 , (z < −zc) −ρ0 + Nc π2 h(z)3 , (z > zc) 0 , (−zc < z < zc) (22) where ρ0 = Nc π2 lim z→zc+ h(z)3 = −Nc π2 lim z→(−zc)− h(z)3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' (23) This solution is identified as the double layer solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Legendre transform to canonical ensemble To determine the phase diagram, one needs to deter- mine the free energy, the energy densities, and the grand potential for the different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Thus, we first need to compute the free energy of the baryonic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' For this purpose we minimize the action for h to determine the location of the discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' It is convenient to work at fixed baryonic charge rather than chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' To this end, we perform a Leg- endre transformation for the action (7): �S = S + � d dz � ˆAt ρ � dz (24) For convenience we rescale ρ as ρ → 4λ4 531441π6 ρ ≡ ˆρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Ex- panding to first nontrivial order in h(z) and h′(z), and using equation (18), we can solve for Φ′(z): Φ′(z) = − ˆρ R(z, ˆρ)(1 + z2)Nc − 2187ˆρ � −4h(z)4(1 + z2) 2 3 + h′(z)2(1 + z2)2R(z, ˆρ)2� 8λ2Nc(1 + z2) 8 3 R(z, ˆρ)3 , (25) where we define R(z, ˆρ) = � 1 + ˆρ2 (1 + z2)5/3 N 2c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' (26) Then the Legendre transformed action is �S = − Nc � d5x � 2λ3(1 + z2) 2 3 R(z, ˆρ) 19683π5 + λ � 4h(z)4(1 + z2) 2 3 + h′(z)2 � (1 + z2)2R(z, ˆρ)2�� 36π5 ((1 + z2)R(z, ˆρ)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' (27) Now we can find the equation of motion for h(z) and solve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' For this purpose, we need to find the appropriate asymptotics of the field h at the boundary: h(z) ≃ h1 z , (28) 6 with h1 remaining as a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' HOMOGENEOUS NUCLEAR MATTER IN THE V-QCD MODEL V-QCD is bottom-up holographic model which con- tains both glue and flavor sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The glue sector is given by the improved holographic QCD framework [81, 82] in which a dilaton field and the potential de- pending on it are used to implement the essential fea- tures of the related QCD sector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' asymptotic freedom, and confinement to deconfinement phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The flavour sector arises from a pair of dynamical space filling flavor branes [83, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In V-QCD, the full back-reaction of the flavor branes is taken into account via the Veneziano limit [85] in which both Nc and Nf are large but their ratio is kept O(1) as it is in real QCD [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In the V-QCD flavor sector, a tachyon field is used to realize the break- ing/restoration of the chiral symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In both sectors, the model parameters are also fixed by considering per- turbative QCD results (running of coupling constant and quark mass) at weak coupling [76, 81, 82], by requiring qualitative agreement with QCD (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' confinement and discrete spectrum) at strong coupling [86], and by fitting to QCD data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' meson and glueball masses and the equation of state at finite temperature) [6, 31, 87–89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' For the more complete review about the construction of the V-QCD model, the fit to fix the potentials and com- parison with the data, we refer the reader to [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In this article, we use one of the models defined in [6] (potentials 7a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The parameter b appearing in the Chern-Simons ac- tion [11] is set to b = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' There are two possible geometries in V-QCD: a horizon-less geometry ending in a “good” kind of singu- larity [90] (dual to a chirally broken confined phase) and a geometry of charged “planar” black hole [91, 92] (dual to a chirally symmetric deconfined phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In this arti- cle, we focus on the former geometry which is the relevant geometry for cold and dense nuclear matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This phase also includes chiral symmetry breaking which is induced by the condensate os a scalar field τ (the “tachyon”) in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In order to discuss nuclear matter, we will employ here an approach which is essentially the same as the homoge- neous approach introduced for the WSS model above [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This approach has been improved by combining the predictions of V-QCD with other models [93–96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The re- sultant equation of states have been widely investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' It was shown that the resultant equations of state are fea- sible in the sense of being consistent with neutron star observations [93, 94, 96–100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' They were also used in phenomenological applications such as modeling spinning neutrons stars [98] and neutron star merger simulations [93, 99, 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In the first two subsections below, we outline the imple- mentation of homogeneous Ansatz in V-QCD and discuss the single layer solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' For more details we refer to [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In the last two subsections, we present the generalization to double layer solution and investigate the possibility of a transition from the single layer to a double layer configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The homogeneous Ansatz For V-QCD, we use the action with finite baryon den- sity which can be written as SV −QCD = Sglue + SDBI + SCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' (29) The explicit expression for the action can be found in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The renormalization group flow of QCD is modeled through a nontrivial evolution of the geometry between the weak coupling (ultraviolet, UV) and strong coupling (infrared, IR) regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We will be using here the confor- mal coordinate r in the holographic direction [81, 82] for which the UV boundary is located at r = 0 while the IR singularity is at r = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' As in the case of the WSS model above, we separate the gauge field into non-Abelian and Abelian components: AL/R = AL/R + ˆAL/R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' (30) Here the left and right handed fields arise from D4 and D4 branes, respectively [83, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Similarly as in the case of the WSS model above, we turn on temporal component of the vectorial Abelian gauge field ˆAL = ˆAR = INf ×Nf Φ(r)dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' (31) Then on top this background, the non-Abelian baryonic terms are treated as a perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We expand the DBI action up to a first nontrivial order in the non-Abelian fields (quadratic in the field strengths F(L/R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' After the expansion, we insert the homogeneous Ansatz for non-Abelian gauge field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Ai L = −Ai R = h(r)σi (32) where h(r) is a smooth function and σi are Pauli matri- ces introducing non-trivial flavor dependence, SU(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' As result, the action for the flavour sector is written as Sh = S(0) DBI + S(1) DBI + SCS (33) where S(0) DBI is the DBI action in the absence of solitons, S(1) DBI is the expansion of the DBI action with homoge- neous Ansatz at the second order, SCS is Chern-Simons term with the homogeneous Ansatz (the explicit expres- sions are given in [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 7 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The single layer solution The solution for the bulk charge density is found by considering the Φ equation of motion [11] ρ′ = − d dr δSh δΦ′ = −δSh δΦ = 2Nc π2 d dr � e−bτ 2h3(1 − 2bτ 2) � , (34) where b is a parameter in the Chern-Simons term, ρ is the bulk charge density, and τ is the tachyon field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' However, the solution for ρ implied by this equation vanishes both in the UV and in the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' That is to say, diverging tachyon in the IR set the solution to zero via exponential factor and boundary condition for h in UV requires it to vanish (since there is no external baryon source).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Therefore, as was the case in with the WSS model above, the baryon density is zero, unless we impose an abrupt discontinuity in the field h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Motivated by these considerations, we write the “single layer” solution for V-QCD as [11] ρ = � ρ0 + 2Nc π2 e−bτ 2h3(1 − 2bτ 2), (r < rc) 2Nc π2 e−bτ 2h3(1 − 2bτ 2), (r > rc) (35) where ρ0 is boundary baryon charge density (the physical density) and rc is the location of the discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The explicit expression for ρ0 is ρ0 = 2Nc π2 e−bτ(rc)2(1 − 2bτ 2(rc))Disc(h3(rc)), (36) where we use the notation Disc(g(rc)) ≡ limϵ→0+(g(r + ϵ) − g(r − ϵ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' For future convenience, we briefly discuss the asymp- totics of the field h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In the UV, h has the asymptotics typical for gauge fields: h ≃ h1 + h2r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' (37) We require that non-Abelian sources vanish and there- fore h1 = 0, but h2 remains as a free parameter (which also determines rc for given ρ0, see appendix A 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Fol- lowing [11], we set h(r) = 0 for r > rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The double layer solution In this subsection, we generalize the single layer solu- tion for baryon field h to have two discontinuities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' ρ = � � � ρ01 + 2Nc π2 e−bτ 2h3(1 − 2bτ 2), (r < rc1) ρ02 + 2Nc π2 e−bτ 2h3(1 − 2bτ 2), (rc1 < r < rc2) 2Nc π2 e−bτ 2h3(1 − 2bτ 2), (r > rc2) (38) which we will be calling the double layer solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' There is also a continuity condition on ρ that is to be satisfied, which is given as ρ02 = −2Nc π2 (1 − 2bτ 2)e−bτ 2Disc h3|r=rc2 , ρ01 − ρ02 = −2Nc π2 (1 − 2bτ 2)e−bτ 2Disc h3|r=rc1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' (39) Therefore, summing the equalities above, we identify the boundary baryon charge density ρ0 as ρ01: ρ0 =ρ(r = 0) =ρ01 = −2Nc π2 2 � i=1 (1 − 2bτ 2)e−bτ 2Disc h3|r=rci .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' (40) We stress however that even though we call this solu- tion by the same name as the double layer solution for the WSS model, these solutions are quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In partic- ular, the double layer V-QCD solution has discontinuities at two values of the holographic coordinate whereas the WSS solution only has discontinuities at a single value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Actually the single layer solution of the V-QCD model is closer to the double layer solution of the WSS model than the double layer solution of the V-QCD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We will discuss this difference in more detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The double layer solution depends on four parameters at fixed ρ0: there is one additional parameter from the location of the extra discontinuity with respect to the single layer solution, and as the solution for h in the sec- ond interval rc1 < r < rc2 is independent of the solution in the first interval, there are two additional integration constants from the solution of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Finally, the generaliza- tion to triple layer or even to a solution with a higher number of flavors is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' One only needs to modify the piecewise solution for the charge density ρ with addition of new intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This will introduce three new parameters for each interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Legendre transform to canonical ensemble As in the analysis of the WSS model above, it is con- venient to work in the canonical ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The Legendre transformed action for V-QCD becomes [11] �Sh = − � d5xVρG � 1 + ρ2 (Vρwe−2A)2 × � 1 + 6w2e−4Ah4 + 6κτ 2e−2Ah2 1 + ρ2(Vρwe−2A)−2 + 3w2e−4Af(h′)2 2G2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' (41) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Second order transition in the Witten-Sakai-Sugimoto model We start by analyzing the configurations in the WSS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We set λ = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='63 [46] and analyse the solutions numerically (see appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' As a function of the chem- ical potential, we find three phases: 8 ρ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0002 ρ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0003 ρ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0005 ρ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0010 1 2 3 4 5 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='05 h 1 2 3 4 5 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 ρ/ρ0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The profile of the gauge field h(z) (left) and the bulk charge density ρ(z) (right) for the single layer (solid curves) and double layer (dashed curves) configurations for various values of the charge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The vertical dashed lines in the left hand plot denote the discontinuities of the double layer solutions at z = zc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Vacuum for µ < µc with µc ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Single layer phase for µc < µ < µl with µl ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='342 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Double layer phase for µ > µl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The phase transition at µ = µc (µ = µl) is of first (sec- ond) order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Here the second order transition (at the higher value of the chemical potential, µ = µl), is iden- tified as the popcorn transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Notice that in the ap- proach of [50], which used a different variation of the homogeneous approach, both the vacuum to nuclear and popcorn transitions were of first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Even though we are not attempting a serious comparison to QCD data, we note that setting MKK = 949 MeV as determined by the mass of the ρ meson [46], we have (for the quark chemical potential) µc ≃ 195 MeV and µl ≃ 325 MeV, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=', num- bers in the correct ballpark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We note that µl/µc ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Denoting the density of the single layer configuration at µ = µc as ρc (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=', the analogue of the saturation den- sity), the density ρl at the second order transition satis- fies ρl/ρc ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Here we are mostly interested in the second order tran- sition from the single to double layer phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We show the relevant configurations in figure 2 for a choice of densi- ties ρ0 around the critical value ρl ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='52 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Recall that the single layer configuration is unique for fixed ρ0, whereas the double layer configuration also depends on zc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We show here the double layer profiles which mini- mize the free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' They are separate from the single layer configuration only for ρ0 > ρl (the three highest values in the figure), where they have lower free energies than the single layer solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Interestingly, the single and double layer solutions at the same ρ0 are close: The functions h(z) deviate by at most a few percent in the region z > zc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The deviations for ρ(z) are slightly higher, and the single layer solution can be viewed as a smoothed out version of the double layer solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' That is, even if we were not considering the double layer solutions explic- itly, their presence could be guessed from the single layer solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In both cases, deviation is largest close to zc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We also remark that the single layer profiles h(z) appear to be qualitatively similar to the solutions found in the approach of [50] (see figure 4 in this reference), up to a shift by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Analysis of configurations in V-QCD We construct the double layer and single layer solution by the procedure which is outlined in appendix A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The essence of the procedure is the minimization of the free energy density at fixed ρ0 depending on the free parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In the case of the single layer, there is only one pa- rameter: rc or equivalently h2, and it is straightforward to solve the equation of state in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' For the dou- ble layer solution, there are four parameters which would make the numerical minimization procedure challenging in contrast to single layer solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Therefore, while we perform minimization of single layer solution for large domain of ρ0 values, we investigate presence of lower the free energy density of the double layer solution only for solutions obtained by gluing together single layer solu- tions for some representative values of ρ0 changing from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='8 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Denoting ∆hi = Disc h(rci), we investigate three qual- itatively different configurations: we consider ∆h1 < 0 and ∆h1 > 0 for double layer solution and ∆h1 > 0 , ∆h2 > 0 for a triple layer solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' For bound- ary baryon number charge we consider the values of ρ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5, ρ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='8 and ρ0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5, which will correspond to 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 r h rc1 rc2rc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='8 r ρ rc1 rc2rc ρ0 =ρ01 ρ02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='8 ρ02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 r h rc1 rc2 rc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='8 r ρ rc1 rc2rc3 rc ρ0 =ρ01 ρ02 ρ03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='8 ρ02 ρ03 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The profile of the gauge field h(r) (left) and the bulk charge density ρ(r) (right) for double layer with ∆h1 < 0 (first row), double layer with ∆h1 > 0 (second row), triple layer solution with ∆h1 > 0 and ∆h2 > 0 (third row) configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The single layer configuration with same boundary charge density ρ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='8 is showed with the gray dashed curve in each plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The parameters rci and ρ0i that characterize the multilayer configurations are shown by blobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The values of rci, h2i, ρ0i and f are given in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' µ/µc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='65, µ/µc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='04 and µ/µc = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='57 for the ther- modynamics determined by the single layer solution, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' While the first choice roughly corresponds to chemical potential values in which double layer solutions is WSS become dominant (as it is seen from figure 3), the other two choices are even much larger than that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In figure 3, the results for the three representative case are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The baryon field profile h(r) and correspond- ing baryon number densities ρ0(r) in the bulk are shown in the first and second column respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In each plot, the single layer solution minimizing the free energy is shown with gray dashed curves whose parameters are 10 rc h2 ρ0 f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='570 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='991 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='80 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='626 {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='476, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='498, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='533} {2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='90, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='20, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='90} {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='80, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='74, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='64} 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='642 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The values of {rc, h2, ρ0, f} for the single layer configuration (first row) and {rci, h2i, ρ0i, f} for the multi layer configurations (second-forth rows) that is shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' given in the first row of table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The red, blue and green solid curves show ∆h1 < 0 and ∆h1 > 0 double layer and (∆h1 > 0, ∆h2 > 0) triple layer solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The parame- ters rci, h2i, ρ0i, where h2i are the asymptotic constants h2 for the single layer solutions that were glued together to obtain the multilayer solutions, and the corresponding free energy densities f are shown in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The locations of the discontinuities and ρ0i are also shown in the figures with the blobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We were able to find double layer solutions which have lower free energy than the single layer solution for the cases of ∆h1 > 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=', the second row of figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' However, we were not able to find double solutions with ∆h1 < 0 that would have lower free energy than the single layer solution (configurations in the first row of the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Notice that having solutions with ∆h1 > 0 means that contributions to the total charge from the two discontinu- ities have opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This means that in the instan- ton picture, the discontinuities must arise from smear- ing instantons with opposite charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This suggests that proton-antiproton pairs are created, which should be for- bidden due to the large energy required for such a pair creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Therefore the configuration of the first row is not physically sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We suspect that it appears because the homogeneous approximation works poorly with con- figurations with discontinuities at several values of the holographic coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We also show the example of a triple layer configuration with ∆h1 > 0 and ∆h2 > 0 on the third row of the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Speed of sound and polytropic index We now study the physical implications of the phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' To this end, we plot the speed of sound and the polytropic index γ = d log p/d log ϵ for the WSS and V-QCD models in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In these plots, the chemical potential was normalized using the value at the vacuum to nuclear matter transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In both models, the speed of sound is below the value c2 s = 1/3 of conformal theories right above the transi- tion to nuclear matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' When µ increases, however, the speed of sound crosses this value and reaches values well above it [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The speed of sound has a maximum in both model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Even though the location of the maximum is dif- ferent between the models, the maximal values are rather close: the maximum of c2 s is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='463 (at µ/µc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='355) for the WSS model and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='504 (at µ/µc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='246) for V-QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Eventually at higher densities, the speed of sounds de- creases to values closer to the conformal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This is clearer in the WSS than in the V-QCD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In the WSS model, where the popcorn transition from a sin- gle to a double layer configuration is found, the speed of sound drops to a roughly constant value which closely agrees with the conformal value in the double layer phase: the speed of sound squared is about one per cent higher than the conformal value 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Similar results are found for the polytropic index γ in the right hand plot of figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In both models, γ de- creases with µ in the (single layer) nuclear matter phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This decrease is fast in the sense that γ drops below the value of γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='75, which was used as a criterion to separate nuclear matter from quark matter in [67, 68], where equations of state obtained as interpolations be- tween known results from nuclear theory at low density and perturbation theory at high density were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' For µ/µc > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 the results from both model are below this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' At the popcorn transition of the WSS model, γ drops to a value close to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Our findings indicate that the homogeneous holo- graphic nuclear matter behaves approximately confor- mally at high densities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=', at densities well above the nuclear saturation density (see also [101]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This is partic- ularly clear for the WSS model, which becomes approxi- mately conformal at the popcorn transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' These find- ings are consistent with earlier studies of homogeneous nuclear matter in the WSS (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=', [102]) and the V- QCD (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=', [94]) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' They also agree with the results found in the effective theory approach of [62, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This agreement in strikingly good for the WSS model, where the results both for the speed of sound (see [64]) and for the polytropic index (see [65]) have been com- puted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' For example, our results for the maximal value of the speed of sound (our value is cs,max ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='68) and the density at the popcorn transition (we found nl/nc ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='4) agree rather well with those of these references – our value for the speed of sound (transition density) is a bit below (above) the values of the effective theory approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We also remark that the non-monotonic behavior for the speed of sound in the WSS model qualitatively agrees with that found in the point-like instanton gas approach in [15], albeit with a different embedding for the D8 branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The maximal value found in this reference is also close to the maximal value obtained here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This agree- 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 μ/μc cs2 V-QCD s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' WSS s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' WSS d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 μ/μc γ V-QCD s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' WSS s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' WSS d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The speed of sound (left) and the polytropic index γ = d log p/d log ϵ (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The solid red, dashed green, and dot-dashed blue curves are the results for the single layer configuration in the WSS model, double layer configuration in the WSS model, and the V-QCD model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' ment is interesting as it obtained in a completely different approach, which is expected to be reliable at lower den- sities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Moreover we compare our results to the different approach of homogeneous nuclear matter derived in [50] in appendix B, and mostly find qualitative agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' CONCLUSIONS In this article, we analyzed nuclear matter using a ho- mogeneous approach in two different holographic mod- els: in the top-down WSS model and in the bottom-up V-QCD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We focused on two topics: the popcorn transitions, where the layer structure of the nuclear mat- ter changes in the bulk, and approach to conformal be- havior at high densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We found a second order pop- corn transition in the WSS model, and signs of approach to conformality in both holographic setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We have several remarks about our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Firstly, the results in the WSS and V-QCD models appeared to be quite different: in particular, the popcorn transition was only found to take place in the WSS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This is however not surprising at all and can be seen to follow from the differences in the geometry and the realisation of chiral symmetry breaking between the models as we now explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Recall that in the WSS model, the geome- try ends at the tip of the cigar in the confined phase as shown in figure 1, and chiral symmetry breaking is re- alised by the joining of the two branches of flavor branes at the tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In the V-QCD picture there is no cigar struc- ture, and chiral symmetry breaking arises from a con- densate of a bulk scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In the WSS model, nuclear matter at low densities is seen to arise from instantons located at the tip, and it is not possible to assign such instantons to be left or right handed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In V-QCD, how- ever, nuclear matter is stabilized at a nontrivial value of the holographic coordinate due to interaction with the bulk scalar field [11], and by definition always contains left and right handed components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Therefore, in V-QCD separate configurations analogous the single and double layer configurations of the WSS in figure 1 do not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The configurations of this figure map to the same con- figuration in V-QCD, which is what we called the single layer configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The double layer configuration in V- QCD defined in (38) would map to a more complicated configuration in the WSS model where discontinuities of the h field are found at two distinct values of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We found that the results for the equation of state near the popcorn transition of the WSS model closely resem- ble those obtained by the framework of [62, 64], where effective theory was used to analyse the transition of the Skyrmion crystal to a crystal of half-Skyrmions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This suggests that the transition in the holographic model should be identified with the topology changing transi- tion where half-Skyrmions appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='2 It is however dif- ficult to say anything definite about this because the holographic approach which we used does not contain individual instantons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Moreover, in [50] it was argued that the topology changing transition should not be iden- tified as the transition between the single and double layer solutions, but should take place between solutions of qualitatively different behavior within the single layer solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Another point is that chiral symmetry should be restored globally at the topology changing transition (meaning that the averages of the condensate over large regions should vanish).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This however will not happen for any of the configurations in the WSS approach because the D8 brane action is treated in the probe approxima- tion, and the embedding of the brane is independent of the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Nevertheless we remark that, as seen from the expressions for the single and double layer configu- rations in (20) and in (22), the bulk charge density has 2 We thank N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Kovensky and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Schmitt for correspondence on this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 12 support near the tip of the cigar only for the single layer configuration, where the flavor branes join, breaking chi- ral symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Therefore the double layer configuration can also exist in chirally symmetric backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Exam- ples of such chirally symmetric double layer configura- tions were indeed found in [17] (the chirally symmetric quarkyonic matter phase of this reference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Finally, we demonstrated that the homogeneous nu- clear matter becomes approximately conformal at high densities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=', above few times the nuclear saturation density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' That is, the values of the speed of sound lay close to the value c2 s = 1/3 of conformal theories, and similarly γ lay values close to the value γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In par- ticular, the polytropic index reached values well below the value γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='75 both in the V-QCD model and in the WSS model, which has been used to classify equa- tions of state for nuclear and quark matter in the ap- proach of [67, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' That is, the part of the single layer phase and all of the double layer phase would be clas- sified as quark matter in this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This appears consistent with the interpretation that the double layer phase is smoothly connected to quark matter [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In the V-QCD setup, however, there is a separate strong first order phase transition from nuclear to quark mat- ter at higher densities [11, 94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' In the WSS model there is a separate quark matter phase also, but in this case the transition is weak and even continuity between the phases is a possibility [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Mannque Rho for the invitation to con- tribute to the special issue “Symmetries and Ultra Dense Matter in Compact Stars” in Symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We also thank Elias Kiritsis, Nicolas Kovensky, Yong-Liang Ma, and Andreas Schmitt for discussions and correspon- dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This work benefited from discussions during the APCTP focus program “QCD and gauge/gravity dual- ity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' have been supported by an ap- pointment to the JRG Program at the APCTP through the Science and Technology Promotion Fund and Lottery Fund of the Korean Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' have also been supported by the Korean Local Governments – Gyeongsangbuk-do Province and Pohang City – and by the National Research Foundation of Korea (NRF) funded by the Korean government (MSIT) (grant number 2021R1A2C1010834).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' acknowledges the support of the Narodowe Centrum Nauki (NCN) Sonata Bis Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 2019/34/E/ST3/00405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Appendix A: Numerical details 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Constructing the solution in the Witten-Sakai-Sugimoto setup Here we summarize the basic steps we follow to find the free energy and the equation of state for the case of the simple profile for the charge density (20) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We derive from the action (27) the equation of mo- tion for h(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' After plugging the baryon charge density ρ and fixing Nc → 3 and λ → 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='63, the only free parameter is the boundary charge density ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Then we can simply solve the equation for h(z) for fixed ρ0 from the UV boundary (we still need to fix h1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We fix the value of h1 by solving for h for a given fixed ρ0 and chose a value of h1 such that ρ(h) = 0 at z = 0 , we can determine bulk charge density ρ profile by considering (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The free energy density is given by explicit integra- tion of (27) from zero to a large cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' At this step, we (re)normalize the free energy by subtract- ing �S in the absence of baryons from the original �S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' From the tabulated data {ρ0, F}, we can construct F(ρ0) and find at which value of ρ the transition to nuclear matter happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The corresponding chem- ical potential and grand potential can be obtained via µ = dF/dρ0 and Ω = F − ρ0µ = −p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' For the case of the more general solution (22), we need to find the value zc where the charge density vanishes but then the procedure to find the energy as a function of ρ0 is analogous to the single layer solution above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' However one difference with respect to the previous single layer solution is that the value of h1 that minimizes the energy changes for densities larger than a critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' From the comparison of the free energy we can see that there is a second order phase transition at this critical density ρc from the single layer solution to the double layer solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Constructing the solution in the V-QCD setup In this subsection, we summarize and outline the calcu- lation of free energy density and minimization procedure: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We work in the probe limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We first construct the thermal gas background solution for the geom- etry [76] in the absence of the baryons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Then, from (41) we derive equations of motion for h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' After plugging background fields and baryon charge density ρ, the only free parameter is the boundary charge density ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' So we can simply solve 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='6 μ/μc cs2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='5 μ/μc γ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The speed of sound (left) and the polytropic index γ = d log p/d log ϵ (right) for single layer (cyan curves) and double layer (magenta curves) solutions in the approach of [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The gray curves show the WSS and V-QCD results that are presented in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' the equation of motion for h for fixed ρ0 by from UV boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' After solving for h for given fixed ρ0 and chosen h2, we can determine bulk charge density ρ profile by considering (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Note that the vanishing point of bulk density profile gives the location of the soliton, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='e ρ(rc) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The free energy density is given by explicit inte- gration of (41) from boundary to the location of the discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' At this step, we also subtract �Sh in the absence of baryons from original �Sh to (re)normalize the free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Now, we can return to our main purpose of min- imizing free energy at fixed ρ0 depending on free parameter rc or equivalently h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We can simply perform above mention procedure with a loop over h2 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' From the tabulated data, we can construct F(h2) and minimize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The corresponding chemical po- tential and grand potential can be obtained via µ = dF/dρ0 and Ω = F − ρ0µ = −p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' For the case of the multi-layer configurations, the number of parameters which should be used in the minimization procedure increase and this makes the similar analysis numerically expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This is beyond the scope of this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Therefore, we decide to analyze the situation by considering some representative situations (the details of them are given in the main text in subsection IV B) and searching for solutions with lower f than that of single layer configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Appendix B: Comparison to a different homogeneous approach In this appendix we compare our results to those ob- tained by employing the homogeneous approach of [50], where one uses a zero curvature condition before taking the system to be homogeneous in the WSS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We set the parameter Λ = 8λ/(27π) to the value Λ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='568 for consistent comparison with our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' The results are shown in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' We see that the maximal value of the speed of sound is higher in the approach of [50] than in the other approaches, and the value of µ at the pop- corn transition is likewise higher than in the approach we used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' For the ratio of transition densities of single layer nuclear matter, we find ρl/ρc ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Similarly, the value of the polytropic index γ is relatively high when using the approach of [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' This also means that the agreement with the effective theory model of [62, 64], which was discussed in the main text, is less good for this approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' (LIGO Scientific, Virgo, Fermi- GBM, INTEGRAL), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 848, L13 (2017), arXiv:1710.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' 120, 172703 (2018), arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='02644 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' [3] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Rodr´ıguez Fern´andez, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Vuorinen, JHEP 12, 14 078 (2018), arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='06244 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' [6] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content=' Jokela, M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} +page_content='03218 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE1T4oBgHgl3EQfbgRN/content/2301.03173v1.pdf'} diff --git a/XdAyT4oBgHgl3EQf9PqM/content/tmp_files/2301.00871v1.pdf.txt b/XdAyT4oBgHgl3EQf9PqM/content/tmp_files/2301.00871v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..67af69f50556973992d06e34f8b9e020eb55331e --- /dev/null +++ b/XdAyT4oBgHgl3EQf9PqM/content/tmp_files/2301.00871v1.pdf.txt @@ -0,0 +1,746 @@ +Measuring nonlocal three-body spatial correlations with Rydberg trimers in ultracold +quantum gases +S. K. Kanungo,1, 2 Y. Lu,1, 2 F. B. Dunning,1 S. Yoshida,3 J. Burgd¨orfer,3 and T. C. Killian1, 2 +1Department of Physics and Astronomy, Rice University, Houston, TX 77005-1892, USA +2Rice Center for Quantum Materials, Rice University, Houston, TX 77005-1892, USA +3Institute for Theoretical Physics, Vienna University of Technology, Vienna A-1040, Austria, EU +We measure nonlocal third-order spatial correlations in non-degenerate ultracold gases of bosonic +(84Sr) and spin-polarized fermionic (87Sr) strontium through studies of the formation rates for +ultralong-range trimer Rydberg molecules. The trimer production rate is observed to be very sen- +sitive to the effects of quantum statistics with a strong enhancement of up to a factor of six (3!) +in the case of bosonic 84Sr due to bunching, and a marked reduction for spin-polarized fermionic +87Sr due to anti-bunching. The experimental results are compared to theoretical predictions and +good agreement is observed. The present approach opens the way to in situ studies of higher-order +nonlocal spatial correlations in a wide array of ultracold atomic-gas systems. +1. +INTRODUCTION +Measurements of atom-atom spatial correlations have +played an important role in understanding the prop- +erties of quantum many-body systems and their non- +classical behaviors, such as Bose-Einstein condensates[1], +the Mott insulator state[2], quantum spin models and +magnetism[3–6], Efimov physics[7], and strongly interact- +ing gases in one dimension [8–10]. Short-range or local +two- and three-body correlations in quantum gases have +been examined on length scales ≤ 20 nm through studies +of photoassociation and three-body recombination[1, 8– +10]. +Long-range or nonlocal correlations with length +scales on the order of the wavelength of light have been +explored using Bragg spectroscopy[6] and direct imaging +in optical tweezers [3] and quantum gas microscopes[5, +11]. The measurement of two-body correlations at inter- +mediate length scales, ∼ 20 − 200 nm, has been achieved +recently through studies of the formation of ultralong- +range Rydberg molecules (ULRMs)[12]. While this ear- +lier work focussed on the creation of dimer molecules and +two-body correlations, we demonstrate here that this ap- +proach can be extended to examine higher-order corre- +lations and report measurements of nonlocal three-body +spatial correlations in ultracold gases of bosons (84Sr) +and spin-polarized fermions (87Sr). +ULRMs are formed through scattering of the Rydberg +electron from a ground-state atom embedded within the +electron cloud, which results in an attractive “molecu- +lar” potential[13, 14]. A typical example of such a po- +tential for a strontium 5s38s 3S1 − 5s2 1S0 atom pair, +calculated using a Fermi pseudopotential, is shown in +Fig. 1(a) and mirrors the radial electron probability den- +sity distribution. This potential can support a number +of vibrational levels, and the vibrational wavefunctions +associated with the lower levels are included in Fig. 1(a). +Of particular interest here is the ground ν = 0 vibra- +tional state, which is strongly localized in the outermost +potential well at an (n−dependent) internuclear separa- +tion Rn ∼ 1.8(n − δ)2a0, where a0 is the bohr radius and +δ is the s-state quantum defect. +Measurements of two-body correlations using dimer +formation rates [12] exploited the fact that the likelihood +of creating a dimer in the ground vibrational state is pro- +portional to the probability that there are two ground- +state atoms in the initial sample with the appropriate +initial separation, Rn. Thus, by varying n, and hence +Rn, it is possible to probe the probability distribution of +atomic separations and derive the pair-correlation func- +tion g(2)(r). A trimer ULRM in its ground state contains +two ground-state atoms in the vibrational ground state at +a distance Rn from the Rydberg core ion. Measurements +of trimer formation can therefore be used to examine +three-body spatial correlations, although analysis of the +data is more complex than for dimers because the relative +positions of the two bound ground state atoms, charac- +terized by the angle θ shown in Fig. 1(b), is not fixed and +the measured values represent angle-averaged quantities. +Here results are presented for ULRMs with values of n +in the range 29−45, which correspond to values of Rn of +∼ 60 − 170 nm, and sample temperatures of 200 nK to +2 µK. The length scales probed are less than or on the +order of the atomic thermal de Broglie wavelength, λdB, +Rn/λdB ∼ 0.2−1, where the effects of quantum statistics +should be clearly visible in the correlation functions. +2. +EXPERIMENTAL METHODS +In the present work, Rydberg trimer excitation rates +are measured in ultracold gases. Cold samples of 84Sr +(boson, nuclear spin I = 0) and spin-polarized +87Sr +(fermion, I = 9/2) are prepared using standard meth- +ods of laser cooling and trapping[15, 16]. Atoms in an +atomic beam are slowed in a Zeeman slower and cooled +to ∼1 mK in a magneto-optical trap (MOT) using the +5s2 1S0 −→ 5s5p 1P1 transition at 461 nm. +To further +reduce the temperature to a few µK, a MOT operating +on the narrow 5s2 1S0 −→ 5s5p 3P1 transition at 689 nm +is employed. The atoms are then loaded into an opti- +cal dipole trap (ODT) formed by two crossed 1064 nm +laser beams, and evaporative cooling[17] is used to cre- +arXiv:2301.00871v1 [physics.atom-ph] 2 Jan 2023 + +2 +θ +Rn +Rn +Rn +FIG. 1. (a) Calculated molecular potential for a 5s38s 3S1 − +5s2 1S0 strontium atom pair. +The calculated vibrational +wavefunctions, multiplied by the radial coordinate R, for the +ν = 0, 1 and 2 vibrational states are included and the hori- +zontal axis for each indicates its binding energy. (b) Rydberg +excitation spectrum in a cold dense strontium gas in the vicin- +ity of the 5s38s 3S1 Rydberg state, where, the x-axis shows +the laser detuning from the Rydberg atomic line (black). The +features associated with the formation of dimer and trimer +ground state molecules are shown in blue and red respectively. +The other remaining features (gray) correspond to creation of +vibrationally-excited molecular states. Illustrations of dimer +and trimer molecules accompany the states of interest. +ate samples with final temperatures in the range of 200 +nK - 2 µK (evaporative cooling of spin-polarized 87Sr is +performed with 84Sr present in the trap to provide sym- +pathetic cooling). Typically for both isotopes, 2-7 ×105 +atoms remain trapped in the ODT with peak densities +in the range of 0.5 - 2.5 ×1012 cm−3 calculated from the +measured trap oscillation frequencies and atom number. +To obtain cold samples of spin-polarized fermions, the +ground state 87Sr atoms are optically pumped to the mF += 9/2 (F = 9/2) state[12]. A 7.6 G bias magnetic field +is applied after loading atoms into the ODT, which pro- +duces a Zeeman splitting of ∼ 650 kHz between adjacent +magnetic sublevels in the 5s5p 3P1 F = 9/2 manifold. +Population is transferred to the mF = 9/2 ground state +by applying a series of σ(+) polarized 689 nm laser pulses +that are red-detuned from each of the mF −→ mF + 1 +transitions by 50 kHz. The resulting spin-polarized 87Sr +atom sample is then sympathetically cooled with 84Sr +atoms to the desired final temperature and the magnetic +field is lowered to 1 G to maintain a quantization axis +during subsequent measurements. All data for the 84Sr +and unpolarized 87Sr atom samples are recorded in zero +magnetic field. +Strontium Rydberg dimers and trimers are created +by two-photon excitation from the ground state via the +5s5p 3P1 (F = 11/2 for 87Sr) intermediate state. The first +(689 nm) photon is blue-detuned 14 MHz from the inter- +mediate state. The second (320 nm) photon is scanned +to generate molecular excitation spectra. Both the lasers +are switched on for ∼10 µs. +The ground-state trimer +excitation is spectroscopically resolved from excitation +to all other states. Less than one Rydberg molecule is +created per laser shot to avoid Rydberg-Rydberg interac- +tions. The Rydberg molecule is detected by selective field +ionization[18], and the product electrons are directed to a +micro-channel plate (MCP) for detection. ULRM states +with Rydberg principal quantum number 29 ≤ n ≤ 45 +are created in this study, corresponding to Rydberg atom +and ULRM sizes 63 nm ≤ Rn ≤ 165 nm. Larger n and +Rn are not currently accessible because ULRM spectral +features become unresolved at the current spectral res- +olution of 100 kHz. Typically, 1000 experimental cycles +can be performed using a single ultracold sample to build +up statistics. +3. +g(3)(r) AND g(2)(r) CORRELATION +FUNCTIONS +Measurement of correlation functions provide an ef- +fective means to examine the behavior of complex +quantum systems, in particular many-body systems. +G(p)(r1, . . . , rp) (p ≥ 2) represents the diagonal elements +of the reduced p-body density matrix and measures the +likelihood of finding p particles at the specified posi- +tion at a given time. The reduced one-particle density +matrix (RDM) ρ(1)(r1, r2), sometimes also denoted by +G(1)(r1, r2) [19], contains information on coherences and +off-diagonal correlations which contrasts the G(p) for val- +ues of p ≥ 2. The theoretical description of correlation +functions in the atomic physics context was explored by +Glauber et al. [19] and we follow that analysis to de- +rive the angle-averaged three-body correlation function +relevant for trimer ULRM excitation. In the following +we will show that for an ideal gas the ensemble averaged + +3 +three-body correlation function can be expressed solely +in terms of the RDM. +Let ˆΨ†(r) and ˆΨ(r) be the creation and annihilation +operators for an atom at position r, which obey commu- +tation relations appropriate to either bosons or fermions. +The RDM is then given by: +G(1)(r1, r2) = ⟨ˆΨ†(r)ˆΨ(r2)⟩ , +(3.1) +and its diagonal elements represent the density: +ρ(r) = G(1)(r, r). +(3.2) +G(1)(r1, r2) for atoms trapped in a potential V (r) can +be expressed in terms of the generalized Bose function, +gα[19], as +G(1)(r1, r2) = +1 +λ3 +dB +g3/2 +� +exp +�µ − [V (r1) + V (r2)]/2 +kBT +� +, exp +� +−π (r2 − r1)2 +λ2 +dB +�� +, +(3.3) +where gα is given by the series, +gα(x, y) = +∞ +� +k=1 +xky1/k +kα +, +(3.4) +µ is the chemical potential, and λdB is the thermal de- +Broglie wavelength determined by the sample tempera- +ture T. The optical dipole trap in the present experiment +is well approximated by the anisotropic harmonic poten- +tial, +V (r) = m(ω2 +xx2 + ω2 +yy2 + ω2 +zz2) +2 +, +(3.5) +where m is mass of the strontium atom, and ωx, ωy and +ωz are the trap oscillation frequencies. The sizes of the +Rydberg molecules studied here are small compared to +the trap dimensions and it is reasonable to assume that +V (r1) ≈ V (r2) ≈ V (r). Use of Eq. (3.5) in Eq. (3.3), +allows numerical calculation of G(1)(r1, r2). +Expressions for G(2)(r1, r2) and G(3)(r1, r2, r3) can be +written as +G(2)(r1, r2) = ⟨ˆΨ†(r1)ˆΨ†(r2)ˆΨ(r2)ˆΨ(r1)⟩ , +(3.6) +G(3)(r1, r2, r3) = ⟨ˆΨ†(r1)ˆΨ†(r2)ˆΨ†(r3)ˆΨ(r3)ˆΨ(r2)ˆΨ(r1)⟩ , +(3.7) +where r1, r2 and r3 denote the position vectors of the var- +ious particles. For an ideal gas, using Wick’s theorem[20], +G(3)(r1, r2, r3) can be expressed in terms of one-body +density matrices[21] as +G(3)(r1, r2, r3) = G(1)(r1, r1)G(1)(r2, r2)G(1)(r3, r3) +± |G(1)(r1, r2)|2G(1)(r3, r3) +± |G(1)(r2, r3)|2G(1)(r1, r1) +± |G(1)(r3, r1)|2G(1)(r2, r2) ++ 2Re +� +(G(1)(r1, r2)G(1)(r2, r3)G(1)(r3, r1) +� +. +(3.8) +The corresponding expression for G(2)(r1, r2) is: +G(2)(r1, r2) = G(1)(r1, r1)G(1)(r2, r2) ± |G(1)(r1, r2)|2. +(3.9) +In the above expressions, the + (-) sign applies to iden- +tical bosons (fermions) in the same internal state. +Consider a dimer molecule where the position vectors +r1 and r2 correspond to the atoms that comprise the +Rydberg-bound ground-state atom pair of the dimer, and +r = r2−r1 denotes the position of the ground-state atom +relative to the core ion. R = (r2 + r1)/2 denotes the po- +sition of the center of mass (COM) of the pair. G(2) may +then be expressed in terms of these new variables. The +normalized and trap-averaged pair correlation function is +then given by +g(2)(r) = +� +dR G(2)(R, r) +� +dR ρ(R)2 +, +(3.10) +where we have exploited the fact that in the length scale +r probed by the dimer, G(2) can be approximated as in- +dependent of the orientation of r because Rn is much +smaller than the scale of variation of the trapping poten- +tial. +It is less straightforward to define such averaged core- +lation functions when three bodies are involved, as is the +case for a Rydberg trimer. In a ground state Rydberg +trimer both of the two ground-state atoms are bound at +the same inter-nuclear distance r ≈ Rn from the core +ion. The measured G(3) depends on the angle between +the relative orientations of ground-state atoms, defined +as the polar angle θ in Fig. 1, but not on the absolute +orientation of the molecule nor on the azimuthal angle. +We consider the relevant three-body correlation function +for the experiments described here, which is normalized, +trap-averaged, and averaged over θ, +g(3)(r) = +� +dR +� +G(3)(R, r, θ) +� +θ +� +dR ρ(R)3 +. +(3.11) + +4 +Equations (3.10) and (3.11) can be numerically evalu- +ated and the results for a trap containing a 500 nK sample +at a fugacity of 0.99 are shown in Fig. 2(a). Fifty terms +are retained in the expansion for gα(x, y), which is suffi- +cient for convergence of Eq. (3.3) even for a fugacity this +close to degeneracy (i.e., 1). As shown in Fig 2, the val- +ues of +� +G(3)(R, r, θ) +� +θ/ρ(R)3 and G(2)(R, r)/ρ(R)2 cal- +culated for the trap center, i.e. R = 0, do not deviate sig- +nificantly from the trap-volume-averaged values for the +present experimental trap parameters. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +0 +1 +2 +3 +4 +5 +6 +Correlation functions +(a) +g(3)(r) Boson +g(3)(R=0,r) Boson +g(2)(r) Boson +g(2)(R=0,r) Boson +g(3)(r) Fermion +g(3)(R=0,r) Fermion +g(2)(r) Fermion +g(2)(R=0,r) Fermion +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +r/ dB +0 +2 +4 +6 +G (3)(R=0,r, )/ (R=0)3 +(b) +=0 += /3 +=2 /3 += +FIG. 2. +(a) Numerically calculated g(3)(r) and g(2)(r) +and ˜g(3)(R, r) ≡ +� +G(3)(R, r, θ) +� +θ/ρ(R)3 and ˜g(2)(R, r) ≡ +G(2)(R, r)/ρ(R)2 evaluated at the trap center (R = 0), for +Bose and Fermi gases. +(b) Plot of G(3)(R, r, θ)/ρ(R)3 at +R = 0 as a function of r/λdB for an ideal gas of bosons at +various values of θ. At θ = 0, bunching in bosons (or anti- +bunching in fermions) is expected. In particular, at large r, +the third atom becomes uncorrelated from the other two and +the θ−dependence of the three-body correlation represents +the two-body correlation. +Fig. 2(b) shows G(3)(R = 0, r, θ)/ρ(R = 0)3 for rep- +resentative values of θ, for an ideal gas evaluated at the +coordinates corresponding to a Rydberg trimer. It is in- +teresting to note that the three-body correlation function +approaches 2 at large r as θ −→ 0 and the two atoms that +will become the ground-state atoms bound to the Ryd- +berg core come closer to each other. +4. +RESULTS AND DISCUSSION +Generalizing the formalism of [12], the measured dimer +(S(2) +n ) and trimer (S(3) +n ) ULRM signals for principal +quantum number n, which we take as the integrals of +the photoexcitation spectral lines, depend on several ex- +perimental parameters and may be approximated as: +S(2) +n +≃ αI1I2βnCO(2) +n +� +d3R G(2)(R, Rn) += αI1I2N (2)βnCO(2) +n g(2)(Rn). +(4.1) +S(3) +n +≃ αI1I2βnCO(3) +n +� +d3R +� +G(3)(R, Rn, θ) +� +θ += αI1I2N (3)βnCO(3) +n g(3)(Rn). +(4.2) +The MCP detection efficiency is characterized by α and +is independent of isotopes, and I1 and I2 are the inten- +sities of the Rydberg excitation lasers, which are mon- +itored by photo-detectors. The local excitation rate is +assumed to be proportional to G(p), with appropriate +arguments and angle average. This quantity is propor- +tional to the p-th power of the local cold atom density. +The integral over the trap results in the nonlocal spatial +correlation function, g(p)(Rn), and the density scaling +factor N (p) = +� +d3R ρ(R)p. Other factors that influence +the photoexcitation rate are the square of the reduced +two-photon electronic-transition matrix element, repre- +sented by βn, which depends on the principal quantum +number n, the (n−independent) Clebsch-Gordan coeffi- +cients, C, that couple the levels of interest, and the ef- +fective Franck-Condon factor, O(p) +n , given by the overlap +of the initial scattering wavefunction with the molecular +bound state, which is different for dimers and trimers and +also depends on n[12]. Once these factors are taken into +account any remaining variations in the excitation rates +can be attributed to changes in g(p)(Rn). +The n-dependence of the product βnO(p) +n +for trimers +and dimers is experimentally determined by measur- +ing molecular excitation rates in an unpolarized 87Sr +sample (see Fig. +3). +87Sr has ten degenerate ground +states, and an unpolarized sample approximates a clas- +sical gas. +Ancillary calculations suggest that residual +two- and three-body correlations are indeed small, i.e., +g(p) +unpol(R) ≈ 0.9 − 1, and their effects are therefore ne- +glected in the calculation of the n-dependence of βnO(p) +n . +For unpolarized 87Sr, all the factors in Eq. 4.2 that in- +fluence the trimer production rate except βn and O(3) +n +can be measured and taken into account, and thus any +n−dependence seen in the trimer production rate must +be associated with the product βnO(3) +n . In earlier mea- +surements of ground-state dimer production, the prod- +uct βnO(2) +n +was observed to scale as (n − δ)3.5(3)[12], a + +5 +3.4 +3.5 +3.6 +3.7 +Log(n − δ) +4 +3 +2 +1 +0 +Log(S(3) +n /αI1I2CN(3)) +33 +35 +37 +39 +41 +43 +45 +n +FIG. 3. Normalized n-dependence of the trimer ground state +production rate (S(3) +n /αI1I2CN (3)) in an unpolarized Fermi +gas of 87Sr. +The measured trimer signals are well fit by a +power law with exponent 12.3 ± 0.8, which furnishes the scal- +ing of the product βnO(3) +n +for trimer excitation. +Effects of +residual three-body correlations in the unpolarized sample of +87Sr are small (g(3) +unpol(r) ≈1) and are neglected in determining +the normalized trimer excitation rates. +result confirmed in the present work. As illustrated by +Fig 3, measurements of trimer formation showed a much +stronger n−dependence in the product βnO(3) +n , which +scales as (n − δ)12.3(8). +Figures 4(a-h) illustrate the n−dependence of the +dimer and trimer excitation spectra recorded using 84Sr +and spin-polarized 87Sr. In each set of measurements the +results are normalized by laser intensities and ground- +state atom densities as well as the n−dependence in the +product βnO(p) +n . Thus any changes seen in the measured +signal levels must be associated with changes in the cor- +relation functions g(2)(Rn) or g(3)(Rn). +For 84Sr the (normalized) trimer photoexcitation rate +increases dramatically with decreasing n, pointing to a +similar increase in g(3)(r). This results because, as n de- +creases, molecule formation probes correlations on ever +shorter length scales, which for the present sample tem- +peratures become smaller than the atomic de Broglie +wavelength. In this regime bunching in bosons results +in an increase in the spatial correlation. As seen in Fig. +4, the dimer production rate for 84Sr also increases as +n decreases, but this increase is much less pronounced +than that observed for trimers highlighting how much +more sensitive trimer formation is to the effects spatial +correlations. +In contrast, the trimer excitation rate in +spin-polarized 87Sr decreases significantly with decreas- +ing n due to anti-bunching, i.e., Pauli exclusion. This +decrease is more pronounced than that seen in dimer pro- +duction, further demonstrating the greater sensitivity of +trimer production to spatial correlations. +0 +1 +2 +Norm. signal +(a) +n=29 +Boson trimer +(b) +n=39 +0 +1 +2 +Norm. signal +(c) +Boson dimer +(d) +0.0 +0.5 +1.0 +Norm. signal +(e) +n=33 +Fermion trimer +(f) +n=39 +0.5 +0.0 +0.5 +Detuning (MHz) +0.0 +0.5 +1.0 +Norm. signal +(g) +0.5 +0.0 +0.5 +Detuning (MHz) +Fermion dimer +(h) +FIG. 4. +Photoexcitation spectra for ground-state dimer and +trimer molecules in a cold strontium gas as a function of laser +detuning from the dimer or trimer line center. +Each data +set is normalized by laser intensities, atom densities, and the +product βnO(p) +n +(see text). Trimer photoexcitation rates for +bosonic 84Sr are shown in (a,b) and for spin-polarized 87Sr in +(e, f). Note the strong signal enhancement as n decreases for +bosons compared to suppression for fermions. For compari- +son, measurements of dimer formation in the same gases are +plotted in (c, d) and (g, h), which show similar but weaker +variation. The data for dimer and trimer production at n=39 +are all normalized to the same peak height. Note the change +in scale of the vertical axes between the upper and lower data +sets. +Figure 5 shows the principal findings of this paper. + +6 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +r/ dB +0.4 +1.0 +2.0 +6.0 +g(2)(r), g(3)(r) +g(2)(r) - Boson +g(3)(r) - Boson +g(2)(r) - Fermion +g(3)(r) - Fermion +84Sr dimer +84Sr trimer +87Sr dimer +87Sr trimer +FIG. 5. +Measured and calculated values of g(3)(r) and +g(2)(r) for ultracold gases of bosonic (84Sr) and spin-polarized +fermionic (87Sr) atoms. Each set of experimental measure- +ments is fit to the corresponding theoretical predictions using +a single amplitude scaling factor, and the molecular size is +scaled by the atomic thermal de Broglie wavelength. Error +bars denote standard error of the mean of multiple measure- +ments or uncertainty in r/λdB due to uncertainty in sample +temperature. +Dimer and trimer ground state photoexcitation spectra +were recorded for a range of quantum numbers, 29 ≤ +n ≤ 45, at various temperatures. +For each spectrum, +the total integrated molecular signal (S(p) +n ) was obtained +by fitting to a voigt profile. The signal was normalized +by αI1I2CN (p)βnO(p) +n +to remove all dependences other +than g(3)(r) or g(2)(r) [Eqs. (4.1)-(4.2)], where r = Rn is +taken to be the size of the ULRM. To enable direct com- +parison between measurements undertaken at different +sample temperatures, the length is scaled by the atomic +thermal de Broglie wavelength, λdB. A single amplitude +scaling parameter is fit for each experimental data set, +which normalizes the data to match the corresponding +theoretical curves calculated from Eqs. (3.10) and (3.11). +(Fermion dimer data is taken from [12]). +As evident from Fig. +5, experimental observations +match theoretical predictions well. For a Bose gas and +small values of r/λdB, g(2)(r) approaches 2, in agreement +with earlier work [12]. The predicted 3! increase is ob- +served in g(3)(r). In contrast, for the Fermi gas, molecule +formation is strongly suppressed at small values of r/λdB, +and more strongly so for trimers. Correlations decay to- +wards unity on the length scale of λdB as expected. +5. +CONCLUSIONS +We have demonstrated that measurements of the for- +mation of ground-state trimer ULRMs provide a sensitive +in situ probe of three-body, nonlocal spatial correlations +in ultracold gases, and have applied this probe to observe +bunching and anti-bunching in thermal gases of indistin- +guishable bosons and fermions respectively. Even higher- +order correlations are accessible by observing formation +of tetramers and higher p-mers [22], offering the possibil- +ity of comprehensive characterization of correlations in +many-body quantum systems. It should be possible to +apply this technique to systems where interactions affect +particle correlations, such as in strongly-interacting 1-D +gases. +6. +ACKNOWLEDGEMENTS +Research supported by the AFOSR under Grant No. +FA9550-14-1-0007, the NSF under Grant No. 1600059, +and the FWF (Austria) under Grant No. +FWF SFB- +SFB041 ViCom and the FWF Doctoral College W 1243. +[1] E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, +E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 79, +337 (1997). +[2] C. Carcy, H. Cayla, A. Tenart, A. Aspect, M. Mancini, +and D. Cl´ement, Phys. Rev. X 9, 041028 (2019). +[3] A. Browaeys and T. Lahaye, Nat. Phys. 16, 132 (2020). +[4] G. Semeghini, H. Levine, A. Keesling, S. Ebadi, T. T. +Wang, D. Bluvstein, R. Verresen, H. Pichler, M. 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Rev. A 59, +4595 (1999). +[20] G. C. Wick, Phys. Rev. 80, 268 (1950). +[21] S. S. Hodgman, R. G. Dall, A. G. Manning, K. G. H. +Baldwin, and A. G. Truscott, Science 331, 1046 (2011). +[22] F. Camargo, R. Schmidt, J. D. Whalen, R. Ding, +G. Woehl, S. Yoshida, J. Burgd¨orfer, F. B. Dunning, +H. R. Sadeghpour, E. Demler, and T. C. Killian, Phys. +Rev. Lett. 120, 083401 (2018). + diff --git a/XdAyT4oBgHgl3EQf9PqM/content/tmp_files/load_file.txt b/XdAyT4oBgHgl3EQf9PqM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6da43eb1db1b745b5f3f170c7bf1b9b3f79fe8d9 --- /dev/null +++ b/XdAyT4oBgHgl3EQf9PqM/content/tmp_files/load_file.txt @@ -0,0 +1,542 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf,len=541 +page_content='Measuring nonlocal three-body spatial correlations with Rydberg trimers in ultracold quantum gases S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Kanungo,1, 2 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Lu,1, 2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Dunning,1 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Yoshida,3 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Burgd¨orfer,3 and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Killian1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 2 1Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Rice University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Houston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' TX 77005-1892,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' USA 2Rice Center for Quantum Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Rice University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Houston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' TX 77005-1892,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' USA 3Institute for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Vienna University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Vienna A-1040,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Austria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' EU We measure nonlocal third-order spatial correlations in non-degenerate ultracold gases of bosonic (84Sr) and spin-polarized fermionic (87Sr) strontium through studies of the formation rates for ultralong-range trimer Rydberg molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The trimer production rate is observed to be very sen- sitive to the effects of quantum statistics with a strong enhancement of up to a factor of six (3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=') in the case of bosonic 84Sr due to bunching, and a marked reduction for spin-polarized fermionic 87Sr due to anti-bunching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The experimental results are compared to theoretical predictions and good agreement is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The present approach opens the way to in situ studies of higher-order nonlocal spatial correlations in a wide array of ultracold atomic-gas systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' INTRODUCTION Measurements of atom-atom spatial correlations have played an important role in understanding the prop- erties of quantum many-body systems and their non- classical behaviors, such as Bose-Einstein condensates[1], the Mott insulator state[2], quantum spin models and magnetism[3–6], Efimov physics[7], and strongly interact- ing gases in one dimension [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Short-range or local two- and three-body correlations in quantum gases have been examined on length scales ≤ 20 nm through studies of photoassociation and three-body recombination[1, 8– 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Long-range or nonlocal correlations with length scales on the order of the wavelength of light have been explored using Bragg spectroscopy[6] and direct imaging in optical tweezers [3] and quantum gas microscopes[5, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The measurement of two-body correlations at inter- mediate length scales, ∼ 20 − 200 nm, has been achieved recently through studies of the formation of ultralong- range Rydberg molecules (ULRMs)[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' While this ear- lier work focussed on the creation of dimer molecules and two-body correlations, we demonstrate here that this ap- proach can be extended to examine higher-order corre- lations and report measurements of nonlocal three-body spatial correlations in ultracold gases of bosons (84Sr) and spin-polarized fermions (87Sr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' ULRMs are formed through scattering of the Rydberg electron from a ground-state atom embedded within the electron cloud, which results in an attractive “molecu- lar” potential[13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' A typical example of such a po- tential for a strontium 5s38s 3S1 − 5s2 1S0 atom pair, calculated using a Fermi pseudopotential, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 1(a) and mirrors the radial electron probability den- sity distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' This potential can support a number of vibrational levels, and the vibrational wavefunctions associated with the lower levels are included in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Of particular interest here is the ground ν = 0 vibra- tional state, which is strongly localized in the outermost potential well at an (n−dependent) internuclear separa- tion Rn ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='8(n − δ)2a0, where a0 is the bohr radius and δ is the s-state quantum defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Measurements of two-body correlations using dimer formation rates [12] exploited the fact that the likelihood of creating a dimer in the ground vibrational state is pro- portional to the probability that there are two ground- state atoms in the initial sample with the appropriate initial separation, Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Thus, by varying n, and hence Rn, it is possible to probe the probability distribution of atomic separations and derive the pair-correlation func- tion g(2)(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' A trimer ULRM in its ground state contains two ground-state atoms in the vibrational ground state at a distance Rn from the Rydberg core ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Measurements of trimer formation can therefore be used to examine three-body spatial correlations, although analysis of the data is more complex than for dimers because the relative positions of the two bound ground state atoms, charac- terized by the angle θ shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 1(b), is not fixed and the measured values represent angle-averaged quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Here results are presented for ULRMs with values of n in the range 29−45, which correspond to values of Rn of ∼ 60 − 170 nm, and sample temperatures of 200 nK to 2 µK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The length scales probed are less than or on the order of the atomic thermal de Broglie wavelength, λdB, Rn/λdB ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='2−1, where the effects of quantum statistics should be clearly visible in the correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' EXPERIMENTAL METHODS In the present work, Rydberg trimer excitation rates are measured in ultracold gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Cold samples of 84Sr (boson, nuclear spin I = 0) and spin-polarized 87Sr (fermion, I = 9/2) are prepared using standard meth- ods of laser cooling and trapping[15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Atoms in an atomic beam are slowed in a Zeeman slower and cooled to ∼1 mK in a magneto-optical trap (MOT) using the 5s2 1S0 −→ 5s5p 1P1 transition at 461 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' To further reduce the temperature to a few µK, a MOT operating on the narrow 5s2 1S0 −→ 5s5p 3P1 transition at 689 nm is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The atoms are then loaded into an opti- cal dipole trap (ODT) formed by two crossed 1064 nm laser beams, and evaporative cooling[17] is used to cre- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='00871v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='atom-ph] 2 Jan 2023 2 θ Rn Rn Rn FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' (a) Calculated molecular potential for a 5s38s 3S1 − 5s2 1S0 strontium atom pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The calculated vibrational wavefunctions, multiplied by the radial coordinate R, for the ν = 0, 1 and 2 vibrational states are included and the hori- zontal axis for each indicates its binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' (b) Rydberg excitation spectrum in a cold dense strontium gas in the vicin- ity of the 5s38s 3S1 Rydberg state, where, the x-axis shows the laser detuning from the Rydberg atomic line (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The features associated with the formation of dimer and trimer ground state molecules are shown in blue and red respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The other remaining features (gray) correspond to creation of vibrationally-excited molecular states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Illustrations of dimer and trimer molecules accompany the states of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' ate samples with final temperatures in the range of 200 nK - 2 µK (evaporative cooling of spin-polarized 87Sr is performed with 84Sr present in the trap to provide sym- pathetic cooling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Typically for both isotopes, 2-7 ×105 atoms remain trapped in the ODT with peak densities in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='5 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='5 ×1012 cm−3 calculated from the measured trap oscillation frequencies and atom number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' To obtain cold samples of spin-polarized fermions, the ground state 87Sr atoms are optically pumped to the mF = 9/2 (F = 9/2) state[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' A 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='6 G bias magnetic field is applied after loading atoms into the ODT, which pro- duces a Zeeman splitting of ∼ 650 kHz between adjacent magnetic sublevels in the 5s5p 3P1 F = 9/2 manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Population is transferred to the mF = 9/2 ground state by applying a series of σ(+) polarized 689 nm laser pulses that are red-detuned from each of the mF −→ mF + 1 transitions by 50 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The resulting spin-polarized 87Sr atom sample is then sympathetically cooled with 84Sr atoms to the desired final temperature and the magnetic field is lowered to 1 G to maintain a quantization axis during subsequent measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' All data for the 84Sr and unpolarized 87Sr atom samples are recorded in zero magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Strontium Rydberg dimers and trimers are created by two-photon excitation from the ground state via the 5s5p 3P1 (F = 11/2 for 87Sr) intermediate state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The first (689 nm) photon is blue-detuned 14 MHz from the inter- mediate state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The second (320 nm) photon is scanned to generate molecular excitation spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Both the lasers are switched on for ∼10 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The ground-state trimer excitation is spectroscopically resolved from excitation to all other states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Less than one Rydberg molecule is created per laser shot to avoid Rydberg-Rydberg interac- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The Rydberg molecule is detected by selective field ionization[18], and the product electrons are directed to a micro-channel plate (MCP) for detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' ULRM states with Rydberg principal quantum number 29 ≤ n ≤ 45 are created in this study, corresponding to Rydberg atom and ULRM sizes 63 nm ≤ Rn ≤ 165 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Larger n and Rn are not currently accessible because ULRM spectral features become unresolved at the current spectral res- olution of 100 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Typically, 1000 experimental cycles can be performed using a single ultracold sample to build up statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' g(3)(r) AND g(2)(r) CORRELATION FUNCTIONS Measurement of correlation functions provide an ef- fective means to examine the behavior of complex quantum systems, in particular many-body systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' G(p)(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' , rp) (p ≥ 2) represents the diagonal elements of the reduced p-body density matrix and measures the likelihood of finding p particles at the specified posi- tion at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The reduced one-particle density matrix (RDM) ρ(1)(r1, r2), sometimes also denoted by G(1)(r1, r2) [19], contains information on coherences and off-diagonal correlations which contrasts the G(p) for val- ues of p ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The theoretical description of correlation functions in the atomic physics context was explored by Glauber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' [19] and we follow that analysis to de- rive the angle-averaged three-body correlation function relevant for trimer ULRM excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' In the following we will show that for an ideal gas the ensemble averaged 3 three-body correlation function can be expressed solely in terms of the RDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Let ˆΨ†(r) and ˆΨ(r) be the creation and annihilation operators for an atom at position r, which obey commu- tation relations appropriate to either bosons or fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The RDM is then given by: G(1)(r1, r2) = ⟨ˆΨ†(r)ˆΨ(r2)⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='1) and its diagonal elements represent the density: ρ(r) = G(1)(r, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='2) G(1)(r1, r2) for atoms trapped in a potential V (r) can be expressed in terms of the generalized Bose function, gα[19], as G(1)(r1, r2) = 1 λ3 dB g3/2 � exp �µ − [V (r1) + V (r2)]/2 kBT � , exp � −π (r2 − r1)2 λ2 dB �� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='3) where gα is given by the series, gα(x, y) = ∞ � k=1 xky1/k kα , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='4) µ is the chemical potential, and λdB is the thermal de- Broglie wavelength determined by the sample tempera- ture T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The optical dipole trap in the present experiment is well approximated by the anisotropic harmonic poten- tial, V (r) = m(ω2 xx2 + ω2 yy2 + ω2 zz2) 2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='5) where m is mass of the strontium atom, and ωx, ωy and ωz are the trap oscillation frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The sizes of the Rydberg molecules studied here are small compared to the trap dimensions and it is reasonable to assume that V (r1) ≈ V (r2) ≈ V (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='5) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='3), allows numerical calculation of G(1)(r1, r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Expressions for G(2)(r1, r2) and G(3)(r1, r2, r3) can be written as G(2)(r1, r2) = ⟨ˆΨ†(r1)ˆΨ†(r2)ˆΨ(r2)ˆΨ(r1)⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='6) G(3)(r1, r2, r3) = ⟨ˆΨ†(r1)ˆΨ†(r2)ˆΨ†(r3)ˆΨ(r3)ˆΨ(r2)ˆΨ(r1)⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='7) where r1, r2 and r3 denote the position vectors of the var- ious particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' For an ideal gas, using Wick’s theorem[20], G(3)(r1, r2, r3) can be expressed in terms of one-body density matrices[21] as G(3)(r1, r2, r3) = G(1)(r1, r1)G(1)(r2, r2)G(1)(r3, r3) ± |G(1)(r1, r2)|2G(1)(r3, r3) ± |G(1)(r2, r3)|2G(1)(r1, r1) ± |G(1)(r3, r1)|2G(1)(r2, r2) + 2Re � (G(1)(r1, r2)G(1)(r2, r3)G(1)(r3, r1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='8) The corresponding expression for G(2)(r1, r2) is: G(2)(r1, r2) = G(1)(r1, r1)G(1)(r2, r2) ± |G(1)(r1, r2)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='9) In the above expressions, the + (-) sign applies to iden- tical bosons (fermions) in the same internal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Consider a dimer molecule where the position vectors r1 and r2 correspond to the atoms that comprise the Rydberg-bound ground-state atom pair of the dimer, and r = r2−r1 denotes the position of the ground-state atom relative to the core ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' R = (r2 + r1)/2 denotes the po- sition of the center of mass (COM) of the pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' G(2) may then be expressed in terms of these new variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The normalized and trap-averaged pair correlation function is then given by g(2)(r) = � dR G(2)(R, r) � dR ρ(R)2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='10) where we have exploited the fact that in the length scale r probed by the dimer, G(2) can be approximated as in- dependent of the orientation of r because Rn is much smaller than the scale of variation of the trapping poten- tial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' It is less straightforward to define such averaged core- lation functions when three bodies are involved, as is the case for a Rydberg trimer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' In a ground state Rydberg trimer both of the two ground-state atoms are bound at the same inter-nuclear distance r ≈ Rn from the core ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The measured G(3) depends on the angle between the relative orientations of ground-state atoms, defined as the polar angle θ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 1, but not on the absolute orientation of the molecule nor on the azimuthal angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' We consider the relevant three-body correlation function for the experiments described here, which is normalized, trap-averaged, and averaged over θ, g(3)(r) = � dR � G(3)(R, r, θ) � θ � dR ρ(R)3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='11) 4 Equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='11) can be numerically evalu- ated and the results for a trap containing a 500 nK sample at a fugacity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='99 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Fifty terms are retained in the expansion for gα(x, y), which is suffi- cient for convergence of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='3) even for a fugacity this close to degeneracy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=', 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' As shown in Fig 2, the val- ues of � G(3)(R, r, θ) � θ/ρ(R)3 and G(2)(R, r)/ρ(R)2 cal- culated for the trap center, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' R = 0, do not deviate sig- nificantly from the trap-volume-averaged values for the present experimental trap parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='2 0 1 2 3 4 5 6 Correlation functions (a) g(3)(r) Boson g(3)(R=0,r) Boson g(2)(r) Boson g(2)(R=0,r) Boson g(3)(r) Fermion g(3)(R=0,r) Fermion g(2)(r) Fermion g(2)(R=0,r) Fermion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='2 r/ dB 0 2 4 6 G (3)(R=0,r, )/ (R=0)3 (b) =0 = /3 =2 /3 = FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' (a) Numerically calculated g(3)(r) and g(2)(r) and ˜g(3)(R, r) ≡ � G(3)(R, r, θ) � θ/ρ(R)3 and ˜g(2)(R, r) ≡ G(2)(R, r)/ρ(R)2 evaluated at the trap center (R = 0), for Bose and Fermi gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' (b) Plot of G(3)(R, r, θ)/ρ(R)3 at R = 0 as a function of r/λdB for an ideal gas of bosons at various values of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' At θ = 0, bunching in bosons (or anti- bunching in fermions) is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' In particular, at large r, the third atom becomes uncorrelated from the other two and the θ−dependence of the three-body correlation represents the two-body correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 2(b) shows G(3)(R = 0, r, θ)/ρ(R = 0)3 for rep- resentative values of θ, for an ideal gas evaluated at the coordinates corresponding to a Rydberg trimer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' It is in- teresting to note that the three-body correlation function approaches 2 at large r as θ −→ 0 and the two atoms that will become the ground-state atoms bound to the Ryd- berg core come closer to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' RESULTS AND DISCUSSION Generalizing the formalism of [12], the measured dimer (S(2) n ) and trimer (S(3) n ) ULRM signals for principal quantum number n, which we take as the integrals of the photoexcitation spectral lines, depend on several ex- perimental parameters and may be approximated as: S(2) n ≃ αI1I2βnCO(2) n � d3R G(2)(R, Rn) = αI1I2N (2)βnCO(2) n g(2)(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='1) S(3) n ≃ αI1I2βnCO(3) n � d3R � G(3)(R, Rn, θ) � θ = αI1I2N (3)βnCO(3) n g(3)(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='2) The MCP detection efficiency is characterized by α and is independent of isotopes, and I1 and I2 are the inten- sities of the Rydberg excitation lasers, which are mon- itored by photo-detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The local excitation rate is assumed to be proportional to G(p), with appropriate arguments and angle average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' This quantity is propor- tional to the p-th power of the local cold atom density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The integral over the trap results in the nonlocal spatial correlation function, g(p)(Rn), and the density scaling factor N (p) = � d3R ρ(R)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Other factors that influence the photoexcitation rate are the square of the reduced two-photon electronic-transition matrix element,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' repre- sented by βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' which depends on the principal quantum number n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' the (n−independent) Clebsch-Gordan coeffi- cients,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' that couple the levels of interest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' and the ef- fective Franck-Condon factor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' O(p) n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' given by the overlap of the initial scattering wavefunction with the molecular bound state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' which is different for dimers and trimers and also depends on n[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Once these factors are taken into account any remaining variations in the excitation rates can be attributed to changes in g(p)(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The n-dependence of the product βnO(p) n for trimers and dimers is experimentally determined by measur- ing molecular excitation rates in an unpolarized 87Sr sample (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 87Sr has ten degenerate ground states, and an unpolarized sample approximates a clas- sical gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Ancillary calculations suggest that residual two- and three-body correlations are indeed small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=', g(p) unpol(R) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='9 − 1, and their effects are therefore ne- glected in the calculation of the n-dependence of βnO(p) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' For unpolarized 87Sr, all the factors in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='2 that in- fluence the trimer production rate except βn and O(3) n can be measured and taken into account, and thus any n−dependence seen in the trimer production rate must be associated with the product βnO(3) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' In earlier mea- surements of ground-state dimer production, the prod- uct βnO(2) n was observed to scale as (n − δ)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='5(3)[12], a 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='7 Log(n − δ) 4 3 2 1 0 Log(S(3) n /αI1I2CN(3)) 33 35 37 39 41 43 45 n FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Normalized n-dependence of the trimer ground state production rate (S(3) n /αI1I2CN (3)) in an unpolarized Fermi gas of 87Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The measured trimer signals are well fit by a power law with exponent 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='8, which furnishes the scal- ing of the product βnO(3) n for trimer excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Effects of residual three-body correlations in the unpolarized sample of 87Sr are small (g(3) unpol(r) ≈1) and are neglected in determining the normalized trimer excitation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' result confirmed in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' As illustrated by Fig 3, measurements of trimer formation showed a much stronger n−dependence in the product βnO(3) n , which scales as (n − δ)12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='3(8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Figures 4(a-h) illustrate the n−dependence of the dimer and trimer excitation spectra recorded using 84Sr and spin-polarized 87Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' In each set of measurements the results are normalized by laser intensities and ground- state atom densities as well as the n−dependence in the product βnO(p) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Thus any changes seen in the measured signal levels must be associated with changes in the cor- relation functions g(2)(Rn) or g(3)(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' For 84Sr the (normalized) trimer photoexcitation rate increases dramatically with decreasing n, pointing to a similar increase in g(3)(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' This results because, as n de- creases, molecule formation probes correlations on ever shorter length scales, which for the present sample tem- peratures become smaller than the atomic de Broglie wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' In this regime bunching in bosons results in an increase in the spatial correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 4, the dimer production rate for 84Sr also increases as n decreases, but this increase is much less pronounced than that observed for trimers highlighting how much more sensitive trimer formation is to the effects spatial correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' In contrast, the trimer excitation rate in spin-polarized 87Sr decreases significantly with decreas- ing n due to anti-bunching, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=', Pauli exclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' This decrease is more pronounced than that seen in dimer pro- duction, further demonstrating the greater sensitivity of trimer production to spatial correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 0 1 2 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' signal (a) n=29 Boson trimer (b) n=39 0 1 2 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' signal (c) Boson dimer (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='0 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' signal (e) n=33 Fermion trimer (f) n=39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='5 Detuning (MHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='0 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' signal (g) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='5 Detuning (MHz) Fermion dimer (h) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Photoexcitation spectra for ground-state dimer and trimer molecules in a cold strontium gas as a function of laser detuning from the dimer or trimer line center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Each data set is normalized by laser intensities, atom densities, and the product βnO(p) n (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Trimer photoexcitation rates for bosonic 84Sr are shown in (a,b) and for spin-polarized 87Sr in (e, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Note the strong signal enhancement as n decreases for bosons compared to suppression for fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' For compari- son, measurements of dimer formation in the same gases are plotted in (c, d) and (g, h), which show similar but weaker variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The data for dimer and trimer production at n=39 are all normalized to the same peak height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Note the change in scale of the vertical axes between the upper and lower data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Figure 5 shows the principal findings of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='25 r/ dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='0 g(2)(r), g(3)(r) g(2)(r) - Boson g(3)(r) - Boson g(2)(r) - Fermion g(3)(r) - Fermion 84Sr dimer 84Sr trimer 87Sr dimer 87Sr trimer FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Measured and calculated values of g(3)(r) and g(2)(r) for ultracold gases of bosonic (84Sr) and spin-polarized fermionic (87Sr) atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Each set of experimental measure- ments is fit to the corresponding theoretical predictions using a single amplitude scaling factor, and the molecular size is scaled by the atomic thermal de Broglie wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Error bars denote standard error of the mean of multiple measure- ments or uncertainty in r/λdB due to uncertainty in sample temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Dimer and trimer ground state photoexcitation spectra were recorded for a range of quantum numbers, 29 ≤ n ≤ 45, at various temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' For each spectrum, the total integrated molecular signal (S(p) n ) was obtained by fitting to a voigt profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The signal was normalized by αI1I2CN (p)βnO(p) n to remove all dependences other than g(3)(r) or g(2)(r) [Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='1)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='2)], where r = Rn is taken to be the size of the ULRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' To enable direct com- parison between measurements undertaken at different sample temperatures, the length is scaled by the atomic thermal de Broglie wavelength, λdB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' A single amplitude scaling parameter is fit for each experimental data set, which normalizes the data to match the corresponding theoretical curves calculated from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' (Fermion dimer data is taken from [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' As evident from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 5, experimental observations match theoretical predictions well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' For a Bose gas and small values of r/λdB, g(2)(r) approaches 2, in agreement with earlier work [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' The predicted 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' increase is ob- served in g(3)(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' In contrast, for the Fermi gas, molecule formation is strongly suppressed at small values of r/λdB, and more strongly so for trimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Correlations decay to- wards unity on the length scale of λdB as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' CONCLUSIONS We have demonstrated that measurements of the for- mation of ground-state trimer ULRMs provide a sensitive in situ probe of three-body, nonlocal spatial correlations in ultracold gases, and have applied this probe to observe bunching and anti-bunching in thermal gases of indistin- guishable bosons and fermions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Even higher- order correlations are accessible by observing formation of tetramers and higher p-mers [22], offering the possibil- ity of comprehensive characterization of correlations in many-body quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' It should be possible to apply this technique to systems where interactions affect particle correlations, such as in strongly-interacting 1-D gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' ACKNOWLEDGEMENTS Research supported by the AFOSR under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' FA9550-14-1-0007, the NSF under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 1600059, and the FWF (Austria) under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' FWF SFB- SFB041 ViCom and the FWF Doctoral College W 1243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Burt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' W.' metadata={'source': 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(2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Browaeys and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Lahaye, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 16, 132 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' [4] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Semeghini, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Levine, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Keesling, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Ebadi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Bluvstein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Verresen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Pichler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Kali- nowski, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Samajdar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Omran, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Sachdev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Vish- 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' 95, 190406 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' [10] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Haller, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Rabie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' Mark, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAyT4oBgHgl3EQf9PqM/content/2301.00871v1.pdf'} +page_content=' G.' 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sha256:1470e72277d6c68b33eb5bbf1d49011bd56ccd7772db01952ed693d86eede6d7 +size 88668 diff --git a/aNFQT4oBgHgl3EQffjaD/content/tmp_files/2301.13340v1.pdf.txt b/aNFQT4oBgHgl3EQffjaD/content/tmp_files/2301.13340v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7915060a877b240a6d6f2582189dd4f179e5a9fb --- /dev/null +++ b/aNFQT4oBgHgl3EQffjaD/content/tmp_files/2301.13340v1.pdf.txt @@ -0,0 +1,1867 @@ +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +1 +Affinity Uncertainty-based Hard Negative Mining in +Graph Contrastive Learning +Chaoxi Niu, Guansong Pang, Member, IEEE, Ling Chen, Senior Member, IEEE +Abstract—Hard negative mining has shown effective in en- +hancing self-supervised contrastive learning (CL) on diverse +data types, including graph contrastive learning (GCL). Existing +hardness-aware CL methods typically treat negative instances +that are most similar to the anchor instance as hard negatives, +which helps improve the CL performance, especially on image +data. However, this approach often fails to identify the hard +negatives but leads to many false negatives on graph data. This +is mainly due to that the learned graph representations are +not sufficiently discriminative due to over-smooth representations +and/or non-i.i.d. issues in graph data. To tackle this problem, this +paper proposes a novel approach that builds a discriminative +model on collective affinity information (i.e, two sets of pairwise +affinities between the negative instances and the anchor instance) +to mine hard negatives in GCL. In particular, the proposed +approach evaluates how confident/uncertain the discriminative +model is about the affinity of each negative instance to an anchor +instance to determine its hardness weight relative to the anchor +instance. This uncertainty information is then incorporated into +existing GCL loss functions via a weighting term to enhance their +performance. The enhanced GCL is theoretically grounded that +the resulting GCL loss is equivalent to a triplet loss with an +adaptive margin being exponentially proportional to the learned +uncertainty of each negative instance. Extensive experiments on +10 graph datasets show that our approach i) consistently enhances +different state-of-the-art GCL methods in both graph and node +classification tasks, and ii) significantly improves their robustness +against adversarial attacks. +Index Terms—Graph contrastive learning, Hard negative min- +ing, Uncertainty estimation, Affinity learning. +I. INTRODUCTION +G +RAPH is ubiquitous and plays an important role in +various fields, such as social networks, bioinformatics, +chemistry, etc. Due to its non-Euclidean nature, learning +expressive graph representations is one crucial foundation of +different graph mining tasks, such as graph classification and +node classification. In recent years, graph neural networks +(GNNs) have become predominant in achieving this goal. +Most existing GNNs focus on supervised or semi-supervised +learning settings [1]–[4], where class label information is +required for training the GNNs. However, obtaining such +information is hard or costly, especially for graph data which +is at large scale and/or demands strong domain knowledge +Chaoxi Niu and Ling Chen are with the Australian Artificial Intelligence +Institute, University of Technology Sydney, Sydney, NSW 2007, Australia. +(email: Chaoxi.Niu@student.uts.edu.au; Ling.Chen@uts.edu.au). +Guansong Pang is with School of Computing and Information Systems, +Singapore Management University, 178902, Singapore. (email: pangguan- +song@gmail.com). +to accurately perform the data annotation. Recently, self- +supervised learning of GNNs [5], [6] which can learn graph +representations without accessing ground truth labels was +introduced to tackle this issue and has attracted significant +research interests. +Graph contrastive learning (GCL) has become one of the +most popular self-supervised methods for graph representation +learning [7]–[14]. It focuses on learning representations by +maximizing the mutual information between augmentations of +the same instance, in which the augmentations of the same +graph/node are often treated as positive instances, with the +other graphs/nodes as negative instances [5], [6]. +Despite the impressive successes achieved by current GCL +methods, their learning capability can be largely limited by the +way they choose negative samples [15]–[18]. One commonly- +used negative selection approach is to randomly select negative +instances from a sufficiently large batch or a memory bank, +and then treat all negative instances equally in contrastive +learning. However, this approach cannot exploit negative in- +stances that can provide more information for the contrastive +learning than the others. Particularly, many prior studies [15], +[16], [18] have shown that hard negative instances which are +difficult to discriminate from the positive are more crucial than +the counterparts (e.g., easy negatives that are distant from the +positive in both semantics and representations) to the learning +of discriminative features. +Many recent contrastive learning (CL) methods [15]–[17], +[19], [20] thus incorporate hard negative mining methods +into their training process to leverage these hard negative +instances. These hardness-aware CL methods typically treat +negative instances that are most similar to the anchor instance +as the hard negatives, which helps further improve the CL +performance, especially on image data [15]–[17], [19], [20]. +However, this hard negative mining approach often performs +poorly on graph data, as shown in some recent studies [18], +[21] and our experiments. This is mainly because the learned +graph representations are not sufficiently discriminative due +to i) the non-i.i.d. (independent and identically distributed) +nature of graph data, e.g., nodes with the same label tend +to be densely connected in graph data, and ii) the over- +smooth graph representations resulting from the iterative mes- +sage passing mechanism. Consequently, for graph data, the +most similar negatives to the anchor can be false negatives +with high probability. To address this issue, the very recent +method ProGCL [18] imposed a beta mixture model on the +pairwise similarities between the negatives and the anchor to +0000–0000/00$00.00 © 2021 IEEE +arXiv:2301.13340v1 [cs.LG] 31 Jan 2023 + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +2 +Data Instances +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 +25 +26 +27 +(a) +0 +3 +6 +9 12 15 18 21 24 27 +Instance ID +0.0 +0.2 +0.4 +0.6 +0.8 +Uncertainty-based Hardness +(b) +Anchor: 11 +Anchor: 26 +0.7 +0.8 +0.9 +1.0 +Similarity w.r.t Anchor 11 +0 +2 +4 +6 +8 +10 +Density +(c) +Flase Negative +True Negative +0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 +Uncertainty w.r.t Anchor 11 +0 +2 +4 +6 +8 +10 +12 +14 +Density +(d) +Flase Negative +True Negative +Fig. 1. (a): Two groups of data instances in blue and orange. (b): The affinity +uncertainty-based hardness results learned by our approach using instance 11 +or 26 as the anchor instance. Instances with a larger uncertainty are more likely +to be hard negative samples w.r.t. the anchor instance. (c): The histograms +of the similarity of the instances to the anchor instance 11. It is clear that +treating the most similar instances to the anchor as the hard negatives can lead +to many false negatives. (d): The uncertainty results learned by our approach +for the instances w.r.t the anchor instance 11, where true negatives including +hard negatives have large uncertainty values (and thus large hardness weights) +while false negative cases receive very small uncertainty values. +estimate the probability of a negative being true one, and +it subsequently combined the estimated probability and the +pairwise similarity to measure the hardness of the negatives. +The method relies on the prior that the similarity distribution +of negatives w.rt. positive is bimodal and works well in node +classification tasks. It fails to work when its prior is not fully +met. As shown in our experiments (Table III in Sec. IV-B), +such failure cases occur in most graph classification datasets +where ProGCL has very marginal improvement, or even worse +performance, compared to the original GCL methods. +This paper introduces a novel approach, dubbed AUGCL, +to tackle this problem. AUGCL learns a data-driven, affinity- +based uncertainty estimator to evaluate the hardness of nega- +tive instances relative to each anchor instance, meaning that +the hardness of an instance is dependent on the given anchor +instance, as shown by an example in Fig. 1(a-b). Particularly, +AUGCL builds a discriminative model on collective affin- +ity information (i.e, two sets of pairwise affinities between +the negative instances and the anchor instance) to evaluate +how confident/uncertain the discriminative model is about +the affinity of each negative instance to the anchor instance. +Instances that have a larger affinity uncertainty would be +more likely to be hard negatives, and they are subsequently +assigned with a larger hard-negative weight to receive more +attention from the GCL models. By doing so, AUGCL learns +discriminative affinity uncertainties for the negative instances +relative to each anchor instance, as shown by the results of +the anchor instance 11 in Fig. 1(b) and (d), where small and +large uncertainty-based hardness values are assigned to false +negatives and true negatives, respectively. By contrast, the +current similarity-based methods that regard the most similar +negative instances to the anchor instance as hard negatives +fail to identify the truly hard negatives but lead to many false +negatives, as shown in Fig. 1(c). Those learned hardness results +can then be seamlessly incorporated into popular GCL models +(e.g., InfoNCE-based models [22]) as a hardness weight to +enhance their performance. AUGCL addresses a similar issue +as ProGCL, but it eliminates the prior information posited +in ProGCL, enabling AUGCL to work more effectively on +diverse node-level and graph-level datasets. +In summary, this work makes the following three main +contributions. +• We propose a novel approach AUGCL that utilizes the +modeling of collective affinities to take account of the +non-i.i.d. and over-smooth representations issues in graph +data via the learning of an uncertainty-based hardness +measure. To the best of our knowledge, it is the first work +that addresses the problem using an uncertainty learning +framework. +• We show theoretically that our approach transforms popu- +lar GCL losses such as InfoNCE into a triplet loss with an +adaptive hardness-based margin, enforcing a large margin +for hard negatives while pulling false negatives close to +anchor instances. +• Extensive experiments on 10 graph datasets demonstrate +the superiority of AUGCL in consistently enhancing +different state-of-the-art GCL methods in both graph +and node classification tasks (having maximal classi- +fication accuracy improvement by ∼2% and ∼1.5%, +respectively), and the robustness against graph adversarial +attacks (maximal improvement by ∼8%). +II. RELATED WORKS +A. Graph Contrastive Learning +Recently, contrastive learning [22]–[25] has become a +prominent technique in self-supervised learning. It has been +successfully adapted into diverse domains, including the graph +domain. A number of GCL methods [7]–[13] have been +proposed. DGI [7] is an early attempt that obtained node rep- +resentations by maximizing the mutual information between +node embeddings and high-level graph information. MVGRL +[8] improved DGI by introducing different structural views +to learn node and graph-level representations. InfoGraph [9] +performed contrastive learning by directly maximizing the +consistency between sampled subgraphs and pooled graph +representations. Additionally, GraphCL [10] systematically +explored the influence of different augmentations on graph- +level contrastive learning. GCA [11] proposed to perform con- +trastive learning with adaptive augmentation on the topology +and node attribute level for node classification. Besides, some +studies have proposed to enhance the GCL by automating data +augmentations [12] or discarding explicit data augmentations +[13]. The main differences among these methods lie on the +way they obtain positive pairs. By contrast, our approach +AUGCL is focused on hard negative mining, which is orthog- +onal to these GCL methods and can be plugged into their +loss function to improve their performance on graph/node- +level tasks. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +3 +B. Hard Negative Mining in Contrastive Learning +Hard negative mining refers to generating or mining the +negatives which are difficult to discriminate from the positive. +Various methods have been proposed to perform hard negative +mining to facilitate contrastive learning, including employing +mixup strategy [26] to mix the anchor instance and negative +instance to synthesize hard negatives [15], [20], [27], [28], and +developing unsupervised sampling methods for selecting hard +negative samples [16], [17]. Recent state-of-the-art methods +in this line of research include DCL [17] and HCL [16]. +These methods are mainly focused on image data and they +often treat negative instances that are most similar to the +anchor instance as the hard negatives. However, for graph +data, the similar negatives could be false negatives relative to +the anchor, and the GCL performance would be degraded by +employing these hard negative mining methods [18], [21]. To +address this issue, ProGCL [18] exploited a two-component +beta mixture model to estimate the probability of negative +instances being true for an anchor and then measured the +hardness of negative instances by integrating the estimated +probability and the similarity between the negative and the +anchor. Similarly, our method also measures the hardness of +negatives for each anchor instance. However, we employ the +uncertainty estimation model to directly learn the negative +instance hardness. The learned hardness is then incorporated +into the contrastive loss via a weighting term, resulting in an +anchor-instance-adaptive contrastive learning framework with +good theoretical support. +C. Uncertainty Estimation +Numerous methods and theories have been introduced to +measure the prediction uncertainty, e.g., by using the maxi- +mum of predicted probabilities [29]–[31], the prediction en- +tropy/energy [30], [32]–[34], or an extra (void/background) +class [34]–[36]. These methods focus on calibrating predic- +tion confidence in supervised learning, whereas we utilize +uncertainty estimation under the self-supervised setting to +empower contrastive learning. Our work is motivated by the +observation that hard samples are typically the instances at the +decision boundary between the positive and negative instances, +which are also the samples that learning models are uncertain +about. Thus, uncertainty estimation offers an effective way to +measure the hardness of negative instances. To be applicable in +graph contrastive learning, AUGCL is designed in a novel way +by using an anchor-instance-dependent uncertainty learning +approach. +III. AUGCL: AFFINITY UNCERTAINTY-BASED GRAPH +CONTRASTIVE LEARNING +A. Preliminaries +Self-supervised graph representation learning has demon- +strated promising performance in empowering diverse graph +learning tasks. This work focuses on node-level and graph- +level tasks. Particularly, let G = (V, E) denote a graph where V +and E denote the set of nodes and edges respectively, then for +a node-level task, the goal of self-supervised graph representa- +tion learning is to leverage a single graph G to learn an encoder +ψ(V, E) without using the labels of nodes so that ψ(V, E) can +yield an expressive low-dimensional embedding zi for each +node in V. The resulting node embeddings Z = {zi}|V| +i=1 can +then be used in various downstream node-level tasks, such as +node classification. For a graph-level task, the goal instead is +to learn a graph encoder ψ(Vi, Ei) given a set of N graphs +{Gi = (Vi, Ei)}N +i=1, where the encoder ψ(Vi, Ei) outputs a +low-dimensional embedding zi for each graph Gi, and the +graph embeddings Z = {zi}N +i=1 can then be used in various +downstream graph-level tasks, e.g., graph classification. Our +approach can be used to improve the self-supervised learning +of graph representations and node representations, as shown +in Sec. IV. Without loss of generality, we use the graph-level +tasks to introduce our approach below. +B. The Proposed Approach AUGCL +1) Popular Graph Contrastive Learning Methods and Their +Weaknesses: Graph contrastive learning is one of the most +popular approaches for self-supervised graph representation +learning. As an instance-wise discriminative approach, it aims +to pull two different augmentations of the same graph closer +and push augmentations of different graphs apart [8], [10]. +InfoNCE [22] is among the most popular contrastive learning +loss functions to achieve this goal. Specifically, given a mini- +batch of randomly sampled graphs {Gi}N +i=1, two augmentation +functions t1 and t2 are first sampled from the augmentation +pool T which consists of all possible augmentations. Then, +two graph views { �Gi}N +i=1 and { �Gi}N +i=1 are generated by +applying t1, t2 to each graph. The embeddings {�zi}N +i=1 and +{�zi}N +i=1 of the augmented graphs are obtained by feeding the +augmented graphs into a shared GNN encoder ψ(·), followed +by a projection head (2-layer perceptron) [24]. For an anchor +instance �Gi – a graph augmented from Gi using t1, the positive +is �Gi – a graph augmented from the same graph Gi but +using a different augmentation t2, while the source of the +negative instances is { �Gj}N +j=1, from which negative instances +are sampled. To enforce the maximization of the consistency +between positive embeddings, the pairwise objective for a +positive pair (�zi, �zi) is formulated as: +ℓInfoNCE(�zi, �zi) = − log +e(h(�zi,�zi)/τ) +e(h(�zi,�zi)/τ) + +N +� +j,j̸=i +e(h(�zi,�zj)/τ) +, (1) +where τ denotes the temperature parameter and h(�zi, �zj) is +the cosine similarity function measuring similarity between �zi +and �zj. +Although these graph contrastive learning methods have +achieved great success in graph representation learning, they +often fail to consider the semantics of negatives in { �Gj}N +j=1. +Consequently, instances that share the same semantics with +the positive can be sampled and treated as negatives in (1). +This false negative sampling issue, also known as sampling +bias in [17], would hinder the learning of contrastive repre- +sentations between positive instances and negative instances. +More importantly, the contrastive learning cannot exploit hard +negatives, i.e., instances that are similar to but semantically + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +4 +Graph/Node Embeddings +Graph Augmentation +GNN +Encoder +GNN +Encoder +Shared +Contrastive Learning +Anchors +Pos +Neg +1 +... +0 +Collective +Affinity +Labels +0 +Uncertainty +Estimation Model +Binary Partition +Uncertianty of Negatives +... +Node Features +Push +Pull +Input +Affinity Uncertainty Learning +Weights of Negatives +... +Neg +Anchor +Similar +Dissimilar +Fig. 2. Overview of our approach AUGCL. Left: AUGCL-based graph contrastive learning. The objective and the general procedures are the same as existing +GCL methods, but AUGCL leverages affinity uncertainty to learn anchor-instance-dependent hardness-based instance weights {wi1, wi2, · · · , wiN} for all +negative instances to improve existing GCL methods. Right: The proposed affinity uncertainty learning approach to obtain the weights. For an anchor �zi, +AUGCL first obtains collective affinity information (i.e, pairwise affinity across the instances) via binary partition of its negative instances. It then utilizes +those affinity information to learn an uncertainty estimator that evaluates how confident the estimator is about the affinity of each negative instance �zj relative +to the anchor instance �zi. A larger affinity uncertainty value uij indicates more likely of �zj being a hard negative, and thus, a larger weight wij (wij = αuij +where α is a hyperparameter). +different from the anchor, which are driving force for con- +trastive learning to learn substantially enhanced discriminative +representations, as shown in the literature empirically and +theoretically [15], [16], [18]. +2) Our Affinity Uncertainty-enabled Approach for Over- +coming the Weaknesses: To address the negative sampling +weaknesses discussed in Sec. III-B1, we propose a novel +framework for learning an Affinity Uncertainty-based hard- +ness measure for enhancing current state-of-the-art Graph +Contrastive Learning methods, termed AUGCL. The key idea +is to first learn the hardness of a negative instance relative to +each anchor instance by comparing the affinity between them +to the affinities of the anchor instance to the other instances. +The hardness results can then be plugged into a contrastive +loss, e.g., InfoNCE, to improve the effectiveness of current +GCL methods in utilizing the hard negatives. +Overview of AUGCL. Since the hardness of a negative +instance varies largely w.r.t. different anchor instances, our +approach AUGCL aims to learn a hardness measure based on +the relative affinity between the negative instance and each +anchor instance. That is, for an anchor instance �zi and its +negative instance candidate set � +Zi = {�zj}N +j=1, we learn a +single hardness measure function φ(�zj|�zi; Θ) : � +Zi → R that +yields a hardness value for each �z ∈ � +Zi relative to �zi. Note that +the function φ parameterized by Θ is trained across all anchor +instances; yet the hardness it yields for the negative instance +�zj is dependent on the anchor �zi. For brevity, φ(�zj|�zi; Θ) is +denoted as φi(�zj; Θ) hereafter. +Unlike current hardness measures that define the hardness +of a negative instance based on its individual relation to the +anchor instance (e.g., the similarity between them), one key +novelty in AUGCL is that it defines the hardness based on two +groups of pairwise affinities between the negative instances +and the anchor instance. More specifically, we introduce the +concept of affinity uncertainty below to achieve this goal: +Definition 1 (Affinity Uncertainty). Given an anchor instance +�zi and its negative instance candidate set � +Zi = {�zj}N +j=1, and +let Ci +1 and Ci +2 be two disjoint groups of instances in � +Zi such +that: one group Ci +1 includes the instances that are closely +aligned and distributed around the anchor �zi, while the other +group Ci +2 contains the rest of other instances; and � +Zi= Ci +1∪Ci +2. +Then the affinity uncertainty of each �z ∈ � +Zi w.r.t. �zi is defined +as: +φi(�z) = g(�z, Ci +1, Ci +2), +(2) +where g is an uncertainty estimator that evaluates how confi- +dent the estimator is about the affinity of �z to the instances in +the anchor instance-centered group Ci +1 compared to the other +group Ci +2. +The affinity uncertainty in (2) takes a holistic approach that +considers diverse affinities of the negative instances within and +across the two groups Ci +1 and Ci +2 to learn an accurate hardness +for each negative instance �z. As shown in the literature +(e.g., [36]) and Fig. 1, instances which are ambiguous to +distinguish are assigned to large uncertainty values. These +instances typically have a poor affinity to both groups Ci +1 and +Ci +2, such as those located on the boundary between the two +groups. By contrast, if the instances are coherently aligned +within Ci +1 or Ci +2, their uncertainty would be small. Thus, this +type of uncertainty can be naturally used to define the hardness +of the negative instances. +The obtained hardness can then be easily plugged into +existing contrastive losses, such as the InfoNCE loss, via a +weighting term for the negative instances. Particularly, the + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +5 +AUGCL-enhanced InfoNCE is given as follows: +ℓAUGCL(�zi, �zi) = − log +e(h(�zi,�zi)/τ) +e(h(�zi,�zi)/τ) + +N +� +j,j̸=i +wije(h(�zi,�zj)/τ) +, +(3) +where wij = αφi(�zj; Θ) is the hardness-based weight added to +�zj relative to �zi. φi(�zj; Θ) is the hardness learned by AUGCL +for the negative instance �zj w.r.t. the anchor instance �zi and α +is a hyperparameter. This enables effective exploitation of the +hard negatives, as large weights are expected for hard negatives +while small weights are expected for the other instances, e.g., +false negatives. +The overall procedure of AUGCL is illustrated in Fig. +2. It follows the standard graph contrastive learning in the +graph augmentation and contrastive learning except that we +incorporate the affinity uncertainty-based hardness through +a weighting term into the contrastive loss as in (3). The +right panel in Fig. 2 shows the steps of learning an anchor- +dependent hardness measure φ for each anchor �zi, consisting +of instance partition and uncertainty estimation as indicated in +Def. 1. Before introducing the details of these two components +in Sec. III-C, below we demonstrate the theoretical motivation +of the proposed method. +Theoretical Motivation. We show below that (3) is equiv- +alent to a triplet loss with an adaptive margin exponentially +proportional to the learned hardness-based weight φi(�zj; Θ). +This provides a more straightforward explanation of the work- +ing mechanism of the proposed weighting method. +Theorem 1. Let uij = φi(�zj; Θ) be the affinity uncertainty- +based hardness of a negative instance �zj w.r.t. the anchor in- +stance �zi. When the projection function is an identity function +and assumes the positive instance is more similar to the anchor +than the negative instances, then minimizing the proposed +objective in (3) is equivalent to minimizing a modified triplet +loss with an adaptive margin mij = τ +2 log(αuij) , i.e., +ℓAUGCL(�zi, �zi) ∝ +1 +2τ +N +� +j,j̸=i +� +∥�z +′ +i − �z +′ +i∥ − ∥�z +′ +i − �z +′ +j∥ + mij +� +, +(4) +where �z +′ +i is the normalized embedding. +The proof for this theorem is detailed in the Appendix B. +From the theorem, we can see that the optimal embeddings to +(4) should satisfy the following inequality: +∥�z +′ +i − �z +′ +i∥ ≪ ∥�z +′ +i − �z +′ +j∥ − mij, +(5) +where mij = τ +2 log(αuij) and uij = φi(�zj; Θ). Thus, mij is +equivalent to a transformed affinity uncertainty-based hardness +of the negative instance �zj relative to the anchor �zi, satisfying: +� +mij ≥ 0, +if αuij ≥ 1; +mij < 0, +otherwise. +(6) +If �zj is a hard negative for �zi, the large uncertainty uij +between �zi and �zj makes the inequality (5) hard to satisfy +through mij > 0, enforcing better representation learning. On +the contrary, if the uncertainty uij is small, (5) can be easily +satisfied with mij ≪ 0, reducing the impact of the possible +false negative instances. +C. Instantiation of AUGCL +We introduce an instantiation of our AUGCL framework +in this subsection. As demonstrated in Def. 1, the affinity +uncertainty-based hardness function φ parameterized with Θ +can be decomposed into two modules, including a binary +clustering function f : {�zj}N +j=1 → {0, 1} parameterized by Θf +and an uncertainty estimation function g : {�zj}N +j=1 ×{0, 1} → +R parameterized by Θg, i.e., Θ = {Θf, Θg}. AUGCL is +a generic framework. Different clustering and uncertainty +estimation methods can be adopted in AUGCL to implement a +specific model, as shown by our empirical results in Sec. IV-D. +Below we describe the two modules of the best instantiated +AUGCL model based on our experiments. +1) Anchor-dependent Binary Partition of Negatives: Given +an anchor �zi, binary clustering is used to partition the negative +samples into two coherent groups – Ci +1 and Ci +2 – for subsequent +affinity uncertainty estimation. Without having access to label +information, clustering is often adopted on the full dataset to +divide instances into several clusters [37]–[42], and instances +from clusters other than the anchor-based cluster are directly +treated as negatives. +Our clustering differs from these existing methods from +two main ways. First, we perform an anchor-dependent binary +partition on only the negative instances in each batch of +instances rather than the full dataset. Specifically, given a +batch of node/graph embeddings {�zj}N +j=1, for each anchor +�zi ∈ {�zi}N +i=1, we perform a binary partition on the nega- +tive instance candidates {�zj}N +j=1 using an existing clustering +method (e.g., k-means), i.e., fk−means : {�zj}N +j=1 → {0, 1}, +where Ci +1 = 1 is the label of the cluster centered around �zi +and Ci +2 = 0 is the other cluster. That is, the clustering assigns +the cluster label Cij +1 += 1 if the instance �zj is sufficiently +similar to �zi, and a different cluster label Cij +2 = 0 is assigned +otherwise. +The second difference is that the obtained partitions are +used to gain a sense of the affinity of an instance to the +other instances, rather than being the direct negative sample +clusters. Those affinity information would be used to evaluate +the hardness of each negative instance through an uncertainty +estimation model in AUGCL. +2) Affinity Uncertainty Estimation: For an anchor �zi, the +binary cluster labels {Cij +1|2}N +j=1 carry the affinity semantics of +the instances {�zj}N +j=1 w.r.t. the anchor instance �zi. We further +propose to perform an uncertainty estimation upon these +affinity semantic-based labels for each anchor �zi ∈ {�zi}N +i=1, +and use this uncertainty to measure the hardness of instances +{�zj}N +j=1. By doing so, a large uncertainty-based hardness is +assigned to fringe instances that locate around the boundary +between the two clusters; these instances are typically hard +negatives w.r.t. �zi. A small hardness is assigned otherwise. +Different uncertainty estimation methods can be used to +specify this component. We found that the recently proposed +method Deep Gambler (DG) [36] worked best in our exper- +iments, so DG is used in AUGCL by default. Specifically, + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +6 +DG extends a multi-class classification task to a problem +that learns an extra class to represent the uncertainty of +instances, in addition to guaranteeing the classification of the +original classes. For an anchor instance �zi, given its associated +negative instance candidates { �zj}N +j=1 and their affinity labels +{Cij +1|2}N +j=1, the DG-based uncertainty estimation is trained by +minimizing the following loss: +ℓDG +i += − +N +� +j +log(pCij +1|2 ∗ o + uij), +(7) +where pCij +1|2 is the predicted class probability on class Cij +1|2 +from a multi-layer perceptrons-based (MLP-based) DG model +g(�z, Ci +1, Ci +2; Θg) parameterized by Θg, uij is the uncertainty +that the model g generates for the instance �zj, and o is a +reward parameter with a larger o encouraging g to be more +confident in inferring and vice versa. The final loss of DG is +computed across all anchor instances �zi ∈ {�zi}N +i=1. +After the DG model is trained, for each anchor �zi, we +calculate uij for its negative instance �zj and obtain a uncer- +tainty matrix U ∈ RN×(N−1) where each row ui contains +the uncertainty of all negative instances w.r.t. the anchor �zi. +These uncertainty values are then used in (3) to improve the +contrastive learning. +D. Time Complexity Analysis +We take k-means and Deep Gambler [36] as the partition +and uncertainty estimation methods, respectively, to analyze +the additional time complexity introduced by AUGCL. Specif- +ically, let L be the number of MLP layers in DG and d +be the number of hidden units for all layers. For the graph +classification task, given a graph dataset with N graphs and the +batch size is set as B, the time complexities of partition and the +uncertainty modeling are O(2( N +B )B2T) and O(KL( N +B )B2d2) +respectively, where T is the number of iterations for k-means +and K is the number of training epochs for the uncertainty es- +timation model. For the node classification task, given a graph +with N nodes, in order to reduce the computation cost, we only +sample M (M ≪ N) negatives for an anchor when training +AUGCL. The resulting time complexities of partition and +training uncertainty model are O(2NMT) and O(KLNMd2) +respectively. In experiments, we use the well-established k- +means clustering implementation from scikit-learn [43], as it +runs very fast in practice. Besides, the values of K, L, M and +d are relatively small and the uncertainty estimation model is +only trained once. Therefore, the computational overhead over +the base model is not significant. +IV. EXPERIMENTS +A. Experimental Setup +1) Datasets: Seven commonly used graph classification +datasets are used in our experiments. They come from +two popular application domains: bioinformatics (MUTAG, +DD, NCI1, and PROTEINS) and social networks (COLLAB, +REDDIT-BINARY, and IMDB-BINARY). For node classifi- +cation task, we use three widely used datasets, i.e., Wiki-CS +[44], Amazon-Computers and Amazon-Photo [45]. Wiki-CS is +a reference network constructed based on Wikipedia. Amazon- +Computers and Amazon-Photo are two co-purchase networks +constructed from Amazon. The statistics of the datasets are +summarized in Table I. +TABLE I +KEY STATISTICS OF DATASETS USED. +Task +Dataset +Graphs +Avg.Nodes +Avg.Edges +Graph +Classification +NCI1 +4,110 +29.87 +32.30 +PROTEINS +1,113 +39.06 +72.82 +DD +1,178 +284.32 +715.66 +MUTAG +188 +17.93 +19.79 +COLLAB +5,000 +74.49 +2,457.78 +RDT-B +2,000 +429.63 +497.75 +IMDB-B +1,000 +19.77 +96.53 +Task +Dataset +Graphs +Nodes +Edges +Node +Classification +Wiki-CS +1 +11,701 +216,123 +Aamazon-Computers +1 +13,752 +245,861 +Aamazon-Photo +1 +7,650 +119,081 +2) Implementation Details and Evaluation Protocol: For +graph classification task, GraphCL [10], a recent SOTA +InfoNCE-based contrastive learning method for graph classifi- +cation, is used as our base, into which our affinity uncertainty- +based hardness learning method is plugged. For a fair compar- +ison, the network backbone, the graph augmentation methods +and the hyper-parameters of our AUGCL-enabled GraphCL +are kept exactly the same as the original GraphCL. We follow a +widely-used two-stage evaluation protocol in the literature [9], +[10], [46], [47], in which we first learn graph representations +in a self-supervised manner and then use the representations +to train a downstream SVM classifier. The 10-fold evaluation +is adopted in classification, and it is repeated five times with +the mean accuracy (%) and standard variation reported. +For node classification task, we adopt GCA [11] as the +base model and plug our AUGCL-based affinity uncertainty +hardness into it. The evaluation protocol for node classification +follows DGI [7] where the model is first trained in an unsu- +pervised manner and then the learned node representations are +used to train and test a simple ℓ2-regularized logistic regression +classifier. On each dataset, the experiment is repeated for 20 +runs with different data splits, and the average classification +accuracy, together with the standard variation, is reported. +For graph and node classification, we use the same architec- +ture in our affinity uncertainty estimation model, i.e., a three- +layer multi-layer-perceptrons (MLP) architecture, containing +128 units per layer with ReLU activation. We adopt the +Stochastic Gradient Descent (SGD) optimizer for the uncer- +tainty estimation model and the learning rate is set to 0.01 +across all the datasets. The uncertainty scaling parameter α is +set to the reciprocal of the mean of uncertainties. The training +epoch number of the uncertainty estimation model is set to 10 +for all datasets. For the reward parameter in the uncertainty +estimation model, it is selected through a grid search, and the +search space is {1.5, 1.6, 1.7, 1.8, 1.9}. +3) Competing Methods: We evaluate the effectiveness of +AUGCL in both graph and node classification tasks. In +both tasks, AUGCL is evaluated against three state-of-the- +art hard negative mining-based contrastive learning methods, +including DCL [17], HCL [16] and ProGCL [18]. In addi- +tion, we also include a set of other relevant state-of-the-art +competing methods, including non-contrastive methods and + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +7 +TABLE II +COMPARISON BETWEEN THE BASELINES AND THEIR AUGCL-ENABLED +COUNTERPARTS. THE BASELINES ARE GRAPHCL [10] AND GCA [11] +FOR GRAPH AND NODE CLASSIFICATION TASKS, RESPECTIVELY. “↑” +REFERS TO THE IMPROVEMENT COMPARED TO THE BASELINES. +Task +Dataset +GraphCL +GraphCLAUGCL +Graph +Classification +NCI1 +78.26 +80.16(↑ 1.90) +PROTEINS +74.36 +75.76(↑ 1.40) +DD +78.01 +80.14 (↑ 2.13) +MUTAG +87.15 +89.20 (↑ 2.05) +COLLAB +71.53 +72.10 (↑ 0.57) +RDT-B +90.09 +91.19 (↑ 1.10) +IMDB-B +71.20 +72.46 (↑ 1.26) +Task +Dataset +GCA +GCAAUGCL +Node +Classification +Wiki-CS +78.08 +78.59 (↑ 0.51) +Amazon-Computers +87.80 +88.94 (↑ 1.14) +Amazon-Photo +91.99 +93.43 (↑ 1.44) +other contrastive methods. Particularly, for graph classification, +the non-contrastive methods include Graphlet Kernel (GK) +[48], Weisfeiler-Lehman Sub-tree Kernel (WL) [49], Deep +Graph Kernels (DGK) [46], node2vec [50], sub2vec [51] and +graph2vec [47], while the GCL methods include InfoGraph +[9], JOAOv2 [12], SimGRACE [13] and GraphCL [10]. +For the node classification task, non-contrastive methods +include node2vec [50], DeepWalk (DW) [52], and Graph +AutoEncoders (GAE and VGAE) [53]. Contrastive methods +include DGI [7], GMI [54], MVGRL [8], and GCA [11]. +Note that ProGCL proposed two strategies to utilize the +estimated hardness results, i.e., weighting and mixup. The +results reported are based on the weighting strategy of ProGCL +to have a direct comparison to our weighting-based AUGCL. +B. Enabling Different GCL Methods on Graph and Node +Classification +1) Performance Improvement over Baselines.: +We first +compare the performance of our proposed method with the +baselines on graph and node classification tasks. The results +are shown in Table II. It is clear that, by incorporating our +affinity uncertainty-based hardness measure, the two baselines +– GraphCL [10] and GCA [11] – are substantially and consis- +tently boosted on all datasets from different domains for both +graph and node classification tasks. This demonstrates that our +method AUGCL can enable these baselines to effectively at- +tend to hard negative instances and learn better representations +of graphs/nodes. +2) Comparison to State-of-the-art Methods: We then com- +pare AUGCL to diverse advanced graph embedding learning +methods. +Graph Classification. The results on graph classification +are reported in Table III. We can observe that graph contrastive +learning methods generally obtain better performance than +non-contrastive methods. Our method further improves the +performance by learning and feeding the affinity uncertainty- +based hardness into the contrastive learning, substantially +outperforming SOTA GCL methods on 6 out of 7 datasets. +Compared to the three recent hardness-aware methods DCL, +HCL and ProGCL, our method AUGCL performs much better +across all seven datasets. Particularly, DCL, HCL and ProGCL +improve GraphCL on some datasets such as PROTEINS, +MUTAG, and IMDB-B, but they fail on the other ones. By +contrast, our method improves over GraphCL by a large +margin across all the seven datasets, indicating the superiority +of our affinity uncertainty-based hardness learning method +over its recent counterparts. +Node Classification. The node classification results are +reported in Table IV. In general, the trends here are similar +to the results in Table III: i) contrastive methods are generally +more effective than the non-contrastive ones, and ii) the +competing hardness-aware methods DCL, HCL and ProGCL +further improve over the contrastive methods on part of the +datasets, while our method AUGCL achieves consistently +better performance on all the three datasets. +General hardness-aware methods DCL and HCL identify +hard negatives by using the individual similarity to the anchor, +which is often ineffective on graph data due to over-smoothed +node representation issues, as also found in the very recent +ProGCL work [18]. ProGCL addresses this issue by positing +a prior model over the pairwise similarity distribution to +learn the hardness. Our method further improves ProGCL +consistently on the three datasets by learning a data-driven +affinity uncertainty estimation model without the prior as- +sumption. Importantly, ProGCL is not generalizable to other +graph mining tasks such as the graph classification task, e.g., +ProGCL fails to work as effectively as the baseline GraphCL +on some datasets in Table III where our method AUGCL +also consistently outperforms the baseline, indicating better +applicability and flexibility of AUGCL on different graph +mining tasks than ProGCL. +C. Improving Robustness against Graph Adversarial Attacks +Self-supervised learning has shown effective performance +in defending against adversarial perturbations [56], [57]. This +subsection investigates whether AUGCL can further improve +over the GCL methods on this important property. In this +experiment, following [58], three different types of graph +adversarial attacks: RandSampling, GradArgmax and RL-S2V +are used, where RandSampling randomly adds or deletes +edges from graphs, GradArgmax performs edge modification +based on gradient information, and RL-S2V is a reinforcement +learning based attack method that learns a generalizable attack +policy. We also use the widely-used evaluation protocol as +in [58] where the graph task is to classify the component +numbers in synthetic graphs and structure2vec [55] is adopted +as the graph encoder, with different depths of structure2vec +considered in the experiments. Both the original structure2vec +trained from scratch (i.e., no pre-training) and the pre-trained +structure2vec [55] using GraphCL [10] are used as base- +lines. The experimental results of our method are obtained +by incorporating our affinity uncertainty-based hardness into +GraphCL to pre-train the structure2vec. The best-competing +method ProGCL is adopted in the same way. The results are +reported in Table V. +From the table, we can observe that: i) all three GCL +methods GraphCL, ProGCL, and AUGCL can largely improve +the robustness against all three graph adversarial attacks, + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +8 +TABLE III +RESULTS OF GRAPH CLASSIFICATION ACCURACY (%). WE OBTAIN THE RESULTS OF GRAPHCL AND ITS FOUR HARDNESS-AWARE VARIANTS UNDER THE +SAME EXPERIMENTAL SETTING AS [10], FROM WHICH THE RESULTS OF THE OTHER METHODS ARE TAKEN. THE RESULTS OF JOAOV2 [12] AND +SIMGRACE [13] ARE TAKEN FROM THEIR OWN PAPER. +Type +Method +NCI1 +PROTEINS +DD +MUTAG +COLLAB +RDT-B +IMDB-B +Non-Contrastive +Methods +GK +- +- +- +81.66±2.11 +- +77.34±0.18 +65.87±0.98 +WL +80.01±0.50 +72.92±0.56 +- +80.72±3.00 +- +68.82±0.41 +72.30±3.44 +DGK +80.31±0.46 +73.30±0.82 +- +87.44±2.72 +- +78.04±0.39 +66.96±0.56 +node2vec +54.89±1.61 +57.49±3.57 +- +72.63±10.20 +- +- +- +sub2vec +52.84±1.47 +53.03±5.55 +- +61.05±15.80 +- +71.48±0.41 +55.26±1.54 +graph2vec +73.22±1.81 +73.30±2.05 +- +83.15±9.25 +- +75.78±1.03 +71.10±0.54 +Contrastive +Methods +InfoGraph +76.20±1.06 +74.44±0.31 +72.85±1.78 +89.01±1.13 +70.65±1.13 +82.50±1.42 +73.03±0.87 +JOAOv2 +78.36±0.53 +74.07±1.10 +77.40±1.15 +87.67±0.79 +69.33±0.34 +86.42±1.45 +70.83±0.25 +SimGRACE +79.12±0.44 +75.35±0.09 +77.44±1.11 +89.01±1.31 +71.72±0.82 +89.51±0.89 +71.30±0.77 +GraphCL +78.26±0.39 +74.36±0.44 +78.01±0.77 +87.15±0.86 +71.53±1.03 +90.09±0.70 +71.20±0.72 +Hardness-aware +Methods +GraphCLDCL +77.62±0.67 +74.73±0.39 +76.84±1.24 +88.28±1.75 +70.36±0.97 +89.88±0.72 +70.62±0.58 +GraphCLHCL +78.16±0.53 +74.39±0.77 +76.83±1.15 +88.94±1.22 +70.37±0.33 +90.05±0.47 +71.38±0.62 +GraphCLProGCL +76.93±0.47 +74.48±0.37 +79.22±0.90 +88.73±1.40 +70.46±0.28 +90.51±0.49 +71.58±0.59 +GraphCLAUGCL (Ours) +80.16±0.34 +75.76±0.39 +80.14±0.54 +89.20±1.08 +72.10±0.65 +91.19±0.44 +72.46±0.80 +TABLE IV +RESULTS OF NODE CLASSIFICATION ACCURACY (%). WE OBTAIN THE +RESULTS OF GCA AND ITS FOUR HARDNESS-AWARE VARIANTS UNDER +THE SAME EXPERIMENTAL SETTING AS [11], FROM WHICH THE RESULTS +OF THE OTHER METHODS ARE TAKEN. +Type +Method +Wiki-CS +Amazon- +Computers +Amazon- +Photo +Non-Contrastive +Methods +Raw feature +71.98±0.00 +73.81±0.00 +78.53±0.00 +node2vec +71.79±0.05 +84.39±0.08 +89.67±0.12 +DW +74.35±0.06 +85.68±0.06 +89.44±0.11 +DW+feature +77.21±0.03 +86.28±0.07 +90.05±0.08 +GAE +70.15±0.01 +85.27±0.19 +91.62±0.13 +VGAE +75.63±0.19 +86.37±0.21 +92.20±0.11 +Contrastive +Methods +DGI +75.35±0.14 +83.95±0.47 +91.61±0.22 +GMI +74.85±0.08 +82.21±0.31 +90.68±0.17 +MVGRL +77.52±0.08 +87.52±0.11 +91.74±0.07 +GCA +78.08±0.58 +87.80±0.42 +91.99±0.39 +Hardness-aware +Methods +GCADCL +78.12±0.60 +86.79±0.48 +91.29±0.32 +GCAHCL +78.19±0.64 +87.64±0.34 +91.79±0.29 +GCAProGCL +78.33±0.64 +88.68±0.35 +93.01±0.29 +GCAAUGCL (Ours) +78.59±0.56 +88.94±0.44 +93.43±0.32 +particularly the more advanced attacks GradArgmax and RL- +S2V, on different network layers, compared with the original +model, ii) the robustness can be further improved by exploiting +hard negative mining techniques used in ProGCL and AUGCL, +compared to GraphCL, and iii) compared with ProGCL, the +better hard negative mining in our method AUGCL generally +results in more remarkably and stably improved robustness +over the GraphCL. Overall, the proposed method AUGCL in- +creases the classification accuracy by up to 8% over GraphCL +and up to 2.7% over ProGCL, and performs very competitively +to the two methods (i.e., around 0.2%-0.8% difference) in the +limited cases where AUGCL is not the best performer. +D. Ablation Studies +This subsection evaluates the impact of using different +clustering and uncertainty estimation methods in f and g, re- +spectively. The GraphCLAUGCL method is used, with GraphCL +as the baseline. +Partition Methods in f. An important module of our +proposed method is the instance-wise partition function f. +k-means is used by default to implement f. Here we also +examine the use of spectral clustering [59] to perform the +binary partition. The results are shown in Table VI. We can +see that AUGCL using either spectral clustering or k-means +achieves similar improvement over GraphCL, suggesting the +stability of our method w.r.t. the generation of the partition +labels. AUGCL with k-means clustering performs consistently +better than the spectral clustering. Hence, k-means clustering +is used by default in our experiments and recommended in +practice. +Uncertainty Estimation Methods in g. The uncertainty +estimation method g is another important module in AUGCL. +In addition to the extra class-based method used by default +in AUGCL, two alternative approaches are used, including +the maximum prediction probability-based method Softmax- +Response [29] and the entropy-based method Predictive En- +tropy [30]. We also include a distance-based method as another +simplified variant of AUGCL. The detailed descriptions of +these uncertainty estimation methods are presented in Ap- +pendix A. +The results are reported in Table VI. It is clear that re- +gardless of the specific uncertainty estimation method used, +all variants of AUGCL can generally improve the baseline +GraphCL on nearly all datasets. This provides further ev- +idence for the effectiveness of our approach. Additionally, +the uncertainty estimation method matters: the default method +(a recently proposed extra class-based method [34], [36]), a +more effective uncertainty estimation model than the other +three methods, shows consistently better performance than the +other three variants, implying that the hardness can be better +captured by more advanced uncertainty estimation methods. +E. Hyperparameter Analysis +We examine the sensitivity of AUGCL w.r.t two key hy- +perparameters, i.e., the uncertainty parameter α in (3) and the +reward parameter o in φ (particularly in (7)). Without loss of +generality, one graph dataset from biochemical molecules and +social networks respectively, i.e, PROTEINS and IMDB-B, are +used. +Uncertainty Parameter α. α is adaptively set, depending +on the uncertainty matrix U, to enable stable performance of +AUGCL. Particularly, given U, we can calculate the mean +µ and standard deviation δ of U, based on which α is + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +9 +TABLE V +CLASSIFICATION ACCURACY UNDER THREE EVASION ATTACKS ON THREE DIFFERENT LAYERS OF STRUCTURE2VEC [55]. PROGCL AND AUGCL BELOW +REPRESENT GRAPHCL METHODS WITH THEIR RESPECTIVE HARD NEGATIVE MINING COMPONENT. +Attacks +2-Layer +3-Layer +4-Layer +Original +GraphCL +ProGCL +AUGCL +Original +GraphCL +ProGCL +AUGCL +Original +GraphCL +ProGCL +AUGCL +Unattack +93.20 +94.73 +94.13 +94.20 +98.20 +98.33 +98.67 +98.87 +98.87 +99.00 +99.47 +99.20 +RandSampling +78.73 +80.68 +82.47 +82.67 +92.27 +92.60 +93.93 +94.67 +95.13 +97.40 +97.13 +97.93 +GradArgmax +69.47 +69.26 +74.80 +77.53 +64.60 +89.33 +94.07 +93.27 +95.80 +97.00 +95.67 +96.47 +RL-S2V +42.93 +42.20 +42.13 +42.47 +41.93 +61.66 +62.07 +63.73 +70.20 +84.86 +86.67 +87.33 +TABLE VI +ABLATION STUDY RESULTS OF GRAPHCLAUGCL USING DIFFERENT CLUSTERING METHODS f OR UNCERTAINTY ESTIMATION METHODS g. +Method +f +g +NCI1 +PROTEINS +DD +MUTAG +COLLAB +RDT-B +IMDB-B +GraphCL (Baseline) +× +× +78.26±0.39 +74.36±0.44 +78.01±0.77 +87.15±0.86 +71.53±1.03 +90.09±0.70 +71.20±0.72 +GraphCLAUGCL +Spectral +ExtraClass +79.79±0.52 +75.57±0.79 +79.05±0.44 +88.86±2.22 +71.74±0.88 +91.01±0.56 +71.86±0.47 +k-means +ExtraClass +80.16±0.34 +75.76±0.39 +80.14±0.54 +89.20±1.08 +72.10±0.65 +91.19±0.44 +72.46±0.80 +k-means +Distance +78.88±0.26 +75.48±1.06 +79.22±0.25 +88.64±0.83 +71.27±0.43 +90.41±0.51 +72.20±0.55 +k-means +Softmax +79.42±0.43 +75.34±0.57 +78.10±0.65 +86.81±1.65 +71.89±0.92 +90.38±0.33 +71.72±0.57 +k-means +Entropy +79.98±0.50 +75.13±0.39 +79.58±0.49 +88.98±1.78 +71.71±0.37 +90.37±0.50 +71.76±0.57 +set to α = +1 +µ. We vary the parameter α in the range of +{ +1 +µ−δ, +1 +µ−0.5δ, 1 +µ, +1 +µ+0.5δ, +1 +µ+δ}. The mean classification accu- +racy (%) under different α are shown in Fig. 3(a) where the +labels in the x-axis denote the coefficient of δ when calculating +α. It is clear that the performance of our model is generally +stable with varying α, and α = 1 +µ is a recommended setting. +Reward Parameter o. We further examine the reward +parameter o in the uncertainty estimation model [36]. With +o varying in {1.5, 1.6, 1.7, 1.8, 1.9}, we report the mean clas- +sification accuracy (%) in Fig. 3(b). The results also show that +AUGCL can achieve reasonably stable performance for a wide +range of the o settings. +1.0 +0.5 +0.0 +0.5 +1.0 +70 +72 +74 +76 +Accuracy +PROTEINS +IMDB-B +(a) Parameter α +1.5 +1.6 +1.7 +1.8 +1.9 +70 +72 +74 +76 +Accuracy +PROTEINS +IMDB-B +(b) Parameter o +Fig. 3. Sensitivity analysis of hyperparameters α and o. +V. CONCLUSION +This paper proposes the idea of affinity uncertainty and +utilizes it to measure the hardness of negative samples to +improve popular GCL models. To this end, we introduce the +affinity uncertainty-based hardness learning approach AUGCL +that synthesizes binary partition and uncertainty estimation +to learn anchor-instance-dependent hardness for all negative +instances, i.e., their hardness results are relative to each anchor +instance. AUGCL is a data-driven approach that eliminates the +prior assumption made in very recent hardness-aware GCL +methods like ProGCL [18], resulting in better applicability and +flexibility on different graph mining tasks, as well as better +robustness to diverse graph adversarial attacks. It also shows +better performance in enabling different GCL loss functions, +compared to a wide range of other state-of-the-art graph +representation methods on graph and node classification tasks. +We also show theoretically that the resulting contrastive loss in +AUGCL is equivalent to a triplet loss with an adaptive margin +that adaptively exploits the hard negatives with a large margin, +with a small margin assigned to the other negative instances. +REFERENCES +[1] K. Xu, W. Hu, J. Leskovec, and S. Jegelka, “How powerful are graph +neural networks?” in International Conference on Learning Represen- +tations, 2019. +[2] T. N. Kipf and M. Welling, “Semi-supervised classification with graph +convolutional networks,” arXiv preprint:1609.02907, 2016. +[3] P. Veliˇckovi´c, G. Cucurull, A. Casanova, A. Romero, P. Li`o, and +Y. Bengio, “Graph attention networks,” in International Conference on +Learning Representations, 2018. +[4] Z. Wu, S. Pan, F. Chen, G. Long, C. Zhang, and S. Y. 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Weiss, “On spectral clustering: Analysis and an +algorithm,” Advances in Neural Information Processing Systems, vol. 14, +2001. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +11 +APPENDIX A +DETAILS OF OTHER UNCERTAINTY ESTIMATION METHODS +A. Softmax-Response +Given the softmax predicted vector p = {p1, . . . , pC} (C +is the number of classes), the uncertainty estimated based on +Softmax-Response [29] is calculated as follows: +u = 1 − +max +c=1,...,C pc +(8) +The uncertainty is large when the predicted probabilities +are more equally distributed, and it is small when predicted +probabilities are concentrated on one specific class. +B. Predictive Entropy +Predictive entropy [30] captures the average amount of +information presented in the predicted probability distribution. +The uncertainty based on predictive entropy is obtained by: +u = − +C +� +c=1 +pc log pc +(9) +Similar to Softmax-Response, the predictive entropy reaches +its maximum value when all predicted classes are equiprobable +and its minimum is attained when there exists one class with +probability one and the probabilities of all other classes are +zero. +C. Distance-Based Uncertainty +For each anchor instance, after obtaining the binary partition +labels of the negative instances {Ci,j +1|2}N +j=1, we calculate the +uncertainty of the negative instance whose cluster labels are +zero. Specially, the distance dij between the anchor instance +and the negative instance �zj (Cij +2 += 0) is first calculated +and then the uncertainty of �zj is set as the reciprocal of dij +weighted by a parameter β i.e., +uij = β 1 +dij +. +(10) +Note that for �zj with Cij +2 = 0, if �zj is close to the anchor �zi, +the uncertainty value uij is large. +APPENDIX B +PROOF OF THEOREM 1 +Proof. +ℓAUGCL(�zi, �zi) += − log +exp(h(�zi, �zi)/τ) +exp(h(�zi, �zi)/τ) + +N +� +j,j̸=i +αuij exp(h(�zi, �zj)/τ) += log +� +�1 + +N +� +j,j̸=i +αuij exp +�h(�zi, �zj) − h(�zi, �zi) +τ +�� +� += log +� +�1 + +N +� +j,j̸=i +exp +�h(�zi, �zj) − h(�zi, �zi) + 2mij +τ +�� +� +(11) +where h(�zi, �zj) = +�zT +i �zj +∥�zi∥∥�zj∥ (−1 ⩽ h(�zi, �zj) ⩽ 1) and +mij = τ +2 log(αuij). Let �z +′ +i = +�zi +∥�zi∥, h(�zi, �zj) can be rewrited +as h(�zi, �zj) = (�z +′ +i)T �z +′ +j. Since the positive instance is more +similar than the negative instances to the anchor, the value of +h(�zi, �zj) − h(�zi, �zi) tends to be −2. +Based on the above analysis, we can apply the Taylor +expansion of first order and get the following approximation: +ℓAUGCL(�zi, �zi) += log +� +�1 + +N +� +j,j̸=i +exp +�h(�zi, �zj) − h(�zi, �zi) + 2mij +τ +�� +� +≈ +N +� +j,j̸=i +exp +�h(�zi, �zj) − h(�zi, �zi) + 2mij +τ +� +≈ e− 2 +τ +� +�(N − 1) + 1 +τ +N +� +j,j̸=i +(h(�zi, �zj) − h(�zi, �zi) + 2mij + 2) +� +� +∝ 1 +τ +N +� +j,j̸=i +(h(�zi, �zj) − h(�zi, �zi) + 2mij) += 1 +2τ +N +� +j,j̸=i +� +∥�z +′ +i − �z +′ +i∥ − ∥�z +′ +i − �z +′ +j∥ + mij +� +(12) +which concludes the proof. + diff --git a/aNFQT4oBgHgl3EQffjaD/content/tmp_files/load_file.txt b/aNFQT4oBgHgl3EQffjaD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7e0a587693e23ecafa7a88c66403d7dfa9772c2e --- /dev/null +++ b/aNFQT4oBgHgl3EQffjaD/content/tmp_files/load_file.txt @@ -0,0 +1,1402 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf,len=1401 +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 8, AUGUST 2021 1 Affinity Uncertainty-based Hard Negative Mining in Graph Contrastive Learning Chaoxi Niu, Guansong Pang, Member, IEEE, Ling Chen, Senior Member, IEEE Abstract—Hard negative mining has shown effective in en- hancing self-supervised contrastive learning (CL) on diverse data types, including graph contrastive learning (GCL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Existing hardness-aware CL methods typically treat negative instances that are most similar to the anchor instance as hard negatives, which helps improve the CL performance, especially on image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' However, this approach often fails to identify the hard negatives but leads to many false negatives on graph data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' This is mainly due to that the learned graph representations are not sufficiently discriminative due to over-smooth representations and/or non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' issues in graph data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' To tackle this problem, this paper proposes a novel approach that builds a discriminative model on collective affinity information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='e, two sets of pairwise affinities between the negative instances and the anchor instance) to mine hard negatives in GCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' In particular, the proposed approach evaluates how confident/uncertain the discriminative model is about the affinity of each negative instance to an anchor instance to determine its hardness weight relative to the anchor instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' This uncertainty information is then incorporated into existing GCL loss functions via a weighting term to enhance their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The enhanced GCL is theoretically grounded that the resulting GCL loss is equivalent to a triplet loss with an adaptive margin being exponentially proportional to the learned uncertainty of each negative instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Extensive experiments on 10 graph datasets show that our approach i) consistently enhances different state-of-the-art GCL methods in both graph and node classification tasks, and ii) significantly improves their robustness against adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Index Terms—Graph contrastive learning, Hard negative min- ing, Uncertainty estimation, Affinity learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' INTRODUCTION G RAPH is ubiquitous and plays an important role in various fields, such as social networks, bioinformatics, chemistry, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Due to its non-Euclidean nature, learning expressive graph representations is one crucial foundation of different graph mining tasks, such as graph classification and node classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' In recent years, graph neural networks (GNNs) have become predominant in achieving this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Most existing GNNs focus on supervised or semi-supervised learning settings [1]–[4], where class label information is required for training the GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' However, obtaining such information is hard or costly, especially for graph data which is at large scale and/or demands strong domain knowledge Chaoxi Niu and Ling Chen are with the Australian Artificial Intelligence Institute, University of Technology Sydney, Sydney, NSW 2007, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' (email: Chaoxi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='Niu@student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='uts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='au;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Ling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='Chen@uts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Guansong Pang is with School of Computing and Information Systems, Singapore Management University, 178902, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' (email: pangguan- song@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' to accurately perform the data annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Recently, self- supervised learning of GNNs [5], [6] which can learn graph representations without accessing ground truth labels was introduced to tackle this issue and has attracted significant research interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Graph contrastive learning (GCL) has become one of the most popular self-supervised methods for graph representation learning [7]–[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' It focuses on learning representations by maximizing the mutual information between augmentations of the same instance, in which the augmentations of the same graph/node are often treated as positive instances, with the other graphs/nodes as negative instances [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Despite the impressive successes achieved by current GCL methods, their learning capability can be largely limited by the way they choose negative samples [15]–[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' One commonly- used negative selection approach is to randomly select negative instances from a sufficiently large batch or a memory bank, and then treat all negative instances equally in contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' However, this approach cannot exploit negative in- stances that can provide more information for the contrastive learning than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Particularly, many prior studies [15], [16], [18] have shown that hard negative instances which are difficult to discriminate from the positive are more crucial than the counterparts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', easy negatives that are distant from the positive in both semantics and representations) to the learning of discriminative features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Many recent contrastive learning (CL) methods [15]–[17], [19], [20] thus incorporate hard negative mining methods into their training process to leverage these hard negative instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' These hardness-aware CL methods typically treat negative instances that are most similar to the anchor instance as the hard negatives, which helps further improve the CL performance, especially on image data [15]–[17], [19], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' However, this hard negative mining approach often performs poorly on graph data, as shown in some recent studies [18], [21] and our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' This is mainly because the learned graph representations are not sufficiently discriminative due to i) the non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' (independent and identically distributed) nature of graph data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', nodes with the same label tend to be densely connected in graph data, and ii) the over- smooth graph representations resulting from the iterative mes- sage passing mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Consequently, for graph data, the most similar negatives to the anchor can be false negatives with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' To address this issue, the very recent method ProGCL [18] imposed a beta mixture model on the pairwise similarities between the negatives and the anchor to 0000–0000/00$00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='00 © 2021 IEEE arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='13340v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='LG] 31 Jan 2023 JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 8, AUGUST 2021 2 Data Instances 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 (a) 0 3 6 9 12 15 18 21 24 27 Instance ID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='8 Uncertainty-based Hardness (b) Anchor: 11 Anchor: 26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='0 Similarity w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='t Anchor 11 0 2 4 6 8 10 Density (c) Flase Negative True Negative 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='8 Uncertainty w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='t Anchor 11 0 2 4 6 8 10 12 14 Density (d) Flase Negative True Negative Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' (a): Two groups of data instances in blue and orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' (b): The affinity uncertainty-based hardness results learned by our approach using instance 11 or 26 as the anchor instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Instances with a larger uncertainty are more likely to be hard negative samples w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' the anchor instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' (c): The histograms of the similarity of the instances to the anchor instance 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' It is clear that treating the most similar instances to the anchor as the hard negatives can lead to many false negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' (d): The uncertainty results learned by our approach for the instances w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='t the anchor instance 11, where true negatives including hard negatives have large uncertainty values (and thus large hardness weights) while false negative cases receive very small uncertainty values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' estimate the probability of a negative being true one, and it subsequently combined the estimated probability and the pairwise similarity to measure the hardness of the negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The method relies on the prior that the similarity distribution of negatives w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' positive is bimodal and works well in node classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' It fails to work when its prior is not fully met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' As shown in our experiments (Table III in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' IV-B), such failure cases occur in most graph classification datasets where ProGCL has very marginal improvement, or even worse performance, compared to the original GCL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' This paper introduces a novel approach, dubbed AUGCL, to tackle this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' AUGCL learns a data-driven, affinity- based uncertainty estimator to evaluate the hardness of nega- tive instances relative to each anchor instance, meaning that the hardness of an instance is dependent on the given anchor instance, as shown by an example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 1(a-b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Particularly, AUGCL builds a discriminative model on collective affin- ity information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='e, two sets of pairwise affinities between the negative instances and the anchor instance) to evaluate how confident/uncertain the discriminative model is about the affinity of each negative instance to the anchor instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Instances that have a larger affinity uncertainty would be more likely to be hard negatives, and they are subsequently assigned with a larger hard-negative weight to receive more attention from the GCL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' By doing so, AUGCL learns discriminative affinity uncertainties for the negative instances relative to each anchor instance, as shown by the results of the anchor instance 11 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 1(b) and (d), where small and large uncertainty-based hardness values are assigned to false negatives and true negatives, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' By contrast, the current similarity-based methods that regard the most similar negative instances to the anchor instance as hard negatives fail to identify the truly hard negatives but lead to many false negatives, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Those learned hardness results can then be seamlessly incorporated into popular GCL models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', InfoNCE-based models [22]) as a hardness weight to enhance their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' AUGCL addresses a similar issue as ProGCL, but it eliminates the prior information posited in ProGCL, enabling AUGCL to work more effectively on diverse node-level and graph-level datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' In summary, this work makes the following three main contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' We propose a novel approach AUGCL that utilizes the modeling of collective affinities to take account of the non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' and over-smooth representations issues in graph data via the learning of an uncertainty-based hardness measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' To the best of our knowledge, it is the first work that addresses the problem using an uncertainty learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' We show theoretically that our approach transforms popu- lar GCL losses such as InfoNCE into a triplet loss with an adaptive hardness-based margin, enforcing a large margin for hard negatives while pulling false negatives close to anchor instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Extensive experiments on 10 graph datasets demonstrate the superiority of AUGCL in consistently enhancing different state-of-the-art GCL methods in both graph and node classification tasks (having maximal classi- fication accuracy improvement by ∼2% and ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='5%, respectively), and the robustness against graph adversarial attacks (maximal improvement by ∼8%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' RELATED WORKS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Graph Contrastive Learning Recently, contrastive learning [22]–[25] has become a prominent technique in self-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' It has been successfully adapted into diverse domains, including the graph domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' A number of GCL methods [7]–[13] have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' DGI [7] is an early attempt that obtained node rep- resentations by maximizing the mutual information between node embeddings and high-level graph information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' MVGRL [8] improved DGI by introducing different structural views to learn node and graph-level representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' InfoGraph [9] performed contrastive learning by directly maximizing the consistency between sampled subgraphs and pooled graph representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Additionally, GraphCL [10] systematically explored the influence of different augmentations on graph- level contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' GCA [11] proposed to perform con- trastive learning with adaptive augmentation on the topology and node attribute level for node classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Besides, some studies have proposed to enhance the GCL by automating data augmentations [12] or discarding explicit data augmentations [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The main differences among these methods lie on the way they obtain positive pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' By contrast, our approach AUGCL is focused on hard negative mining, which is orthog- onal to these GCL methods and can be plugged into their loss function to improve their performance on graph/node- level tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 8, AUGUST 2021 3 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Hard Negative Mining in Contrastive Learning Hard negative mining refers to generating or mining the negatives which are difficult to discriminate from the positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Various methods have been proposed to perform hard negative mining to facilitate contrastive learning, including employing mixup strategy [26] to mix the anchor instance and negative instance to synthesize hard negatives [15], [20], [27], [28], and developing unsupervised sampling methods for selecting hard negative samples [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Recent state-of-the-art methods in this line of research include DCL [17] and HCL [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' These methods are mainly focused on image data and they often treat negative instances that are most similar to the anchor instance as the hard negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' However, for graph data, the similar negatives could be false negatives relative to the anchor, and the GCL performance would be degraded by employing these hard negative mining methods [18], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' To address this issue, ProGCL [18] exploited a two-component beta mixture model to estimate the probability of negative instances being true for an anchor and then measured the hardness of negative instances by integrating the estimated probability and the similarity between the negative and the anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Similarly, our method also measures the hardness of negatives for each anchor instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' However, we employ the uncertainty estimation model to directly learn the negative instance hardness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The learned hardness is then incorporated into the contrastive loss via a weighting term, resulting in an anchor-instance-adaptive contrastive learning framework with good theoretical support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Uncertainty Estimation Numerous methods and theories have been introduced to measure the prediction uncertainty, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', by using the maxi- mum of predicted probabilities [29]–[31], the prediction en- tropy/energy [30], [32]–[34], or an extra (void/background) class [34]–[36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' These methods focus on calibrating predic- tion confidence in supervised learning, whereas we utilize uncertainty estimation under the self-supervised setting to empower contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Our work is motivated by the observation that hard samples are typically the instances at the decision boundary between the positive and negative instances, which are also the samples that learning models are uncertain about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Thus, uncertainty estimation offers an effective way to measure the hardness of negative instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' To be applicable in graph contrastive learning, AUGCL is designed in a novel way by using an anchor-instance-dependent uncertainty learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' AUGCL: AFFINITY UNCERTAINTY-BASED GRAPH CONTRASTIVE LEARNING A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Preliminaries Self-supervised graph representation learning has demon- strated promising performance in empowering diverse graph learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' This work focuses on node-level and graph- level tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Particularly, let G = (V, E) denote a graph where V and E denote the set of nodes and edges respectively, then for a node-level task, the goal of self-supervised graph representa- tion learning is to leverage a single graph G to learn an encoder ψ(V, E) without using the labels of nodes so that ψ(V, E) can yield an expressive low-dimensional embedding zi for each node in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The resulting node embeddings Z = {zi}|V| i=1 can then be used in various downstream node-level tasks, such as node classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' For a graph-level task, the goal instead is to learn a graph encoder ψ(Vi, Ei) given a set of N graphs {Gi = (Vi, Ei)}N i=1, where the encoder ψ(Vi, Ei) outputs a low-dimensional embedding zi for each graph Gi, and the graph embeddings Z = {zi}N i=1 can then be used in various downstream graph-level tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', graph classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Our approach can be used to improve the self-supervised learning of graph representations and node representations, as shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Without loss of generality, we use the graph-level tasks to introduce our approach below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The Proposed Approach AUGCL 1) Popular Graph Contrastive Learning Methods and Their Weaknesses: Graph contrastive learning is one of the most popular approaches for self-supervised graph representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' As an instance-wise discriminative approach, it aims to pull two different augmentations of the same graph closer and push augmentations of different graphs apart [8], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' InfoNCE [22] is among the most popular contrastive learning loss functions to achieve this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Specifically, given a mini- batch of randomly sampled graphs {Gi}N i=1, two augmentation functions t1 and t2 are first sampled from the augmentation pool T which consists of all possible augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Then, two graph views { �Gi}N i=1 and { �Gi}N i=1 are generated by applying t1, t2 to each graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The embeddings {�zi}N i=1 and {�zi}N i=1 of the augmented graphs are obtained by feeding the augmented graphs into a shared GNN encoder ψ(·), followed by a projection head (2-layer perceptron) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' For an anchor instance �Gi – a graph augmented from Gi using t1, the positive is �Gi – a graph augmented from the same graph Gi but using a different augmentation t2, while the source of the negative instances is { �Gj}N j=1, from which negative instances are sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' To enforce the maximization of the consistency between positive embeddings, the pairwise objective for a positive pair (�zi, �zi) is formulated as: ℓInfoNCE(�zi, �zi) = − log e(h(�zi,�zi)/τ) e(h(�zi,�zi)/τ) + N � j,j̸=i e(h(�zi,�zj)/τ) , (1) where τ denotes the temperature parameter and h(�zi, �zj) is the cosine similarity function measuring similarity between �zi and �zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Although these graph contrastive learning methods have achieved great success in graph representation learning, they often fail to consider the semantics of negatives in { �Gj}N j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Consequently, instances that share the same semantics with the positive can be sampled and treated as negatives in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' This false negative sampling issue, also known as sampling bias in [17], would hinder the learning of contrastive repre- sentations between positive instances and negative instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' More importantly, the contrastive learning cannot exploit hard negatives, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', instances that are similar to but semantically JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 8, AUGUST 2021 4 Graph/Node Embeddings Graph Augmentation GNN Encoder GNN Encoder Shared Contrastive Learning Anchors Pos Neg 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 0 Collective Affinity Labels 0 Uncertainty Estimation Model Binary Partition Uncertianty of Negatives .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Node Features Push Pull Input Affinity Uncertainty Learning Weights of Negatives .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Neg Anchor Similar Dissimilar Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Overview of our approach AUGCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Left: AUGCL-based graph contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The objective and the general procedures are the same as existing GCL methods, but AUGCL leverages affinity uncertainty to learn anchor-instance-dependent hardness-based instance weights {wi1, wi2, · · · , wiN} for all negative instances to improve existing GCL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Right: The proposed affinity uncertainty learning approach to obtain the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' For an anchor �zi, AUGCL first obtains collective affinity information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='e, pairwise affinity across the instances) via binary partition of its negative instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' It then utilizes those affinity information to learn an uncertainty estimator that evaluates how confident the estimator is about the affinity of each negative instance �zj relative to the anchor instance �zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' A larger affinity uncertainty value uij indicates more likely of �zj being a hard negative, and thus, a larger weight wij (wij = αuij where α is a hyperparameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' different from the anchor, which are driving force for con- trastive learning to learn substantially enhanced discriminative representations, as shown in the literature empirically and theoretically [15], [16], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 2) Our Affinity Uncertainty-enabled Approach for Over- coming the Weaknesses: To address the negative sampling weaknesses discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' III-B1, we propose a novel framework for learning an Affinity Uncertainty-based hard- ness measure for enhancing current state-of-the-art Graph Contrastive Learning methods, termed AUGCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The key idea is to first learn the hardness of a negative instance relative to each anchor instance by comparing the affinity between them to the affinities of the anchor instance to the other instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The hardness results can then be plugged into a contrastive loss, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', InfoNCE, to improve the effectiveness of current GCL methods in utilizing the hard negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Overview of AUGCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Since the hardness of a negative instance varies largely w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' different anchor instances, our approach AUGCL aims to learn a hardness measure based on the relative affinity between the negative instance and each anchor instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' That is, for an anchor instance �zi and its negative instance candidate set � Zi = {�zj}N j=1, we learn a single hardness measure function φ(�zj|�zi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Θ) : � Zi → R that yields a hardness value for each �z ∈ � Zi relative to �zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Note that the function φ parameterized by Θ is trained across all anchor instances;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' yet the hardness it yields for the negative instance �zj is dependent on the anchor �zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' For brevity, φ(�zj|�zi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Θ) is denoted as φi(�zj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Θ) hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Unlike current hardness measures that define the hardness of a negative instance based on its individual relation to the anchor instance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', the similarity between them), one key novelty in AUGCL is that it defines the hardness based on two groups of pairwise affinities between the negative instances and the anchor instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' More specifically, we introduce the concept of affinity uncertainty below to achieve this goal: Definition 1 (Affinity Uncertainty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Given an anchor instance �zi and its negative instance candidate set � Zi = {�zj}N j=1, and let Ci 1 and Ci 2 be two disjoint groups of instances in � Zi such that: one group Ci 1 includes the instances that are closely aligned and distributed around the anchor �zi, while the other group Ci 2 contains the rest of other instances;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' and � Zi= Ci 1∪Ci 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Then the affinity uncertainty of each �z ∈ � Zi w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' �zi is defined as: φi(�z) = g(�z, Ci 1, Ci 2), (2) where g is an uncertainty estimator that evaluates how confi- dent the estimator is about the affinity of �z to the instances in the anchor instance-centered group Ci 1 compared to the other group Ci 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The affinity uncertainty in (2) takes a holistic approach that considers diverse affinities of the negative instances within and across the two groups Ci 1 and Ci 2 to learn an accurate hardness for each negative instance �z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' As shown in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', [36]) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 1, instances which are ambiguous to distinguish are assigned to large uncertainty values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' These instances typically have a poor affinity to both groups Ci 1 and Ci 2, such as those located on the boundary between the two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' By contrast, if the instances are coherently aligned within Ci 1 or Ci 2, their uncertainty would be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Thus, this type of uncertainty can be naturally used to define the hardness of the negative instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The obtained hardness can then be easily plugged into existing contrastive losses, such as the InfoNCE loss, via a weighting term for the negative instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Particularly, the JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 8, AUGUST 2021 5 AUGCL-enhanced InfoNCE is given as follows: ℓAUGCL(�zi, �zi) = − log e(h(�zi,�zi)/τ) e(h(�zi,�zi)/τ) + N � j,j̸=i wije(h(�zi,�zj)/τ) , (3) where wij = αφi(�zj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Θ) is the hardness-based weight added to �zj relative to �zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' φi(�zj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Θ) is the hardness learned by AUGCL for the negative instance �zj w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' the anchor instance �zi and α is a hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' This enables effective exploitation of the hard negatives, as large weights are expected for hard negatives while small weights are expected for the other instances, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', false negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The overall procedure of AUGCL is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' It follows the standard graph contrastive learning in the graph augmentation and contrastive learning except that we incorporate the affinity uncertainty-based hardness through a weighting term into the contrastive loss as in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The right panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 2 shows the steps of learning an anchor- dependent hardness measure φ for each anchor �zi, consisting of instance partition and uncertainty estimation as indicated in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Before introducing the details of these two components in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' III-C, below we demonstrate the theoretical motivation of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Theoretical Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' We show below that (3) is equiv- alent to a triplet loss with an adaptive margin exponentially proportional to the learned hardness-based weight φi(�zj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' This provides a more straightforward explanation of the work- ing mechanism of the proposed weighting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Let uij = φi(�zj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Θ) be the affinity uncertainty- based hardness of a negative instance �zj w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' the anchor in- stance �zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' When the projection function is an identity function and assumes the positive instance is more similar to the anchor than the negative instances, then minimizing the proposed objective in (3) is equivalent to minimizing a modified triplet loss with an adaptive margin mij = τ 2 log(αuij) , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', ℓAUGCL(�zi, �zi) ∝ 1 2τ N � j,j̸=i � ∥�z ′ i − �z ′ i∥ − ∥�z ′ i − �z ′ j∥ + mij � , (4) where �z ′ i is the normalized embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The proof for this theorem is detailed in the Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' From the theorem, we can see that the optimal embeddings to (4) should satisfy the following inequality: ∥�z ′ i − �z ′ i∥ ≪ ∥�z ′ i − �z ′ j∥ − mij, (5) where mij = τ 2 log(αuij) and uij = φi(�zj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Thus, mij is equivalent to a transformed affinity uncertainty-based hardness of the negative instance �zj relative to the anchor �zi, satisfying: � mij ≥ 0, if αuij ≥ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' mij < 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' (6) If �zj is a hard negative for �zi, the large uncertainty uij between �zi and �zj makes the inequality (5) hard to satisfy through mij > 0, enforcing better representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' On the contrary, if the uncertainty uij is small, (5) can be easily satisfied with mij ≪ 0, reducing the impact of the possible false negative instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Instantiation of AUGCL We introduce an instantiation of our AUGCL framework in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' As demonstrated in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 1, the affinity uncertainty-based hardness function φ parameterized with Θ can be decomposed into two modules, including a binary clustering function f : {�zj}N j=1 → {0, 1} parameterized by Θf and an uncertainty estimation function g : {�zj}N j=1 ×{0, 1} → R parameterized by Θg, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', Θ = {Θf, Θg}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' AUGCL is a generic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Different clustering and uncertainty estimation methods can be adopted in AUGCL to implement a specific model, as shown by our empirical results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' IV-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Below we describe the two modules of the best instantiated AUGCL model based on our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 1) Anchor-dependent Binary Partition of Negatives: Given an anchor �zi, binary clustering is used to partition the negative samples into two coherent groups – Ci 1 and Ci 2 – for subsequent affinity uncertainty estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Without having access to label information, clustering is often adopted on the full dataset to divide instances into several clusters [37]–[42], and instances from clusters other than the anchor-based cluster are directly treated as negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Our clustering differs from these existing methods from two main ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' First, we perform an anchor-dependent binary partition on only the negative instances in each batch of instances rather than the full dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Specifically, given a batch of node/graph embeddings {�zj}N j=1, for each anchor �zi ∈ {�zi}N i=1, we perform a binary partition on the nega- tive instance candidates {�zj}N j=1 using an existing clustering method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', k-means), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', fk−means : {�zj}N j=1 → {0, 1}, where Ci 1 = 1 is the label of the cluster centered around �zi and Ci 2 = 0 is the other cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' That is, the clustering assigns the cluster label Cij 1 = 1 if the instance �zj is sufficiently similar to �zi, and a different cluster label Cij 2 = 0 is assigned otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The second difference is that the obtained partitions are used to gain a sense of the affinity of an instance to the other instances, rather than being the direct negative sample clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Those affinity information would be used to evaluate the hardness of each negative instance through an uncertainty estimation model in AUGCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 2) Affinity Uncertainty Estimation: For an anchor �zi, the binary cluster labels {Cij 1|2}N j=1 carry the affinity semantics of the instances {�zj}N j=1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' the anchor instance �zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' We further propose to perform an uncertainty estimation upon these affinity semantic-based labels for each anchor �zi ∈ {�zi}N i=1, and use this uncertainty to measure the hardness of instances {�zj}N j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' By doing so, a large uncertainty-based hardness is assigned to fringe instances that locate around the boundary between the two clusters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' these instances are typically hard negatives w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' �zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' A small hardness is assigned otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Different uncertainty estimation methods can be used to specify this component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' We found that the recently proposed method Deep Gambler (DG) [36] worked best in our exper- iments, so DG is used in AUGCL by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Specifically, JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 8, AUGUST 2021 6 DG extends a multi-class classification task to a problem that learns an extra class to represent the uncertainty of instances, in addition to guaranteeing the classification of the original classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' For an anchor instance �zi, given its associated negative instance candidates { �zj}N j=1 and their affinity labels {Cij 1|2}N j=1, the DG-based uncertainty estimation is trained by minimizing the following loss: ℓDG i = − N � j log(pCij 1|2 ∗ o + uij), (7) where pCij 1|2 is the predicted class probability on class Cij 1|2 from a multi-layer perceptrons-based (MLP-based) DG model g(�z, Ci 1, Ci 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Θg) parameterized by Θg, uij is the uncertainty that the model g generates for the instance �zj, and o is a reward parameter with a larger o encouraging g to be more confident in inferring and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The final loss of DG is computed across all anchor instances �zi ∈ {�zi}N i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' After the DG model is trained, for each anchor �zi, we calculate uij for its negative instance �zj and obtain a uncer- tainty matrix U ∈ RN×(N−1) where each row ui contains the uncertainty of all negative instances w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' the anchor �zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' These uncertainty values are then used in (3) to improve the contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Time Complexity Analysis We take k-means and Deep Gambler [36] as the partition and uncertainty estimation methods, respectively, to analyze the additional time complexity introduced by AUGCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Specif- ically, let L be the number of MLP layers in DG and d be the number of hidden units for all layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' For the graph classification task, given a graph dataset with N graphs and the batch size is set as B, the time complexities of partition and the uncertainty modeling are O(2( N B )B2T) and O(KL( N B )B2d2) respectively, where T is the number of iterations for k-means and K is the number of training epochs for the uncertainty es- timation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' For the node classification task, given a graph with N nodes, in order to reduce the computation cost, we only sample M (M ≪ N) negatives for an anchor when training AUGCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The resulting time complexities of partition and training uncertainty model are O(2NMT) and O(KLNMd2) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' In experiments, we use the well-established k- means clustering implementation from scikit-learn [43], as it runs very fast in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Besides, the values of K, L, M and d are relatively small and the uncertainty estimation model is only trained once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Therefore, the computational overhead over the base model is not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' EXPERIMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Experimental Setup 1) Datasets: Seven commonly used graph classification datasets are used in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' They come from two popular application domains: bioinformatics (MUTAG, DD, NCI1, and PROTEINS) and social networks (COLLAB, REDDIT-BINARY, and IMDB-BINARY).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' For node classifi- cation task, we use three widely used datasets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', Wiki-CS [44], Amazon-Computers and Amazon-Photo [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Wiki-CS is a reference network constructed based on Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Amazon- Computers and Amazon-Photo are two co-purchase networks constructed from Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The statistics of the datasets are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' TABLE I KEY STATISTICS OF DATASETS USED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Task Dataset Graphs Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='Nodes Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='Edges Graph Classification NCI1 4,110 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='87 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='30 PROTEINS 1,113 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='06 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='82 DD 1,178 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='32 715.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='66 MUTAG 188 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='93 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='79 COLLAB 5,000 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='49 2,457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='78 RDT-B 2,000 429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='63 497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='75 IMDB-B 1,000 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='77 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='53 Task Dataset Graphs Nodes Edges Node Classification Wiki-CS 1 11,701 216,123 Aamazon-Computers 1 13,752 245,861 Aamazon-Photo 1 7,650 119,081 2) Implementation Details and Evaluation Protocol: For graph classification task, GraphCL [10], a recent SOTA InfoNCE-based contrastive learning method for graph classifi- cation, is used as our base, into which our affinity uncertainty- based hardness learning method is plugged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' For a fair compar- ison, the network backbone, the graph augmentation methods and the hyper-parameters of our AUGCL-enabled GraphCL are kept exactly the same as the original GraphCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' We follow a widely-used two-stage evaluation protocol in the literature [9], [10], [46], [47], in which we first learn graph representations in a self-supervised manner and then use the representations to train a downstream SVM classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The 10-fold evaluation is adopted in classification, and it is repeated five times with the mean accuracy (%) and standard variation reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' For node classification task, we adopt GCA [11] as the base model and plug our AUGCL-based affinity uncertainty hardness into it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The evaluation protocol for node classification follows DGI [7] where the model is first trained in an unsu- pervised manner and then the learned node representations are used to train and test a simple ℓ2-regularized logistic regression classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' On each dataset, the experiment is repeated for 20 runs with different data splits, and the average classification accuracy, together with the standard variation, is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' For graph and node classification, we use the same architec- ture in our affinity uncertainty estimation model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', a three- layer multi-layer-perceptrons (MLP) architecture, containing 128 units per layer with ReLU activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' We adopt the Stochastic Gradient Descent (SGD) optimizer for the uncer- tainty estimation model and the learning rate is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='01 across all the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The uncertainty scaling parameter α is set to the reciprocal of the mean of uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The training epoch number of the uncertainty estimation model is set to 10 for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' For the reward parameter in the uncertainty estimation model, it is selected through a grid search, and the search space is {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='7, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 3) Competing Methods: We evaluate the effectiveness of AUGCL in both graph and node classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' In both tasks, AUGCL is evaluated against three state-of-the- art hard negative mining-based contrastive learning methods, including DCL [17], HCL [16] and ProGCL [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' In addi- tion, we also include a set of other relevant state-of-the-art competing methods, including non-contrastive methods and JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 8, AUGUST 2021 7 TABLE II COMPARISON BETWEEN THE BASELINES AND THEIR AUGCL-ENABLED COUNTERPARTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' THE BASELINES ARE GRAPHCL [10] AND GCA [11] FOR GRAPH AND NODE CLASSIFICATION TASKS, RESPECTIVELY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' “↑” REFERS TO THE IMPROVEMENT COMPARED TO THE BASELINES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Task Dataset GraphCL GraphCLAUGCL Graph Classification NCI1 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='26 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='16(↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='90) PROTEINS 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='36 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='76(↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='40) DD 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='01 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='14 (↑ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='13) MUTAG 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='15 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='20 (↑ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='05) COLLAB 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='53 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='10 (↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='57) RDT-B 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='09 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='19 (↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='10) IMDB-B 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='20 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='46 (↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='26) Task Dataset GCA GCAAUGCL Node Classification Wiki-CS 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='08 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='59 (↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='51) Amazon-Computers 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='80 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='94 (↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='14) Amazon-Photo 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='99 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='43 (↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='44) other contrastive methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Particularly, for graph classification, the non-contrastive methods include Graphlet Kernel (GK) [48], Weisfeiler-Lehman Sub-tree Kernel (WL) [49], Deep Graph Kernels (DGK) [46], node2vec [50], sub2vec [51] and graph2vec [47], while the GCL methods include InfoGraph [9], JOAOv2 [12], SimGRACE [13] and GraphCL [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' For the node classification task, non-contrastive methods include node2vec [50], DeepWalk (DW) [52], and Graph AutoEncoders (GAE and VGAE) [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Contrastive methods include DGI [7], GMI [54], MVGRL [8], and GCA [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Note that ProGCL proposed two strategies to utilize the estimated hardness results, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', weighting and mixup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The results reported are based on the weighting strategy of ProGCL to have a direct comparison to our weighting-based AUGCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Enabling Different GCL Methods on Graph and Node Classification 1) Performance Improvement over Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' : We first compare the performance of our proposed method with the baselines on graph and node classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The results are shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' It is clear that, by incorporating our affinity uncertainty-based hardness measure, the two baselines – GraphCL [10] and GCA [11] – are substantially and consis- tently boosted on all datasets from different domains for both graph and node classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' This demonstrates that our method AUGCL can enable these baselines to effectively at- tend to hard negative instances and learn better representations of graphs/nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 2) Comparison to State-of-the-art Methods: We then com- pare AUGCL to diverse advanced graph embedding learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Graph Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The results on graph classification are reported in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' We can observe that graph contrastive learning methods generally obtain better performance than non-contrastive methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Our method further improves the performance by learning and feeding the affinity uncertainty- based hardness into the contrastive learning, substantially outperforming SOTA GCL methods on 6 out of 7 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Compared to the three recent hardness-aware methods DCL, HCL and ProGCL, our method AUGCL performs much better across all seven datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Particularly, DCL, HCL and ProGCL improve GraphCL on some datasets such as PROTEINS, MUTAG, and IMDB-B, but they fail on the other ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' By contrast, our method improves over GraphCL by a large margin across all the seven datasets, indicating the superiority of our affinity uncertainty-based hardness learning method over its recent counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Node Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The node classification results are reported in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' In general, the trends here are similar to the results in Table III: i) contrastive methods are generally more effective than the non-contrastive ones, and ii) the competing hardness-aware methods DCL, HCL and ProGCL further improve over the contrastive methods on part of the datasets, while our method AUGCL achieves consistently better performance on all the three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' General hardness-aware methods DCL and HCL identify hard negatives by using the individual similarity to the anchor, which is often ineffective on graph data due to over-smoothed node representation issues, as also found in the very recent ProGCL work [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' ProGCL addresses this issue by positing a prior model over the pairwise similarity distribution to learn the hardness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Our method further improves ProGCL consistently on the three datasets by learning a data-driven affinity uncertainty estimation model without the prior as- sumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Importantly, ProGCL is not generalizable to other graph mining tasks such as the graph classification task, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', ProGCL fails to work as effectively as the baseline GraphCL on some datasets in Table III where our method AUGCL also consistently outperforms the baseline, indicating better applicability and flexibility of AUGCL on different graph mining tasks than ProGCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Improving Robustness against Graph Adversarial Attacks Self-supervised learning has shown effective performance in defending against adversarial perturbations [56], [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' This subsection investigates whether AUGCL can further improve over the GCL methods on this important property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' In this experiment, following [58], three different types of graph adversarial attacks: RandSampling, GradArgmax and RL-S2V are used, where RandSampling randomly adds or deletes edges from graphs, GradArgmax performs edge modification based on gradient information, and RL-S2V is a reinforcement learning based attack method that learns a generalizable attack policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' We also use the widely-used evaluation protocol as in [58] where the graph task is to classify the component numbers in synthetic graphs and structure2vec [55] is adopted as the graph encoder, with different depths of structure2vec considered in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Both the original structure2vec trained from scratch (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', no pre-training) and the pre-trained structure2vec [55] using GraphCL [10] are used as base- lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The experimental results of our method are obtained by incorporating our affinity uncertainty-based hardness into GraphCL to pre-train the structure2vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The best-competing method ProGCL is adopted in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The results are reported in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' From the table, we can observe that: i) all three GCL methods GraphCL, ProGCL, and AUGCL can largely improve the robustness against all three graph adversarial attacks, JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 8, AUGUST 2021 8 TABLE III RESULTS OF GRAPH CLASSIFICATION ACCURACY (%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' WE OBTAIN THE RESULTS OF GRAPHCL AND ITS FOUR HARDNESS-AWARE VARIANTS UNDER THE SAME EXPERIMENTAL SETTING AS [10], FROM WHICH THE RESULTS OF THE OTHER METHODS ARE TAKEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' THE RESULTS OF JOAOV2 [12] AND SIMGRACE [13] ARE TAKEN FROM THEIR OWN PAPER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Type Method NCI1 PROTEINS DD MUTAG COLLAB RDT-B IMDB-B Non-Contrastive Methods GK 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='66±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='11 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='34±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='18 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='87±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='98 WL 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='50 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='56 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='72±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='00 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='82±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='41 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='30±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='44 DGK 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='31±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='46 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='30±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='82 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='44±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='72 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='04±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='39 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='96±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='56 node2vec 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='89±1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='19±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='44 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='46±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='80 TABLE IV RESULTS OF NODE CLASSIFICATION ACCURACY (%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' WE OBTAIN THE RESULTS OF GCA AND ITS FOUR HARDNESS-AWARE VARIANTS UNDER THE SAME EXPERIMENTAL SETTING AS [11], FROM WHICH THE RESULTS OF THE OTHER METHODS ARE TAKEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Type Method Wiki-CS Amazon- Computers Amazon- Photo Non-Contrastive Methods Raw feature 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='98±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='00 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='81±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='00 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='53±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='00 node2vec 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='79±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='05 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='08 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='67±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='12 DW 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='35±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='06 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='68±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='06 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='44±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='11 DW+feature 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='21±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='03 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='28±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='07 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='08 GAE 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='01 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='27±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='19 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='62±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='13 VGAE 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='63±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='19 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='37±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='21 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='11 Contrastive Methods DGI 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='35±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='14 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='95±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='47 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='61±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='22 GMI 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='85±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='08 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='21±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='31 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='68±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='17 MVGRL 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='08 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='11 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='74±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='07 GCA 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='08±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='58 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='80±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='42 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='99±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='39 Hardness-aware Methods GCADCL 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='60 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='79±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='48 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='29±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='32 GCAHCL 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='19±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='64 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='64±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='34 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='79±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='29 GCAProGCL 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='33±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='64 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='68±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='35 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='29 GCAAUGCL (Ours) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='59±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='56 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='94±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='44 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='43±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='32 particularly the more advanced attacks GradArgmax and RL- S2V, on different network layers, compared with the original model, ii) the robustness can be further improved by exploiting hard negative mining techniques used in ProGCL and AUGCL, compared to GraphCL, and iii) compared with ProGCL, the better hard negative mining in our method AUGCL generally results in more remarkably and stably improved robustness over the GraphCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Overall, the proposed method AUGCL in- creases the classification accuracy by up to 8% over GraphCL and up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='7% over ProGCL, and performs very competitively to the two methods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='2%-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='8% difference) in the limited cases where AUGCL is not the best performer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Ablation Studies This subsection evaluates the impact of using different clustering and uncertainty estimation methods in f and g, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The GraphCLAUGCL method is used, with GraphCL as the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Partition Methods in f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' An important module of our proposed method is the instance-wise partition function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' k-means is used by default to implement f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Here we also examine the use of spectral clustering [59] to perform the binary partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The results are shown in Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' We can see that AUGCL using either spectral clustering or k-means achieves similar improvement over GraphCL, suggesting the stability of our method w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' the generation of the partition labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' AUGCL with k-means clustering performs consistently better than the spectral clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Hence, k-means clustering is used by default in our experiments and recommended in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Uncertainty Estimation Methods in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The uncertainty estimation method g is another important module in AUGCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' In addition to the extra class-based method used by default in AUGCL, two alternative approaches are used, including the maximum prediction probability-based method Softmax- Response [29] and the entropy-based method Predictive En- tropy [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' We also include a distance-based method as another simplified variant of AUGCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The detailed descriptions of these uncertainty estimation methods are presented in Ap- pendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The results are reported in Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' It is clear that re- gardless of the specific uncertainty estimation method used, all variants of AUGCL can generally improve the baseline GraphCL on nearly all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' This provides further ev- idence for the effectiveness of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Additionally, the uncertainty estimation method matters: the default method (a recently proposed extra class-based method [34], [36]), a more effective uncertainty estimation model than the other three methods, shows consistently better performance than the other three variants, implying that the hardness can be better captured by more advanced uncertainty estimation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Hyperparameter Analysis We examine the sensitivity of AUGCL w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='t two key hy- perparameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', the uncertainty parameter α in (3) and the reward parameter o in φ (particularly in (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Without loss of generality, one graph dataset from biochemical molecules and social networks respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='e, PROTEINS and IMDB-B, are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Uncertainty Parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' α is adaptively set, depending on the uncertainty matrix U, to enable stable performance of AUGCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Particularly, given U, we can calculate the mean µ and standard deviation δ of U, based on which α is JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 8, AUGUST 2021 9 TABLE V CLASSIFICATION ACCURACY UNDER THREE EVASION ATTACKS ON THREE DIFFERENT LAYERS OF STRUCTURE2VEC [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' PROGCL AND AUGCL BELOW REPRESENT GRAPHCL METHODS WITH THEIR RESPECTIVE HARD NEGATIVE MINING COMPONENT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Attacks 2-Layer 3-Layer 4-Layer Original GraphCL ProGCL AUGCL Original GraphCL ProGCL AUGCL Original GraphCL ProGCL AUGCL Unattack 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='20 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='73 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='13 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='20 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='20 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='33 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='67 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='87 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='87 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='00 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='47 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='20 RandSampling 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='73 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='68 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='47 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='67 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='27 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='60 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='93 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='67 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='13 97.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='47 RL-S2V 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='93 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='20 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='13 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='47 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='93 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='66 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='07 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='73 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='20 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='86 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='67 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='33 TABLE VI ABLATION STUDY RESULTS OF GRAPHCLAUGCL USING DIFFERENT CLUSTERING METHODS f OR UNCERTAINTY ESTIMATION METHODS g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Method f g NCI1 PROTEINS DD MUTAG COLLAB RDT-B IMDB-B GraphCL (Baseline) × × 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='26±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='39 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='36±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='44 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='01±0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='72 GraphCLAUGCL Spectral ExtraClass 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='79±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='52 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='57±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='79 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='44 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='86±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='22 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='74±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='88 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='56 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='86±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='47 k-means ExtraClass 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='34 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='76±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='39 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='14±0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='46±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='80 k-means Distance 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='88±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='26 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='48±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='06 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='22±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='25 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='64±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='83 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='27±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='43 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='41±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='51 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='55 k-means Softmax 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='42±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='43 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='34±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='57 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='65 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='81±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='65 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='89±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='92 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='38±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='33 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='72±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='57 k-means Entropy 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='98±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='50 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='13±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='39 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='58±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='49 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='98±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='78 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='71±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='37 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='37±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='50 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='76±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='57 set to α = 1 µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' We vary the parameter α in the range of { 1 µ−δ, 1 µ−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='5δ, 1 µ, 1 µ+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='5δ, 1 µ+δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The mean classification accu- racy (%) under different α are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 3(a) where the labels in the x-axis denote the coefficient of δ when calculating α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' It is clear that the performance of our model is generally stable with varying α, and α = 1 µ is a recommended setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Reward Parameter o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' We further examine the reward parameter o in the uncertainty estimation model [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' With o varying in {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='7, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='9}, we report the mean clas- sification accuracy (%) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The results also show that AUGCL can achieve reasonably stable performance for a wide range of the o settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='0 70 72 74 76 Accuracy PROTEINS IMDB-B (a) Parameter α 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='9 70 72 74 76 Accuracy PROTEINS IMDB-B (b) Parameter o Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Sensitivity analysis of hyperparameters α and o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' CONCLUSION This paper proposes the idea of affinity uncertainty and utilizes it to measure the hardness of negative samples to improve popular GCL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' To this end, we introduce the affinity uncertainty-based hardness learning approach AUGCL that synthesizes binary partition and uncertainty estimation to learn anchor-instance-dependent hardness for all negative instances, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', their hardness results are relative to each anchor instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' AUGCL is a data-driven approach that eliminates the prior assumption made in very recent hardness-aware GCL methods like ProGCL [18], resulting in better applicability and flexibility on different graph mining tasks, as well as better robustness to diverse graph adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' It also shows better performance in enabling different GCL loss functions, compared to a wide range of other state-of-the-art graph representation methods on graph and node classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' We also show theoretically that the resulting contrastive loss in AUGCL is equivalent to a triplet loss with an adaptive margin that adaptively exploits the hard negatives with a large margin, with a small margin assigned to the other negative instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' REFERENCES [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Xu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Hu, J.' metadata={'source': 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Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 14, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' 8, AUGUST 2021 11 APPENDIX A DETAILS OF OTHER UNCERTAINTY ESTIMATION METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Softmax-Response Given the softmax predicted vector p = {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' , pC} (C is the number of classes), the uncertainty estimated based on Softmax-Response [29] is calculated as follows: u = 1 − max c=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=',C pc (8) The uncertainty is large when the predicted probabilities are more equally distributed, and it is small when predicted probabilities are concentrated on one specific class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Predictive Entropy Predictive entropy [30] captures the average amount of information presented in the predicted probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' The uncertainty based on predictive entropy is obtained by: u = − C � c=1 pc log pc (9) Similar to Softmax-Response, the predictive entropy reaches its maximum value when all predicted classes are equiprobable and its minimum is attained when there exists one class with probability one and the probabilities of all other classes are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Distance-Based Uncertainty For each anchor instance, after obtaining the binary partition labels of the negative instances {Ci,j 1|2}N j=1, we calculate the uncertainty of the negative instance whose cluster labels are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Specially, the distance dij between the anchor instance and the negative instance �zj (Cij 2 = 0) is first calculated and then the uncertainty of �zj is set as the reciprocal of dij weighted by a parameter β i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=', uij = β 1 dij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' (10) Note that for �zj with Cij 2 = 0, if �zj is close to the anchor �zi, the uncertainty value uij is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' APPENDIX B PROOF OF THEOREM 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' ℓAUGCL(�zi, �zi) = − log exp(h(�zi, �zi)/τ) exp(h(�zi, �zi)/τ) + N � j,j̸=i αuij exp(h(�zi, �zj)/τ) = log � �1 + N � j,j̸=i αuij exp �h(�zi, �zj) − h(�zi, �zi) τ �� � = log � �1 + N � j,j̸=i exp �h(�zi, �zj) − h(�zi, �zi) + 2mij τ �� � (11) where h(�zi, �zj) = �zT i �zj ∥�zi∥∥�zj∥ (−1 ⩽ h(�zi, �zj) ⩽ 1) and mij = τ 2 log(αuij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Let �z ′ i = �zi ∥�zi∥, h(�zi, �zj) can be rewrited as h(�zi, �zj) = (�z ′ i)T �z ′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Since the positive instance is more similar than the negative instances to the anchor, the value of h(�zi, �zj) − h(�zi, �zi) tends to be −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' Based on the above analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' we can apply the Taylor expansion of first order and get the following approximation: ℓAUGCL(�zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' �zi) = log � �1 + N � j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='j̸=i exp �h(�zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' �zj) − h(�zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' �zi) + 2mij τ �� � ≈ N � j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='j̸=i exp �h(�zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' �zj) − h(�zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' �zi) + 2mij τ � ≈ e− 2 τ � �(N − 1) + 1 τ N � j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='j̸=i (h(�zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' �zj) − h(�zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' �zi) + 2mij + 2) � � ∝ 1 τ N � j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='j̸=i (h(�zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' �zj) − h(�zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content=' �zi) + 2mij) = 1 2τ N � j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} +page_content='j̸=i � ∥�z ′ i − �z ′ i∥ − ∥�z ′ i − �z ′ j∥ + mij � (12) which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFQT4oBgHgl3EQffjaD/content/2301.13340v1.pdf'} diff --git a/adAzT4oBgHgl3EQf2v4D/vector_store/index.faiss b/adAzT4oBgHgl3EQf2v4D/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..c71e2bcf1a388ac63db1c4de84b8d55047d76ddd --- /dev/null +++ b/adAzT4oBgHgl3EQf2v4D/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:75d1cdc8147728d324886faba74db0442baefe8dee0055929cac2f8178b2eb01 +size 4128813 diff --git a/cdE0T4oBgHgl3EQfnwHq/vector_store/index.faiss b/cdE0T4oBgHgl3EQfnwHq/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..f106d57cd3edfb4ef02123f968d9f6ba50af02dc --- /dev/null +++ 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The +effective viscosity (µeff) of MC beneath MORs is recognized as the crucial factor in modulating +their axial high versus flat topography. Based on a two-step viscosity calculation (suspension +and solid-melt mixture rheology), we provide a theoretical estimate of µeff as a function of +melt suspension characteristics (crystal content, polymodality, polydispersity and strain-rate), +and its volume fraction in the MC region. We then develop a numerical model to show the +control of µeff on the axial topography. Using an enthalpy-porosity-based fluid-formulation +of uppermost mantle the model implements a one-way fluid-structure interaction (FSI) that +transmits viscous forces of the MC region to the overlying upper crust. The limiting non-rifted +topographic elevations (-0.06 km to 1.27 km) of model MORs are found to occur in the viscosity +range: µeff = 1012 to 1014 Pa s. The higher-end (1013 to 1014) Pa s of this spectrum produce +axial highs, which are replaced by flat or slightly negative topography as µeff ≤ 5 × 1012 Pa s. +We discuss a number of major natural MORs to validate the model findings. +1 +Introduction +Many mid-ocean ridges (MORs) evolve with complex 3D axial topography, which is hard to explain +with standard tectonic models. +Their spatially varying axial topography, such as high, flat or +valley, is generally attributed to the spreading rate [106, 110], the magma availability [59, 74, 108], +in a particular ridge-segment, and upper crustal faulting [12]. +However, these contrasting axial +morphologies are often found in MORs, e.g., South-East Indian Ridge (SEIR), where the spreading +rate shows practically no variations [16], and ultra-slow South-West Indian Ridges (SWIR) displaying +typical axial valley topography, where they have large magma availability [55]. +A direction of +MOR studies explains the rift morphology as a product of the two competing processes- tectonic +and magmatic, conceived as horizontal spreading and dike opening, respectively [12, 71]. A non- +dimensional parameter, called the M factor (a ratio of the dike intrusion to plate-spreading driven +widening rates), has been used to reproduce the axial structures in numerical models. M = 1, i.e., +a condition of dike intrusion rate to completely balance with the plate spreading rate, gives rise to +an axial high, whose height depends on the magma density. In contrast, M < 1, i.e., a condition +of less effective diking than the spreading rate, yields faulted axial valley [12, 54]. Some studies +have shown the axial morphology as a function of the extension rate and inherent short-wavelength +seafloor heterogeneities (e.g., [110]). However, their interpretation faces disagreement with the Mid- +Atlantic ridge model, which proposes the magma supply as a critical factor in determining the axial +morphology [71]. Although these models well integrate the axial morphological spectrum by a single +factor- M, the modelling approach does not account for sub-crustal melt processes. It is noteworthy +that many recent MOR studies demonstrated how the latter could significantly control the MOR +evolution [16, 78], albeit a comprehensive model is still unavailable. Our present article aims to +bridge this gap, treating the axial morphology in the thermo-mechanical framework of an ideal +three-dimensional melt upwelling system, where divergence force components act along and across +the ridge axis. This modelling approach allows us to investigate the extent of magmatic control on +3D axial morphology. +∗senjoyjeet@gmail.com +†shamiksrakar@gmail.com +‡nibir.mandal@jadavpuruniversity.in +1 +arXiv:2301.04979v1 [physics.flu-dyn] 12 Jan 2023 + +While emphasizing magmatic roots, several workers considered magma buoyancy as the principal +factor to elucidate the origin of axial-high topography [11, 121]. [31] provided a condition of the +sub-ridge viscosity distribution required for buoyancy-driven axial high topography. On the other +hand, [82] predicted that mantle viscosities beneath the ridge must be at least two orders higher than +the generally accepted values to form an axial valley. [109] also indicated viscosity as the key factor, +but it is ultimately the plate velocity to regulate the sub-crustal density or viscosity that determine +the axial morphology. Here, the most critical question is – how the plate velocity regulates the +sub-crustal viscosity? [24] showed from a 2D numerical model that the mantle viscosity at shallow +depths (< 20 km) beneath the ridge should be low (∼ 1018 Pa s) to form axial high topography, but +it should be high enough (∼ 1021 Pa s) to form a low axial relief. According to their model, high- +viscosity melt flows lower the hydrostatic pressure beneath the ridge, reducing the melt upwelling +height. +However, none of these studies explicitly accounts for the viscosity effect of sub-crustal +melt-rich zones on the axial morphology. +-2000 +-2400 +-2600 +-2800 +-2200 +-3000 +B* +B +A +A* +EPR +V.E. = 2 +C +C* +SEIR +V.E. = 1.5 +B +B* +JDF +V.E. = 2 +-2300 +-2400 +-2500 +-2600 +-2700 +-2900 +-3000 +-2800 +A +A* +C +C* +-2500 +-2300 +-2400 +-2700 +-2800 +-2900 +-2600 +25 km +25 km +25 km +Depth (m) +Depth (m) +Depth (m) +(a) +(b) +} +Suspension rheology ++ +1-2 km +0.1-1 km +magma conduit +melt/magma bodies + +Magma conduit/melt/magma bodies +~ +; Mantle ~ +1 +9 +19 +10 - 10 Pa s +10 Pa s + +Viscosity Values: +Melt flow +Oceanic upper crust +Mantle +Mush complex (MC) +MOR axis +10 km +20 km +Decompression melting stops at this level +Figure 1: (a) Bathymetric profiles across the East Pacific Rise (EPR), Juan De Fuca (JDF) and +South-Eastern Indian Ridge (SEIR). They show high (EPR-AA*), moderately high (JDF-BB*) +and plateau dominated (SEIR-CC*) ridge-axis topography, respectively. +Data source: GeoMa- +pApp (http://www.geomapapp.org/)/CC BY. (b) A conceptual cartoon diagram of the sub-ridge +melt/magma settings considered for the topographic modelling in this study. +The magma scale +(Table1) covers narrow melt conduits and magma bodies containing suspended crystals. The mush +complex (MC) represents a distinct zone consisting of melt bodies and conduits within a high- +viscosity host rock matrix. +The problem of sub-crustal melt transport mechanisms has recently rejuvenated the MOR re- +search in new directions [14, 15, 32, 112]. It is now evident that melts start to localize in discrete +zones during their ascent that eventually mediates for a heterogeneous magma supply to the ridge +axes. Earlier numerical models [74, 99] showed melt fraction as a function of spreading rates, sug- +gesting that the melt fraction is substantially reduced from fast- to slow-spreading ridges. Secondly, +the melt upwelling processes participate in solidification at the shallow level to form isolated mushy +bodies, as reported by many earlier workers [107,108,111]. The crystal content in the mushy melts +can largely vary depending on the degree of crystallization, and their varying relative volume ra- +tios would determine the viscosity of the melt-bearing sub-ridge regions. [9] provided a depth-wise +viscosity profile based on melt content (∼ 3%), dehydration, and grain boundary sliding. +This +model predicts an increase in overall viscosity with height, mainly due to water extraction during +partial melting. However, later experimental studies suggested that such dehydration can hardly +affect viscosity at shallower depths as partial melting generally ceases to occur at a deeper level [52]. +The olivine rich high-fluid channels in subcrustal magma mush at a shallower depth [58] indicates +crystal transport as suspension. A detailed viscosity analysis of the sub-crustal regions containing +2 + +crystal-bearing melts beneath MORs, especially in view of the axial morphology, is yet to be fully +explored. +This article introduces a novel approach to model sub-crustal/lower-crustal (hereafter sub-crustal) +viscosity and offers a viscosity-based explanation for the axial morphologies: highs and flat topogra- +phy of MORs, e.g., East Pacific Rise, Juan du Fuca, and South East Indian Ridge (Figure 1a).In the +first step, we provide a series of systematic calculations of the effective viscosity of mush complex +(MC) beneath the ridge axes. The mush complex (MC) is defined here as a constitution of crystal- +bearing-melts (with largely varying crystal contents) and host rocks [112]. Our calculations consider +the following parameters: the process-times, spatial magnitudes, and the constitution of sub-crustal +materials (see the concept diagram, Figure 1b and Section 2). We then develop a three-dimensional +fluid-structure interaction (FSI) approach to model the mechanical connection between the MC and +the overlying crust at a mid-oceanic ridge. The FSI model allows us to investigate how the viscosity +of the sub-crustal mushy region can modulate the flat versus high MOR axial topography. +2 +Sub-crustal mush complexes: viscosity modelling +2.1 +Mush complex in MOR settings +At mid-oceanic ridges, the ascending melts produced by decompression melting − at a depth of +around 40 km − focus to the ridge axis, forming large (∼ 30 km wide) melt-rich regions [80]. Seismic +imaging and theoretical estimates indicate a wide variation in their partial melt content (∼ 10 - 70% +at shallower levels, ≤ 10 km and 5 – 25% at deeper levels, ∼ 30 km) from one ridge to the other or +different segments within the same ridge [7,51,80,100]. It is vital to assess how such variations in +melt content can influence the mechanical strength of sub-crustal melt-rich regions (MC) at shallow +depths and modulate the first-order ridge-axis topography. In submarine systems, the temperature +calculated for the critical depth of partial melting cessation constrains the amount of available melt +in the subcrustal MC system [80]. However, the melts ascend upward with a complex 3D pattern +of their paths, determined by coupled convection-solidification processes [74, 99, 122]. The volume +fraction of melt-crystal aggregates goes up [43,51]as subcrustal magma bodies form at mid-oceanic +ridges. The plot shows a linear regression of the average melt fractions with depth (Figure 2a). +There can be large variations from the linear average at subcrustal regions due to significant spatial +variations in the magma pool populations and their fractional crystallization beneath MORs(Figure +2b). +Geophysical signatures, such as lower seismic velocities and high attenuation suggest the oc- +currence of mushy zones beneath mid-ocean ridges [1, 121], containing suspension-rich melt bodies +(super solidus) as well as subsolidus host rocks [112]. +Based on such sub-ridge mush-melt pat- +terns reported in the literature [49, 59, 68, 106] [80] [79], we consider a mechanically distinct zone, +mush complex (MC), treated as a continuum to implement the dynamic and kinematic coupling +between the underlying mantle and the overlying elastic crust [15,122]. It is now a well-established +fact that ascending mushy melts encounter the lithospheric base that acts as a melt barrier and +forces the melts to focus into the ridge axis, forming a distinct melt-rich regime within the host +rocks [49, 59, 68, 106]. From gravity anomaly data, Lin and Parmentier [68] detected a low-density +zone at the base of the lithosphere at EPR. Mckenzie and Bickle [80] discussed the occurrence of +underlying hot sheets at the decoupling zone between the circulating mantle and the spreading +plates. On the other hand, many geophysical studies found sub-crustal melt lenses, 1-2 km wide and +100 m thick, containing 30-40% crystals as suspension, in several ridges. The lenses are extended +horizontally up to 15-20 km with their melt content decreasing to 30%, forming spatially extensive +mushy regions [13,107,113]. Singh et al. [107] recognized 3-4 km thick axial magma chambers at a +depth of 3 km in slow-spreading, magma-rich Lucky strike. Fast and intermediate spreading ridges +are reported to have magma bodies at shallower depths, ∼ 2.4 km in JdF [13] and ∼ 1.8 km in +EPR [29], where their maximum thickness is ∼ 4 to 6 km [29,60]. In contrast, slow ridges generally +lack such distinct magma bodies, but have mushy regions (low-velocity zones) at the crustal base. +Dunn et el. [30] reported a 6 km thick mushy zone of sparse melt channels at a depth of 4 km at MAR +(35◦N). Besides melt pockets, which are prevalent in fast-spreading ridges, discrete melt channels +in the lower crust and sub-crustal axial zones [13,30] also constitute a typical feature (low-velocity +mushy regions) at the crustal base, which is also considered as a part of the MC [108]. +Based on the available reports on axial melt lenses and melt-rich bodies beneath mid-ocean +ridges, the vertical extent of the mush complex (MC) was chosen in the present model in sub-crustal +regions and in the uppermost mantle. The MC was allowed to evolve with progressively deforming +overlying elastic upper crust under basal stresses that eventually decreased the axial depth and +3 + +Log strain rate (ε) +. +Log suspension viscosity +Melt with crystals +Melt +(b) +(a) +Figure 2: (a) Variation of melt fractions with depth (plots based on available data in literature). +Except Hewitt [51] and McKenzie and Bickle [80], all the data are taken from axial melt lenses, +magma chambers and melt conduits. In the plot, these data points are complemented with rocks +(1% melt, [112]), marked in deep blue. Red straight line shows the overall regression trend. Shaded +area delineates the depth range (2-8 km) of evolved MMC, where the average melt suspension fraction +in MC is 0.3 - 0.4. (b) The plot shows the variation of suspension viscosity as a function of strain +rate and characteristics of the melt suspension (modified from [43]). +increased the MC thickness. The MC is modelled with a triangular cross-section, describing an +along-axis prismatic area in the lower crust, with a maximum thickness of 4 km beneath the ridge +axis (Figure 1b). The upper crust (i.e., solid elastic crust) at the axis is chosen 4 km thick in the +initial model setting, where the MC vertically covers the lower crust and a part of the topmost +mantle region (detailed in, Figure 5). The reason for choosing a larger MC depth, as compared to +the available data in our initial model is that the solid crust progressively thins by elastic strains +during the simulation. For example, the initial crustal thickness at the axis is reduced by 2 km to +finally set the MC depth and thickness at 2 km and 6 km, respectively [108]. +2.2 +Melt-viscosity modelling: parametric considerations +The viscosity of melts and magmas at shallow depths is historically modelled within a framework of +suspension rheology, following the landmark work of [34]. However, natural suspensions show viscous +behaviour more complex than that predicted from Einstein’s theory. +The complexity originates +primarily from the effects of additional factors, such as packing and shapes of solid particles in +suspensions. The packing of solid components in the liquid phase is an influential factor to modify +the effective viscosity of a mixture under the same solid volume fraction [64]. The packing vis-a-vis +viscosity, depends significantly also on the solid particle size distribution in the liquid. For example, +a bimodal size distribution with increasing size ratios up to a threshold point lowers the effective +viscosity [25]. Chang and Powel [20] showed that the viscosity of a suspension decreases initially with +increasing smaller particle volume fraction and then increases monotonously after reaching a critical +volume fraction. Liquid suspensions attain their maximum packing fraction in the case of multimodal +size distributions. Such multimodal (trimodal and tetramodal) particle packing increases viscosity +higher than those for bimodal and unimodal distributions [63]. This study also suggests that packing +with different particle sizes yields a polydispersion effect on the bulk viscosity of suspensions for the +same particle volume fraction. Polydispersity allows smaller particles to pack more closely by forming +layers between larger particles or by occupying the void spaces between larger particles [28]. Such +4 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Depth from seafloor (km) +.15 +-20 +●Hewitt(2010) +-25 +Arnulfetal.(2018) +Mainprice（1997) +Xuetal.(2014) +McKenzie and Bickle (1988) +Gonnerman andManga (2007) +-30 +Rock +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Melt fractionpolymodal and polydisperse particle (heterogeneous packing) distributions can thus significantly +enhance the bulk viscosity of melt suspensions in a mushy region. +Petrological and compositional data indicate mineral phases, e.g., olivine, plagioclase and clinopy- +roxene can crystallize in partially molten zones at successive stages during the melt ascent. However, +their depth correlation becomes weak with decreasing spreading rate [62,67]. Volcanic studies sug- +gest that the polydispersity characteristics of mushy melts and their variations hold a connection +with the spreading rates at MORs. Mushy melts can strongly differ in their crystal and bubble +contents depending upon the spreading rates [32, 57]. Slow-spreading ridges generally show larger +compositional variations in erupting lavas than the fast-spreading ridges. Higher degrees of composi- +tional homogenization in the fast-spreading ridges are commonly attributed to the magma chamber +processes [88], in contrast to slow-spreading ridges, e.g., the Mid Atlantic Ridge (MAR), which +are devoid of large, stable magma chambers and involves fractional melting throughout the whole +melting regime to produce melts with a strong compositional variability [84]. It is noteworthy that +magma chamber processes act as potential sites for hot mafic magma replenishment into the cold +silicic resident magmas (Figures 3a and 3b). Several workers have reported mafic enclaves from +volcanic rocks as an evidence of the magma replenishing process (Figure 3a) [26,77,83]. +Cold resident magma +Hot repleneshing magma +Solid crystal +t� +t� +T (°C) +T (°C) +T (°C) +T (°C) +t� +t� +(a) +(b) +Figure 3: A cartoon presentation of two different replenishment dynamics in the subcrustal magma- +chambers (adapted from [77]). (a) Volumetrically small and slow magma replenishment. In this +case a thin layer of groundmass crystallization forms, and the thermal gradient between the hot +emplaced magma and the host mafic magma is dissipated and reduced. (b) Fast and large magma +replenishment. In this case, the hot magma disperses into a magma chamber like fountains before +crystallizing to form enclaves. Moreover, a large volume of replenishing magma causes a thicker +hot layer, which sustains the thermal gradient for a longer period with convective churning and +multimodal crystal enclaves. +The effects of replenishment dynamics on magma chamber processes depend mainly on the +magma flow conditions. Turbulent fountain-type injections produce dispersed spherical enclaves of +crystals, whereas a non-turbulent slow influx of hot magma (Re ≤ 400) (Figure 3) results in ponding +of groundmass crystals at the magma-chamber base [26,77]. Martin et al. [77] also suggested that +the more replenishing magma volume flux, the larger olivine phenocrysts would form, resulting in +5 + +convection-driven churning within the magma chamber due to a longstanding steeper temperature +gradient (Figure 3b). However, recent geochemical studies focus on the general mushy environment +as a host of crystallization and magma mixing [2]. Geochemical investigations of crystal-bearing +enclaves within erupted lavas in volcanic settings allow us to recognize several factors controlling +shallow magmatic environments beneath mid-oceanic ridges: a) occurrence of magma chamber, b) +melt-upwelling velocity, and c) volume in the mushy regions. +These factors determine whether +magma-dominated, volatile environments can significantly influence the dispersion and multimodal- +ity characteristics of solid suspensions in hot magmas. In contrast, magma-poor mushy regions, +consisting of small and unstable non-convecting magma chambers, can form dominantly monodis- +persed microlith ponds. +Petrogenetic analysis of a recent East Pacific Rise (EPR) eruption suggests the presence of sub- +ridge crystal-bearing mushy regions, which are thought to be a product of consistent replenishment +(Figures 3a and 3b) by evolved magmas from a deeper level source [44]. Rubin et al., [98] reported +homogeneity in lava basalts from fast ridges [Heterogeneity Index (HI) ∼ 1.6 for spreading rate +> 10 cm/yr], but strong heterogeneity from slower ones [HI ∼ 3 for spreading rate < 4 cm/yr]. +They accounted for the relative thermal stability to explain the higher degree of homogeneity in +the fast ridges. Phyric basalts containing mushy zone crystals suggest the injection of primitive +magmas. However, the petrological estimates of mid-ocean ridge basalt (MORB) at MAR point to +larger volumes of aphyric basalt, representing a collection of aphyric magmas within a mushy zone at +shallow depths. The magmas originated from convection-assisted melt segregation. Lange et al. [66] +calculated mush viscosity [41] as a function of plagioclase phenocryst content. According to their +estimate, the viscosity of melt suspensions can increase by eight times with an increase in small-size +(∼ max. 10 mm) plagioclase phenocryst fraction, where the crystallinity is 20%. Their analysis +predicts the maximum size of olivine phenocrysts in erupting plagioclase-ultraphyric-basalts (PUB) +in the range of 1 to 3 mm. They also explain the presence of PUB selectively in slow and intermediate +ridges as a consequence of olivine phenocryst segregation in conduits during the magma ascent rather +than in the magma chambers. Lange et al. [66] hypothesized that during the ascent of melt-crystal +aggregates through conduits, the melt to crystal ratio is high, and their bulk viscosity is thereby low. +However, it is hard for low-viscosity magmas to transport crystals without segregation in the conduit. +Going by these arguments for the petrogenesis of plagioclase-phyric basalts, we infer that the bulk +viscosity of magmas at the time of their ascent through conduits must be low in cases of slow and +moderately spreading ridges, allowing extensive crystal segregation to produce monomodal crystal +packing with little or no polydispersity. In contrast, the melt suspension viscosity in fast-spreading +ridges, or ridges with prominent magma chambers, should be high due to greater polydispersity +and polymodality even the melt volume fraction is relatively large. The polydispersity variation in +sub-ridge magmatic processes is thus an influential factor to modulate the melt suspension viscosity. +Based on the preceding discussions of melt characteristics, we thus consider the packing factors in +the viscosity analysis of crystal-bearing melts in mushy regions. +Table 1: Model Parameters used in determining the viscosity scale +Domain +Properties +Lava Scale +Conduit length (lc) = 0.1 km; Conduit diameter (dc) = 0.1 km; +Transmitted volume (vc) = 0.01 km3; Strain rate ( ˙ε)= 10−5s−1 +Transmitting time (tc) = 5 hrs +Magma Scale +Conduit length (lc) = 1 km; Conduit diameter (dc) = 0.1 km; +Transmitted volume (vc) = 0.01 km3; Strain rate ( ˙ε)= 10−9s−1 +Transmitting time (tc) = 13 yrs +Lava Scale +Conduit length (lc) = 10 km; Conduit diameter (dc) = 1 km; +Transmitted volume (vc) = 0.01 km3; Strain rate ( ˙ε)= 10−14s−1 +Transmitting time (tc) = 100 hrs +6 + +2.3 +Mush complex (MC) viscosity: a two-step calculation +We calculate the viscosity of mush complex (MC) in two steps: 1) viscosity calculation of crystal- +bearing melts, based on the theory of suspension rheology, called melt suspension, and 2) viscosity +of MC (host rock + melt suspensions), based on the theory of two-phase fluid mixtures. To calculate +the viscosity of a crystal-melt aggregate, we assume the solid (crystals) component as a suspension +in the liquid matrix, as described in the preceding section. Again, the solid part in suspension is +treated as rigid particles and the liquid part as a continuous, viscous medium. The calculation is +carried out using the equations and calculations found in the literature for erupted lavas. Figure 1b +provides a cartoon diagram to show its conceptual framework. +Melt fraction (wt%) +2 +4 +6 +8 +10 +12 +14 +16 +18 + Effective viscosity of MC ( log µ ) (Pa s) +eff +Melt fraction (wt%) +Viscosity of melt + (log µ +) (Pa s) +melt + Effective viscosity of MC + ( log µ ) (Pa s) +eff +Melt fraction (wt%) +(a) +(b) +(c) +(d) + Effective viscosity of MC + ( log µ ) (Pa s) +eff +Viscosity of melt + (log µ +) (Pa s) +melt + Effective viscosity of MC + ( log µ ) (Pa s) +eff +Viscosity of melt + (log µ +) (Pa s) +melt + Effective viscosity of MC + ( log µ ) (Pa s) +eff +Viscosity of melt + (log µ +) (Pa s) +melt +Melt fraction (wt%) +Figure 4: Three-dimensional plots of the effective viscosity of MC (µeff) as a function of melt +suspension viscosity (µM) and molar volume fraction (φ), obtained from the two-stage viscosity +calculations for increasing values of the cohesion parameter, α. (a) α = 0.6, characterized by convex +surface plot. (b) α = 1 (an ideal situation). (c) and (d) α = 1.4 and 2, respectively. Note the +transformation of convex to concave curvature of the surface plot with increasing α. Insets show +the full-length plots, highlighting the effective viscosity range of 1010 to 1018 Pa s. Two limiting +effective viscosity (µeff) values, 1012 Pa s and 1014 Pa s are shown to constrain the viscosity range +in our FSI model to reproduce the spectrum of axial high to flat topography. Host rock viscosity is +chosen 1019 Pa s in these calculations, emulating mantle. +7 + +α= 0.6 +α= 1 +18 +18. +17 +17 ~ +16 . +Effective viscosity of +(s ed) (n Bo) sn +mush (log Hefr) (Pa s) +Effective viscosity of +15y +14 +13 +12 、 +12 +11. +11 +10岁 +10 +10.5 g +viscosity +Ato +7 +5 +3 +0.9 +0.5 +0.7 +0.91 +0.5 +0.7 +0.3 +0.1 +0.3 +0 +0.1 +0 +magma fraction +magma fraction +α= 2 +α= 1.4 +53 +1、 +10.5， +18. +18 +17 、 +17 +16、 +16 ~ +15. +Effective viscosity +Effective viscosity +mush (log Hef) (Pa s +14 +13 +12. +12、 +11↓ +10 +10 +7 +5 +5 +3 +0.5 +0.7 +0.9 +1 +3 +0.9 +0.3 +0.5 +0.7 +1 +0.1 +0.1 +0.3 +0 +magma fraction +magma fraction +16 +18 +2 +6 +8 +10 +12 +14 +Effective viscosity (log μueff) (Pa s)18 +16 +14 +12 +10 +8 +6 +4 +22.3.1 +Melt suspension viscosity +We introduce a scale of the magmatic process with characteristic lengths, 0.01 to 1 km, as applicable +to shallow level magma conduit dimensions (diameter and length) in the sub-ridge region [48]. Lava +eruption episodes determine the characteristic time through the conduits. According to the Volcanic +Explosivity Index (VEI) [85] study, covering more than 75% of the documented Holocene eruptions, +almost 50% of them record continuous blast duration of less than 6 hrs. Furthermore, 63% of the +eruptions yield eruptive volumes ranging from 0.001 to 0.1 km3. Considering the median of time +intervals, successive eruptions of a volcano is found to occur in a time-frequency of 13 years [105]. +We use these data to calculate the strain rates associated with magma flows in shallow conduits. +To simplify the calculation, we consider 0.001- 0.1 km3 magma undergoing eruption through a +cylindrical conduit of 0.01-1 km diameter and on a time duration of 13 years. This wider range of +diameters and conduit lengths is chosen because eruptions generally occur through multiple magma +conduits of varying lengths, and a cumulative effect of the aforesaid parameters is required in our +present analysis. For further simplification, we deal with a representative set of their values, where +the magma conduit length (lc) and diameter (dc) are 1 km and 0.1 km, respectively, and the erupted +magma volume (vc) is 0.01 km3, which passes through the conduit on a time scale (tc) of 13 years. +This set of values yields an average characteristic strain rate (ε +′) of shallow-conduit upwelling of +magma, ε +′ = +� +vc/(0.25πdc +2lctc) +� += 10−9s−1. The characteristic strain rate ε +′ can range from 10−6 +to 10−12s−1 if the parametric values were varied to cover the entire range discussed above. It is +noteworthy that this strain rate scale supports suspended crystals to move passively within the +melt phase [17]. It is necessary to treat lava eruption on a different time scale (designated as lava +scale), which represents the duration of a continuous flow event. These events usually take place +in a duration of 1 to 12 hours [105]. We thus choose an average value of 5 hours to represent the +lava scale. Considering the lava conduit length and cumulative diameter as 0.1 km and the erupted +magma volume as 0.01 km3, we obtain the characteristic strain rate 10−3s−1 (Table1). +The suspension parameters, polymodality, and polydispersity can increase the viscosity of melt +suspensions. Now, we calculate the degree of viscosity increase possible in melt suspensions under +a sub-crustal environment at MORs. Several studies have provided empirical relations to express +the melt viscosity as a function of suspension properties [25,27,34,64,75,81,104]. Consider first the +effect of crystal content in melts. Costa [27] enumerated crystal-free melt viscosity, µl = 105 Pa s at +a temperature of 800◦C and a pressure of 300 MPa. The effective viscosity of melts increases with +increasing solid volume fraction (φs) in the suspension. [76] suggested that melts erupt as lava with +a maximum viscosity, µM = 107Pas corresponding to φ(s,c) = 0.55, called a critical solid fraction. +However, [114] suggested that the critical solid fraction can be further large, φ(s,c) = 0.6 ∼ 0.7 at +the time of lava eruption, implying crystal-bearing melt viscosity, µM = 108.5 Pa s, which means +the enhancement of suspension melt viscosity by an order: I1 = 3.5 [27,43]. +Polydispersity (δ) is a measure of the size variation of suspended solid particles in magma. For +packing with particle distribution on radii, P(R), the parameter can be expressed as, +δ = +�� +∆R2� +⟨R⟩ +(1) +where δR = R − ⟨R⟩, and the moments of R is defined by Rn > +� +RnP(R)dR [28]. It is noteworthy +that an increase in δ allows the suspension to increase the maximum limit of critical solid fraction +� +φ(s,c) +� +. The polydispersity, in turn, multiplies the suspension viscosity. The maximum packing +ratio of mono-dispersed spheres accommodates a maximum solid fraction of 0.64, which can increase +to 0.75 for suspensions with a polydispersity of 0.65 [8]. Roscoe [96] derived a couple of equations +using experimental results [33,117], showing that various size distribution of rigid spheres influences +the viscosity of suspensions less than a uniform size distribution. However, we consider here the +theoretical work of Klein et al. [63], who showed that polydispersity would steeply increase the +maximum packing fraction after a threshold limit for monomodal size distributions. This packing +effect results in an exponential increase of the suspension viscosity and multiplies its magnitude 40 +times. Moreover, the experimental study suggested that an increase in crystal polydispersity might +augment volcanic lava viscosity up to 3 orders of magnitude at a higher deformation rate [81, 92]. +We thus consider the maximum viscosity enhancement in the order, I2 = 3, corresponding to the +polydispersity of crystal-bearing melts. +Strain rate is another factor in our viscosity calculation. Experimental studies suggest Newtonian +melt rheology prevails at strain rates lower than 10−5s−1 [18]. But, at higher strain rates, the melts +develop shear thinning behaviour [118], which reduces the viscosity by more than 2.5 orders in case +of larger solid fraction +� +φ(s,c) ∼ 0.8 +� +[17]. Considering the strain rates in the order of 10−6 to 10−12 +8 + +s−1 on the magma scale and 10−3 s−1 on the lava scale, we choose a maximum viscosity enhancement +in the order, I3 = 2.5, solely due to the decreasing strain rate, leaving out other variables [17]. +To summarize, we use a suspension factor (I), taking into account the cumulative effects of +solid crystal fraction (I1), size distribution (polydispersity) (I2), and strain rate (I3), respectively. +Considering pure melt viscosity in the order of 105 Pa s, as an example, the suspension viscosity +(µM) can be enhanced to a maximum extent of 108.5 Pa s for a limiting solid fraction (0.6 to 0.7), +implying that I1 = 3.5 [27,43].On the other hand, an increase in crystal polydispersity can multiply +µM by an order of 103 at a higher deformation rate [81,92]. We thus consider I2 = 3. Finally, for +the strain rate effects, µM can multiply by a factor of 102.5 depending on the variation of strain +rates in the range 10−3 to 10−12 s−1, as applicable to the MC in our model. That means, I3 = 2.5. +Taking their net effects (i.e., I1 + I2 + I3), we obtain I = 9. +2.3.2 +Viscosity of mush complex +We are now estimating the viscosity (µeff) of mush complexes (MC), using the theory of mixture +rheology within a framework of continuum mechanics [123]. Consider a mixture of host rock (µR = +1019 Pa s) and melt suspensions (µM = 100.5 − 1011.5 Pa s), the effective viscosity of MC (µeff) can +be expressed by the Lederer-Roegiers equation for a two-phase liquid system as, +ln µ12 = +x1 +x1 + ax2 +ln (µ1) + +(ax2) +x1 + ax2 +ln (µ2) +(2) +where α is a constant used to represent the difference in intermolecular cohesive energy between the +participating two components, 1 and 2. xi and µi (i = 1, 2) are the mole fraction and the viscosity +of ith component in the mixture, respectively. The Lederer-Roegiers equation provides an accurate +viscosity calculation of multi-phase fluids with contrasting component viscosities [123]. Equation +(2) is close to the Arhenius equation, which can be demonstrated from Roegiers and Zhmud’s [93] +approach. +Fluidity (inverse of viscosity) of a fluid phase depends on the molar flow activation +energy, ∆E (a measure of intermolecular cohesion). The Arrhenius relation describes the fluidity in +the framework of Eyring’s rate process theory ( [42]) as, +1 +µ = K +�h +exp +�∆E +RT +� +(3) +which leads to, +ln µi = C1 + ∆Ei +RT +(4) +where C1 is a constant, �h is Planck’s Constant, T is absolute temperature, R is the universal gas +constant, and K is the ratio of molar volume and Avogadro number. Subscript i refers to the fluid +component. +For a two-phase liquid system, we consider an additive principle to find the net activation energy +of the mixture. According to Eyering’s Rate Process theory of viscosity [42], the relative motion of +one fluid layer over the other demands a molecule to overcome a potential-energy barrier, called flow +activation energy per molecule. The total flow activation energy is obtained by taking a product of +this quantity with the number of molecules in the system, neglecting any energy dissipation during +the molecular transport. Based on this assumption, the total flow activation energy follows, +∆E12 = x1∆E1 + x2∆E2 +(5) +Using equations (4) and (5), we arrive at the Arrhenius equation for the binary mixture viscosity, +ln µ12 = x1 ln µ1 + x2 ln µ2 +(6) +Equation (5) can be generalized with an asymmetric mixing rule (Roegiers and Zhumd 2011) for the +flow activation energy: +∆E12 = +(1 − γ)x1 +(1 − γ)x1 + γx2 +∆E1 + +γx2 +(1 − γ)x1 + γx2 +∆E2 +(7) +where 0 < γ < 1. For γ < 0.5, the contribution of component 1 to the flow activation energy is +greater than that of component 2, and vice-versa for γ > 0.5. Using equations (4) and (7), we +obtain the Roegiers equation (2) by replacing α = γ/(1 − γ) in equation 2. α = 1 implies an equal +contribution of flow activation energy by the components, whereas α ̸= 1 indicates their unequal +9 + +contributions. For asymmetric two-liquid mixtures, Roegiers and Roegiers [95], and Roegiers [94] +considered α as the ratio of the specific intermolecular attraction energies of the components to derive +Equation (2), where α was held constant for an ideal binary system at a particular temperature. The +equation, validated experimentally by Roegiers [94], and later tackled analytically by Zhmud [123] +yields α as the ratio ln(µ12/µ1)/ ln(µ2/µ12) for a two-phase system with equal mole fraction of the +participating components. In the foregoing analysis we use equation (2) with µ1 = µR and µ2 = µM, +x2 = φ (molar volume fraction of melt suspension, and µ12 = µeff (MC viscosity). +A set of 3D graphical plots presents the calculated µeff as a function of φ and µM for α = 0.6, 1.0, +1.4 and 2 (Figures 4a-d). All of them show an inverse relation of the MC viscosity (µeff) with melt +volume fraction (φ) and suspension viscosity (µM), as widely reported in the literature [27,43,81], +for the entire range of α values considered in the present calculations. µeff is reduced by two orders +(1014 to 1012 Pa s) depending on the φ and µM variations. Our model calculations suggest that µeff +can increase with suspension melt fraction in specific conditions, e.g., suspensions with large volume +fractions of crystals, as observed in magmatically robust ridge settings at fast spreading ridges where +magmas are extremely enriched with crystals. This model provides the MC viscosity estimates also +in opposite environments in slow spreading ridges, characterized by magma poor and low in crystal +content, where crystals readily settle down in the course of magma ascent [66]. +3 +Axial topography: fluid-structure interaction (FSI) mod- +elling +3.1 +Numerical methods +This model couples the three-dimensional convective melt upwelling in the melt-bearing mantle +part (fluid region) with the overlying elastic layer (oceanic crust) (Figure 5) in the framework of a +Fluid-Structure Interaction (FSI) theory. The fluid sub-problem is tackled using the finite volume +computational dynamics code Fluent®, where the CFD model idealizes the mechanical setting +as a two-layer system: basal layer (uppermost mantle part), thermomechanically coupled with an +overlying high-viscosity layer (i.e., elastic solid crust) (Figure 5). The model base is subjected to +thermal perturbations to simulate thermo-chemical convection with synchronous Darcy’s (porous +melt flows) and crystallization (phase transition) [74, 99]. We then take an average of the three- +dimensional velocity data, calculated at the interface above the MC region, and use as the fluid +structure interface velocity to set a mechanical (FSI) coupling of the fluid domain with the top +elastic crust in the finite element (FE) simulations. This FSI coupling principally aims to reproduce +finite deformations in the crustal layer, which otherwise cannot be implemented through the control- +volume based fluid simulations used in this study. It is noteworthy that this combined fluid-solid +(CFD-FSI-FE) modelling approach (Figure 5) geophysically conceptualizes the MC (prismatic sub- +axial zone) as a control volume that conserves mass and momentum by a combination of material +influx from below and solidification / recycling / eruption, and partly outflux across the top surface +[74]. The melts in the fluid domain ascend with a complex heterogeneous pattern due to the 3D +convection structures (Figure 6b), and consequently form ellipsoidal magma pockets with circular +plan views, as seen in the FE results (see, Figure 8a). The mechanical properties of MC are allowed +to evolve with the convection in the overall fluid domain, but maintaining both the mass (continuity) +and momentum conservations. The theoretical framework of convection in sub-ridge fluid domains +is developed on the following conservation equations: continuity, momentum and energy equations, +where we introduce a number of source terms: Darcy and buoyancy source terms in the momentum +equation, and an enthalpy source term in the energy equation [10,115]. The continuity, momentum +and energy equations are finally expressed as follows, +∇ · v = 0 +(8) +ρ ∂ +∂tv + ρv.∇v = −∇p + µfd∇2v + Sg + SD +(9) +ρ ∂ +∂t(ρh) + ∇ · (ρvh) = ∇a∇h − Sh +(10) +where p, ρ and µfd denote pressure, density and viscosity of the fluid domain, respectively. T, h +and a represent temperature, enthalpy, and thermal diffusivity (a = k/ρc, k and c are the thermal +conductivity and specific heat, respectively). The fluid velocity, v, is chosen to vary linearly with the +10 + +melt fraction, φ. In this single-phase idealization, the domain viscosity µfd is varied as a power-law +function of temperature [22,99]. In the momentum equation 9 SD regulates the dominance of Darcy +(i.e., porous) flow, whereas Sg implements the buoyancy factor through Boussinesque approximation. +In equation10 acts as an enthalpy factor to incorporate the energy involved in the phase (solid-melt) +transformation. +The mathematical expressions of these source terms are, +SD = −C (1 − φ)2 +(φ3 + ε)v +(11) +Sg = ρgθ∆T +(12) +Sh = ∂ +∂t(ρ∆H) + ∇ · (ρv∆H) +(13) +C and ε in equation11 are constants, whose values are taken as 1e5 and 0.001 respectively, after [99]. +In equation12, ∆T represents temperature fluctuations with respect to the reference temperature, +and θ is the co-efficient of thermal expansion. In equation13, ∆H is the mean latent heat content. +The fluid subdomain base is subjected to a random thermal perturbation (RTP) condition, which +aims to initiate convective flows in the sub-crustal region ( [99]). In this random thermal condition +partial melting occurs in the domains of high temperatures (>solidus), whereas solidification in the +domains of low temperatures ( 0) in MORs. The axial high topography is possible to +develop only when the viscosity of sub-crustal mush complexes exceeds a threshold value (∼ 6×1012 +Pa s), as demonstrated in Figure 7b, showing crossovers from slightly negative or flat to positive +relief at most of the points on the surface. +Model relief (m) ++750 ++500 ++250 +-250 +-500 +0 ++1000 +14 +Effective viscosity (µ ) : 1×10 Pa s +eff +1.5 Myr +2 Myr +4 Myr +6 Myr +Model relief (m) ++105 +0 ++90 ++75 ++60 ++45 ++30 ++15 +-15 +-30 +-45 +-60 +-75 +-90 +-105 +12 +Effective viscosity (µeff) : 5×10 Pa s + drag +1.5 Myr +2 Myr +4 Myr +6 Myr +-0.5 +0.5 +1.5 +2.5 +1.5 +2 +4 +6 +Model relief ( km) + +12 +µ : 1×10 Pa s +eff +Time (Myr) +0.04 +0.00 +-0.04 +-0.08 +1.5 +2 +4 +6 +Model relief ( km) + +14 +µ : 1×10 Pa s +eff +Time (Myr) +Model relief ( km) + +12 +µ : 5×10 Pa s +Drag +eff +0.05 +-0.00 +-0.05 +1.5 +2 +4 +6 +Time (Myr) +Model relief (m) ++105 +0 ++90 ++75 ++60 ++45 ++30 ++15 +-15 +-30 +-45 +-60 +-75 +-90 +-105 +12 +Effective viscosity (µ ) : 1×10 Pa s +eff +1.5 Myr +2 Myr +4 Myr +6 Myr +150 km +(a) +(b) +(c) +(d) +Figure 8: Vertical elevation maps showing contrasting MOR topographic patterns in FE models +for varying MC viscosities: (a) high (µeff = 1014 Pa s) (b) low (µeff = 1012 Pa s) and (c) low +(µeff = 5 × 1012 Pa s), coupled with across-axis drag at the lithospheric base. Model run time: 1.5 +Myr, 2 Myr, 4 Myr, and 6 Myr. (d) Box plots of the topographic reliefs calculated at the nodes of +the evolved axes are shown in the three panels: (a) to (c). It is noteworthy that the median values +change with time. +We independently investigated the additional effects of magmatic drag forces on the growth of +axial highs. As this factor becomes more effective in the case of a low-viscosity MC condition, we +present here two simulations run with low µeff (= 5x1012 Pa s), one with and the other without +basal drag factor. The drag-free model (Figures 8a-b and 9a-b) produces length-wise persistent axial +highs of moderate elevations (WAR = 13 m at 2 Myr), flanked by low-amplitude (WOR = -50 m) +ridge-parallel depressions (Figures 8d and 9d). However, the axial high progressively reduces its +average elevation, forming an almost flat topography (WAR ∼ 0 m, Figures 8d and9d) at 6 Myr. +17 + +m0 +00 +00 +0 +8 +08 +8 +8 +Q +.The off-axis depressions reduce their negative relative relief to flat (WOR = -25 m). The simulation +with basal drag produces axial topography, dominated by a series of depressions, leaving sporadic +small highs but not forming any persistent linear topographic high (Figures 8c and 9c). The axial +depressions (WAR = -22 m at 1.5 Myr, minima = - 75 m) hardly change their negative relief on a run +time of 6 Myr (WAR = - 22 m, minima = - 58 m, Figures 8d and 9d) and form a weak depression, +flanked by a flat topographic belt (WOR ∼ 0) at a distance of 60 km from the ridge axis (Figures +8c-d; 9c-d) +(a) +(b) +(c) +(d) +1.5 Myr +2 Myr +4 Myr +6 Myr +150 km ++750 ++500 ++250 +-250 +-500 +0 ++1000 +1.5 Myr +2 Myr +4 Myr +6 Myr ++105 +0 ++90 ++75 ++60 ++45 ++30 ++15 +-15 +-30 +-45 +-60 +-75 +-90 +-105 ++105 +0 ++90 ++75 ++60 ++45 ++30 ++15 +-15 +-30 +-45 +-60 +-75 +-90 +-105 +1.5 Myr +2 Myr +4 Myr +6 Myr +-0.05 +1.5 +2 +4 +6 +0.02 +-0.02 +-0.06 +1.5 +2 +4 +6 +0.04 +0.00 +-0.06 +1.5 +2 +4 +6 +0.08 +-0.02 +0.05 +0.15 +13 +Effective viscosity (µ ) : 1×10 Pa s +eff +12 +Effective viscosity (µ ) : 2.5×10 Pa s +eff +12 +Effective viscosity (µ ) : 5×10 Pa s + drag +eff +Model relief (m) +Model relief (m) +Model relief ( km) +Model relief ( km) +Model relief ( km) +Time (Myr) +Time (Myr) +Time (Myr) + +13 +µ : 1×10 Pa s +eff + +12 +µ : 2.5×10 Pa s +eff + +12 +µ : 5×10 Pa s+drag +eff +Model relief (m) +Figure 9: Topographic elevation maps of FE model of MOR run with varying MC viscosity (µeff): +(a) moderate (µeff = 1013 Pa s), (b) low (µeff = 2.5 × 1012 Pa s) and (c) low (µeff = 5 × 1012 Pa +s + drag ). Model run time steps: 1.5 Myr, 2 Myr, 4 Myr, and 6 Myr. (d) Box-plots of the reliefs +at the nodes of the evolved axes in the three FE models: (a) – (c). +We examined the effect of densities of fluid subdomain and MC on the MOR topography. The +density of the MC region is varied as a function of the temperature distribution using the thermal +expansion coefficient. The extent of density variation (∼ 100 kg/m3) is used in the fluid-structure +interaction process. Numerical simulations for this variation show little or no effect on ridge to- +pography at low effective viscosities (for example, at 3 × 1012 Pa s, see Figure10). Also, we ran +simulations with a sufficiently high density (upto 2700 kg/m3) and found very little difference in +topography in case of lower effective viscosities (for example at 2 × 1012 Pa s and at 3 × 1012 Pa s) +(Figure10). The observed results can be explained by considering the relative magnitude between +the pressure term (equation21) and the viscous stress term in equation17 (main text). The viscous +stress term that varies with the strain rate largely controls the morphological undulations, when the +pressure terms in Cauchy stress tensor (equation17) remain almost unaffected. The pressure created +by flow velocities at the base is 0.5×density×velocity2, where the magnitude of velocity is extremely +low, as calculated from the strain-rate range. The dynamic pressure part, involving square of the +velocity term, is thus negligible small, as compared to the static pressure (density×gravity×depth). +Again, the effect of static pressure becomes relatively weak in case of high viscosity conditions. For +example, for a MC viscosity of 1013 Pa s the calculated viscous stress (equation21) is in the order of +18 + +818 +C0000 +. +0000 +0hundreds of MPa at the interface for an average strain rate of 10−5s−1, whereas the static pressure +is in the order of tens of MPa at the interface, implying that the viscosity will dominantly control +the process of topography building in the overlying solid crust. For large effective viscosity of the +MC (> 1012 Pa s), crustal deformations at MORs are thus attributed to the rheological conditions +of the subcrustal magmas, rather than the buoyancy conditions in the MC. (Figure 10). Thus, the +axial highs in our models are not a manifestation of the density structure in the MOR system. This +factor only influences the magnitude of flat axial topography under low-viscosity conditions in the +MC. +3 +normal density (2500 kg/ m ) +3 +reduced density (2400 kg/ m ) +3 +enhanced density (2700 kg/ m ) +- 0.05 +- 0.02 + 0.00 +0.05 + 0.10 +Axial relief (km) +10 +10 +11 +10 +11 +5ˣ10 +12 +10 +12 +7ˣ10 +} +} +12 +2ˣ10 +12 +3ˣ10 + Effective viscosity ( µ ) of MC (Pa s) +eff +Figure 10: Semi-log graphical plots (model run time 3 Myr) of the axial relief as a function of +the effective viscosity (µeff) of MC. Note that, little or no fluctuations occur as the MC viscosity +decrease down to a value ≤ 1 × 1012 Pa s. Also, the differential topographic variations are not +sensitive to the MC density at low (µeff). +To summarize, the 3D views of a high (µeff = 1014 Pa s) and a low-viscosity (µeff = 1012 Pa +s) model reveal a spectacular difference in their stable ridge topography produced on a run time of +7 Myr (Figures 11a-b), which broadly agree with those observed in nature. A time-series analysis +of the across-axis profiles of model topography shows that the off-axis troughs continuously migrate +away from the ridge axis, leaving a flat region between the axial high and them (Figures 11a-b). +The FSI model explains the mechanics of MOR topographic modulation by µeff. The Cauchy stress +term in the Neumann condition for the FSI consists of two terms – a) hydrostatic pressure b) viscous +stress (17). The latter is significantly higher than that the density controlled buoyancy pressure (i.e., +the first term). However, the two dynamic terms turn to be in similar orders when the MC viscosity +becomes low. For µeff < 1012 Pa s, the axial topography no longer varies with viscosity; it is the +static pressure term (equation17) that takes the control in producing a flat topography (Figure10). +4 +Discussions +4.1 +Effects of sub-crustal melt accumulation +Using a three-dimensional graphical plot (Figure4b), we have shown the effective viscosity (µeff) of +MC as a function of the suspension viscosity (µM) and the volume fraction of crystal-bearing melts +(φ) in the system. An increment of µM by an order of 7 (2 to 9), accompanied by an increase of +φ from ∼ 40% to 50%, i.e., pure melt fraction anywhere between 8% and 30%, would eventually +increase µeff from 1012 to 1014 Pa s (Figure 12a). This inverse relation of µeff with φ resolves +the apparently contradictory observations, axial highs in the magma-rich EPR ridges ( [61]), and +flat ridge topography in the magma-poor MAR. The same explanation applies to the topographic +transition, high to flat in SEIR at 103◦35′E, where both sides are somehow rifted [16]. +19 + +Model relief (m) ++105 +0 ++90 ++75 ++60 ++45 ++30 ++15 +-15 +-30 +-45 +-60 +-75 +-90 +-105 +Model relief (m) ++750 ++500 ++250 +-250 +-500 +0 ++1000 +14 +Effective viscosity (µ ) : 1×10 Pa s +eff +12 +Effective viscosity (µ ) : 5 ×10 Pa s + drag +eff +(a) +(b) +14 + µ : 1×10 Pa s +eff +12 + µ : 5×10 Pa s + drag +eff +Figure 11: (a) 3D views of the axis topography in models with µeff = 1014 Pa s (upper panel) and +1012 Pa s (lower panel). Model run time: 7 Myr. The low-viscosity model was run with basal drag +force. (b) A time-series analysis of the first-order across-axis topographic profiles from the high- and +low-viscosity simulation runs. The inset shows a magnified view of the axial negative relief in the +lower panel. +It is noteworthy that the inverse µeff – φ relation occurs below a threshold slope of the µM +versus φ curve, as demonstrated in Figure 12b. The threshold regression line shows that increasing +φ initially reduces µeff, followed by a compensatory rise, ultimately attaining the same µeff value. +Under a threshold condition, across-axis asymmetric sub-ridge melt distributions beneath MORs +can thus hardly break their axial topographic symmetry [36]. Figure 12c shows different possible +paths of µeff variations with µM and φ. µM and φ (blue dotted line) can locally fluctuate in ridge +settings due to some variations in the strain rate. This fluctuation results in an unsteady state +of µeff, ultimately leading to a local instability in the axial topography. In specific cases, µeff +can remain steady over a broad range of non-linear µM – φ regression, as shown in Figure 12d. +Such a sub-crustal condition is possibly required for the long-timescale relative stability of axial +morphologies, as reported from many MORs [97]. +The viscosity model also yields µM – φ relations that support contrasting observations from fast +and slow spreading ridges; axial high topography in fast ridges with high melt percentages, whereas +axial flat topography in slower ridges with low melt contents. We now provide simple numerical +estimates to discuss the MC viscosity as a function of crystal-bearing melt viscosity (µM) and molar +volume percentage (φ) of melt suspension using the mixture rheology curve in Figure 12. For a given +value of µM, e.g., 105 Pa s, the MC viscosity can be as low as 1012 Pa s if φ = 50%. Considering the +crystal-free, pure melt viscosity in the order of 102 Pa s, the suspension (i.e., crystal-bearing melts) +must contain solid crystals by 60-70% to attain its viscosity in the order of 105 Pa s (discussed in the +earlier section). It means the pure melt percentage in the complex must be in the range of 15 to 20%. +The graphs (Figure 12) show an inverse relation of the mush viscosity with melt suspension content; +µeff becomes 1014 Pa s as φ decreases to 36%, which corresponds to a pure melt fraction of 11-15%. +This melt fraction estimate would be further low if the polydispersity and polymodality factors were +considered in the calculation. Similarly, an increase in µM (e.g., 105 to 107 Pa s) can yield the MC +viscosity (µeff) in the order of 1014 Pa s for φ = 40% (Figure 12). On the other hand, both µM +and φ in the mush can increase to yield the same mush viscosity, i.e., 1014 Pa s for φ = 57%, and +20 + +1000 + 1.5 Myr +2 Myr +800. +.4 Myr +6 Myr +600 - +Model relief (m) +400 : +200- +0 +-200 - +-400 +-60 +-40 +-20 +0 +20 +40 +60 +Distance from MOR axis (km)1000 +20 +1.5 Myr +2 Myr +10 - +800 - +4 Myr +0 - +6 Myr +-10 +600 - +-20 +Model relief (m) +-30 - +400 +-40. +-50 +200 +-60 +-14-12-10 -8 +0 +2 +4 +6 +8 +10 12 14 +0 +-200. +-400 +-60 +-40 +-20 +0 +20 +40 +60 +Distance from MOR axis (km)Effective viscosity of MC (log µ ) (Pa s) +eff +(a) +(b) +(c) +(d) +Viscosity of crystal-bearing + melt (log µ ) (Pa s) +� +Viscosity of crystal bearing + melt (log µ ) (Pa s) +� +Viscosity of crystal-bearing + melt (log µ ) (Pa s) +� +Crystal-bearing melt fraction +Viscosity of crystal-bearing + melt (log µ ) (Pa s) +� +Crystal-bearing melt fraction +Crystal-bearing melt fraction +Crystal-bearing melt fraction +(e) Fast spreading ridges +Melt flow +Upper crust +Mantle +Mush complex (MC) +Decompression melting stops at this level +Axial high +Melt-rich-crystal-rich MC +Melt-poor-crystal-poor MC +(f) Slow spreading ridges +Melt flow +Mantle +Decompression melting stops at this level +Mush Complex (MC) +Axial flat/low +Figure 12: Projection of the 3D plots for MC viscosity (presented in Figure 4 2) on a 2D frame defined +by volume percentage (φ) and viscosity (µM) of crystal-bearing melt (i.e., melt- suspension). (a) A +specific regression where a large increase in µM (an order of 107 Pa s) with a moderate rise (∼ 10%) +in φ enhances the effective viscosity (µeff) of MC by up to two orders. (b) A linear regression of +µM with φ involving a limited change in µeff, where a zone of a small decrease is followed by a +zone of little increase (shaded with different colours), ultimately leading to the same µeff under a +specific combination of the µM and φ variation. It is to be noted that the red dotted line defines a +critical slope line; a regression below this line would result in a lowering of µeff, whereas above it +would increase µeff. (c) A regression (blue dotted line) of a smaller change in µM (∼ 101 Pa s) as +well as a lower change (∼ 5%) in φ showing a modification of µeff by up to 1 order of magnitude. +Note that a similar order of change in µeff is possible when one of the two parameters: φ or µM +varies, keeping the other constant, as indicated by red dotted lines. (d) A steady-state condition +of µeff in complex with varying φ and µM along a particular regression (dashed red line). +(e) +and (f) Cartoon diagrams of the sub-crustal phenomena at typical fast- and slow-spreading ridges, +showing the possibility of higher and lower effective viscosities in a melt-rich-crystal-rich and a melt- +poor-crystal-poor conditions, respectively. +µM = 1010 Pa s (Figure 12). The two schematics in Figures 12e and 12f show sub-crustal melt bodies +and magma conduits (i.e., MC region) with contrasting suspension characteristics at the two types +of ridges. The mush complexes in faster ridges generally have melts with larger phenocrysts and +groundmasses in larger volume fractions than those in slow-spreading ridges. Consequently, a higher +effective viscosity of MC due to higher crystal content, polydispersity, and polymodality (Figures 12a +and 12e), produces axial high topography in fast ridges. On the other hand, the opposite suspension +characteristics sets in a low viscosity condition (Figures 12a and 12f) beneath slow ridges, which +gives rise to axial flat topography. Melt contents in the MC can, however, fluctuate due to a number +of factors, such as sub-crustal solidifications and numerous volcanic events in the process of new crust +formation. A concerted operation of the following three processes: 1) mid-oceanic ridge eruptions, +2) sub-crustal solidifications [74], and 3) continuous convective upwelling of partial melts [99] can +modulate the µM – φ regression to maintain a steady-state µeff condition (Figure 12d) required for +the stable axial topography. +4.2 +Axial topographic growth: mechanisms and their validation +Several MOR models have attempted to integrate sub-ridge thermomechanical processes in the +mantle-lithosphere, giving a spectrum of competing mechanisms, such as magmatic upwelling versus +hydrothermal cooling [21], tectonic extension versus diking [12], fault-driven collapse versus isostatic +compensation [35, 69], overpressure building versus release of magma chambers [47, 91], and fluid +convection versus matrix compation [56]. The MOR model of our present concern invokes a sub- +ridge mechanism of porous convection with synkinematic melting-solidification processes to describe +21 + +10.5 +μeff = 1016 +1015 +1014 +1013 +10.5. +μeff = 1016 +1015 +1014 +1013 +1012 +1012 +0.2 +0.3 +0.4 +0.5 +0.6 +2'0 +0.8 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Magma fraction +Magma fraction +μeff = 1016 +1014 +μeff = 1016 +1015 +1015 +10.5. +1014 +1013 +10.5. +1013 +1012 +1012 +0.3 +0.2 +0.3 +0.2 +0.4 +0.5 +0.6 +0.7 +0.4 +0.5 +0.6 +0.7 +0.8 +0.8 +Magma fraction +Magma fraction +2 +4 +6 +8 +10 +12 +14 +16 +18 +Effective viscosity (log μef) (Pa s)the FSI mechanics. The convection-driven upwelling occurs at a depth of cessation of the adiabatic +decompression melting. We use this threshold depth to introduce random temperature points (range +800◦C to 1400◦C) on a narrow region beneath the MOR axis (Figure S.3a). This random thermal +perturbation (RTP) initiates the convection in the porous upper mantle, where the porous convective +flow accompanies melting and solidifications in the sub-ridge shallow upper mantle and sub-crustal +regions mediated an enthalpy transfer process. The flows are always geometrically asymmetric due +to concerted effects of the RTP, porous convection [56] and the intermixing of multiple convection +cells and melting-solidification processes. +The convective flows induced in the MC develop normal stresses at its interface with the overlying +crust, as modelled through a fluid-structure interaction (FSI). The effective viscosity of MC and its +kinematic condition determines the magnitude of normal stresses transmitted to the overlying solid +crust. The kinematic conditions of the MC are nominally transmitted to the overlying crust together +with the dynamic conditions as an implementation of the Robin transmission condition in the FSI +mechanism [3]. In the FSI formulation, the shear stresses are generally excluded, considering that the +differential velocity across the interface is small. However, for lower MC viscosities this factor can be +significant due to the existence of strong relative motion. The stress transmission eventually results +in deformations in the elastic crust to produce an axial topography. The effective viscosity of MC +plays a critical role in controlling the magnitude of transmitted stresses that ultimately determine +the vertical reliefs in the overlying elastic crust. Decreasing MC viscosities consequently results in +a transition of high to flat axial topography. However, density takes the lead role to maintain flat +topography at lower MC viscosities (≤ 1012 Pa s) (Figure 10). +We will now use some generic parameters of the natural MOR systems to test the validity of the +FSI mechanism proposed in this study. The flow data of MC support the dynamic magma budget +of mid-oceanic ridge calculated and validated with natural ridge processes in previous studies [74]. +The melt budget suggests eruptible melts amount to 8 − 10% of the total upwelling melt beneath +MORs, which means, 3.7x106 m3/yr in a 500 km long ridge. Secondly, the model presented here +treats MC as a fluid region consisting of liquid mixtures of crystal-bearing melts and host rocks. +Its viscosity analysis yields a value in the range 1012 to 1014 Pa s, which is comparable to that of +lower-crustal magma bodies in MORs. For example, Chenevez et al., [23] estimated the viscosity of +gabbroic mushes within the axial magma chamber as 1015 Pa s from Oman Ophiolites. In addition, +McKenzie [79] considered the effective viscosity of melt-bearing matrix in the order of ∼ 1015 Pa +s. On the other hand, experiments have shown viscosity in the order of 1011 Pa s for lower crusts +containing melts by 20-25% [89]. Similarly, Fontaine et al. [39] estimated an effective viscosity of 1013 +Pa s for sub-crustal regions in the melt-rich fast spreading ridges showing axial highs, as produced +in our model (Figure 11a). The spectrum of melt percentages at MC, as predicted from the present +rheological calculations (30-80%, see Figure 4a-d), is supported by earlier estimates. The strain +rates in the MC (10−3s−1 to 10−11s−1) with a median value of 10−5s−1 also agree well with those +recorded in natural MOR systems. Our FSI model shows that the timescale of mid-oceanic ridge +processes to stabilize (Figure 8) is 6 Myr, which is comparable to those reported in the literature [40]. +4.3 +Axial topography: a model versus nature comparison +In the quantitative analysis of axial topography the median relief of natural ridge systems has +been considered to estimate the vertical anomaly with respect to the average seafloor depth (2600 +m, [101]). +We chose an across-axis section of the EPR at 17◦N [68] to compare its long wave +axial-high topography with those obtained from our model. The 17◦N section displays a first-order +characteristic topography consisting of a sharp axial high (maximum elevation: H ∼ 400 m, axial +width: W ∼ 40 km), flanked by a symmetric pair of flat regions (width ∼ 60 km), and narrow, weak +depression zones away from the high. The axial topography shows a good match with that produced +in the model simulation (µeff = 1014 Pa s, H ∼ 500 m and W ∼ 30 km) (Figures 13a). We also +validated along-axis model topographic patterns with the available natural data. Figure 14a shows +a comparative analysis of the Juan de Fuca (JdF) ridge-segment (44◦30′N to 49◦ N) topography +and the µeff = 1014 Pa s model axis relief (at 7 Myr model run-time). The JdF ridge includes +a seamount, and six major segments form reliefs with their median close to the model value (220 +m). The model and the natural ridge systems show remarkable similarity in terms of the relief +density distribution and scatter (Figure 14a). Geologically, the JdF ridge (JdFR) with moderate +spreading rates (60 mm/yr) receives lateral magma supply from the neighbouring Cobb hotspot +and axial seamounts. Enhanced fractional crystallization with efficient cooling [19] away from the +hotspot regions thus produces a viscous magma-enriched sub-crustal melt-rich system, which in +turn facilitates the growth of axial high topography, as predicted from high-viscosity FSI model +22 + +A* +106� W 15’ +17� N 30’ +17�N +16� N 45’ +16� N 30’ +17� N 15’ +106� W +105� W 45’ +105� W 30’ +105� W 15’ +105� W +104� W 45’ +A +-3400 +-3300 +-3200 +-3100 +-3000 +-2900 +-2800 +-2700 +14 +μ : 1×10 Pa s +eff +Depth (m) +A* +EPR 17�N +A +A* +0 +20 30 40 50 60 70 80 +10 +70 60 50 40 30 20 10 +80 +Across-axis distance (km) +16� E +16� E 30’ +53� S +52� S 45’ +52� S 30’ +52� S 15’ +52� S +17� E +17� E 30’ +18� E +18� E 30’ +19⁰ E +51� S 15’ +-2400 +-2600 +-2800 +-3000 +-3200 +-3400 +-3600 +Depth (m) +S +S* +S +S* +SWIR (17⁰E 27’) +(a) +(b) +0 +0 +20 30 40 50 60 70 80 +10 +70 60 50 40 30 20 10 +80 +Across-axis distance (km) +12 +μ : 2.5×10 Pa s + Drag +eff +10 +0 +-10 +-20 +-30 +-40 +-50 +45� W 50’ +45� W 40’ +45� W 30’ +45� W 20’ +45� W 10’ +44⁰W +44� W 50’ +44� W 40’ +4�� W 30’ 44� W 20’ +14� N 10’ +13� N 20’ +13� N 30’ +13� N 40’ +13� N 50’ +1�� N +MAR ( 13� N 55’) +-2200 +-2400 +-2600 +-2800 +-3000 +-3200 +-3400 +-3600 +-3800 +M +M* +Depth (m) +M +M* +0 +12 +μ : 2.5×10 Pa s + Drag +eff +0 +20 30 40 50 60 70 80 +10 +70 60 50 40 30 20 10 +80 +Across-axis distance (km) +(c) +10 +0 +-10 +-20 +-30 +-40 +-50 +-60 +Figure 13: Comparisons of the cross-axis topographic profiles between nature and model: (a) a high +viscosity (µeff = 1014 Pa s) model versus EPR 17◦N. (b) a low-viscosity (µeff = 2.5 × 1012 Pa s) +model with drag force and SWIR 17◦27’E. (c) a low-viscosity (µeff = 2.5 × 1012 Pa s) model with +drag force and MAR 13◦55’N. Red lines show polynomial fits (higher-order) to the original natural +data (in black). Blue lines indicate a second-degree polynomial fit for the SWIR and MAR. The +EPR section displays a good match with the model topography [maximum ridge elevation (H): 400 +m (model) and 500 m (nature), axial width (W): 40 km (model) and 30 km (nature), similar ‘neck’s +on both sides]. The SWIR section also shows a similarity with the model in their first-order axial +topography, barring quantitative differences [H: - 40 m (model) and - 400 m (nature), W: 20 km +(model) and 40 km (nature)]. The MAR section matches fairly with the first-order model topography, +but with differences in their magnitude [H: - 60 m (model) and - 600 m (nature), W: 15 km (model) +and 10 km (nature)]. MOR data source: GeoMapApp (http://www.geomapapp.org/)/CC BY. +23 + +500 - +400 - +300 - +200 - +100 - +0 - +-100 - +-200Model relieModel relief (m100 +UUZ- +-300 +-400Model relieModel relief (m100 +-100 +-2 +-3 +-400 +-500 +600Model relieModel relief (m(Figure 14a). Additional notable features of JdFR are: 1) the axial seamount does not significantly +differ from the adjoining parts of the axial ridge segment in terms of the crystal content of their +magmas [19], and 2) the excess melt volume is compensated by forming a thick crust. However, +there is a possibility of narrow, focused upward magma flux to the axial seamount’s base, resulting +in enhancement of the normal flux in JdFR by three times [119], as reflected from higher upwelling +velocity/strain rates in this region. The ridge seamount thus represents a local feature to enhance +the vertical strain rates over the common viscous behaviour of the underlying mush and give rise to +topographic characteristics observed in the corresponding model, where the positive relief is larger +than the maxima by 60%, and ten times the median value (Figure 14a). +We compared across-axis model topographic profiles with two sets of nearly flat topography from +ultraslow SWIR and slow-spreading MAR extrapolated directly from the GeoMap database. These +two ridges exhibit predominantly rifted valley topography. We thus chose two narrow segments, +where rifting is not a dominant ridge process, as indicated by thick crust, but they show weak axial +valleys or flat axial topography. A topographic profile from the low-viscosity (µeff = 2.5 × 1012 +Pa s) MC model compares well with the first-order valley geometry (17◦27′E, [45]) in the SWIR, +when the off-axis depressions due to extensional faulting are excluded (Figure 13b). In the case of +MAR, another profile of the same model run grossly reproduces the ridge profile (13◦55′ N, [73]), +which consists of a narrow, shallow valley at the ridge flanked by flat off-axis stretches (Figure 13c). +However, there are large differences in the magnitudes of axial depression topography between the +natural settings and the corresponding models (e.g., ∼30 m in model vs. ∼100 m in SWIR, ∼50 +m in model vs. ∼ 150 m in MAR, 13◦55′ N) but they show a first-order similarity in their axial +zone topography, e.g., across-axis width (∼80 km) of the gentle axial depressions. However, the +higher-order off-axis topographic elements in model and nature do not perfectly match with one +another (Figure 13c). These higher-order mismatches perhaps result from strong effects of tectonic +(tensile) stresses, as compared to relatively weak rheological effects of the underlying mush complex. +We extended our model validation with a number of natural along-axis topographic profiles from +SEIR (87◦30′ E to 93◦30′ E). These profiles show a marked similarity in their relief patterns (Figure +14b) with those in the model run for moderate effective MC viscosity (µeff = 3 × 1013 Pa s at +7 Myr run time). At the western portion the SEIR has a significant influence of melt plumes [4], +and developed a first-order transform discontinuity and a prominent overlapping spreading centre. +Overall, the ridge, deepening towards the east [103], displays along-axis roughness of its relief fairly +in agreement with our model. The western part of SEIR, extending up to 90◦E receives mantle- +derived melts from the Kerguelen-Heard hotspot, as suggested by the 87Sr/ 86Sr ratio enrichment. +Alternatively, there is a possibility for greater availability of partial melts due to a greater mean +depth of melting, which is correlated with the He isotope ratio peak at 88◦E (see Mahoney et al. [72] +and references therein). Both the cases can give rise to a condition of high magma percentage and +low magma viscosity in the portion of SEIR of our present concern, which might retain µeff at +moderate values (∼ 3 × 1013 Pa s), as derived from the viscosity calculations (Figure 12b). +We support our model interpretations with a positive correlation of the along-axis model topog- +raphy with the southern part of the EPR and the northern segment of the PAR (42◦S – 46◦30′S) +(Figure 14b). The latter is thought to be stable for more than 50 Myr [97]. Its median relief matches +well with that of a high (µeff = 7 × 1013 Pa s) model. On the other hand, the EPR ridge segment +shows a lower relief variance than the model (Figure 14c).The overall topographic parity allows us +to predict the viscosity of melt-rich sub-crustal region in the order of 1013 Pa s for this particular +ridge segment. Applying our two-phase viscosity model, we suggest that although this region is rich +in melt content, it gains relatively high viscosity due to a large volume fraction of solid crystals +in the melt suspensions. The relatively lower along-axis variance in the EPR ridge topography, as +compared to that in the corresponding model, results from a number of possible factors, such as +longitudinal stability of the ridge position, continuous deep-seated upwelling, and overwhelming vis- +cous magmatic control [97]. A rate balance between the melt supply and crystallization can account +for a steady-state effective viscosity of the underlying melt-bearing regions to sustain such stable +ridge-axis topography (see Figure 12d). +This discussion leads us to suggest the following. The magma-rich EPR has retained a high- +viscosity condition of the melt-bearing sub-crustal regions to produce narrow axial high, as produced +in our simulation with µeff = 5 × 1013 − 1014 Pa s. Hotspot fed and rapidly cooling melts in the +JDFR has a sub-ridge MC with the highest viscosity (µeff = 1014 Pa s), and their produced an +axial high elevation comparable to that in the model. On the other hand, the western SEIR is rich +in moderately viscous melt suspensions but becomes melt-poor, resulting in deep rifted axial valley +topography in the eastward direction [4,103]. The analysis suggests a moderate sub-crustal viscosity +(µeff) has formed axial high topography in western SEIR, which agrees well with the simulation +24 + +14 + µ : 1 ×10 Pa s +eff +46�N +44�N +45�N +134�W +133�W +132�W +131�W +131�W +130�W +129�W +127�W +49�N +47�N +48�N +Ridge +Model +Relief (km) +0.0 0.5 1.0 1.5 +2 + Density distribution +0.0 +1 +1.5 +0.5 +-0.5 +0.0 +0.5 +1 +1.5 +2 +Model +Ridge +(n = 249; bw= 0.113) +(n = 101; bw = 0.073) +Relief (km) +Model +Ridge +Relief (km) +0.0 0.5 +1.0 +1.5 2.0 +0 +100 +200 +300 +400 +500 +Along axis length (km) +JdFR +(a) +87�E +88�E +89�E +90�E +91�E +92�E +93�E +94�E +42�S +43�S +44�S +45�S +13 + µ : 3×10 Pa s +eff +Ridge +Model +Relief (km) +-0.1 +0.1 +0.3 +0.5 + Density distribution +0 +1 +2 +3 +4 +5 +Model +Ridge +(n = 251; bw= 0.031) +(n = 100; bw = 0.034) +Relief (km) +0 +0.2 +0.4 +0.6 +Model +Ridge +Along axis length (km) +0 +100 +200 +300 +400 +500 +Relief (km) +-0.1 +0.1 +0.3 +0.5 +SEIR +(b) +117�W +116�W +115�W +114�W +113�W +112�W +111�W +110�W 109�W +42�S +43�S +44�S +45�S +46�S +47�S +48�S +13 + µ : 7 ×10 Pa s +eff +Ridge +Model +Relief (km) +0.0 +0.5 +1.0 +Ridge +Model +Relief (km) +0.0 +0.5 +1.0 +Along axis length (km) +0 +100 +200 +300 +400 +500 + Density distribution +Relief (km) +0 +1 +2 +3 +4 +5 +0 +0.5 +1 +1.5 +Model +Ridge +(n = 251; bw= 0.078) +(n = 101; bw = 0.024) +EPR +(c) +Figure 14: Comparison between natural and model along-axis topography: (a) A high viscosity +(µeff = 1014 Pa s) model and JdFR (44◦30’N to 49◦N). (b) A moderate viscosity (µeff = 3 × 1013 +Pa s) model and eastern SEIR (87◦ 30’E to 93◦30’E).(c) A moderately high viscosity (µeff = 7×1013 +Pa s) model and EPR (42◦S – 46◦30’S). The box plots, scatter plots, and density distributions are +also shown, along with the natural ridge bathymetry and the evolved model ridge axis elevations. For +JdFR, the along-axis profiles show matching topography with the model [median relief : 0.2043 km +(natural) and 0.22 km (model); density peak : 0.12 km (natural) and 0.05 km (model)]; SEIR and +EPR profiles are also in good agreement with the model topography: for SEIR, median relief : 0.076 +km (natural) and 0.054 km (model); density peak : 0.05 km (natural) and 0.02 km (model); and for +EPR, median relief : 0.15 km (natural) and 0.15 km (model); density peak : 0.22 km (natural) and +0.04 km (model). Natural data source: GeoMapApp (http://www.geomapapp.org/)/CC BY. +25 + +0 +0 +- +- +8 +-- +-O +00 +8 +00 +e +8 +8 +安 +8.0 +8 +1 +: +-00 +8 +00 +O +Q +08 +RDO0 +0 +F- : +- +- +- +- +- +008 +8 +0 +?oo +O +8 +0 +00result for µeff = 1 − 5 × 1013 Pa s. Our FSI model for µeff ∼ 1012 Pa s points to an appreciable +mismatch on the along-axis model topography with those observed in the magma poor MAR and +SWIR (both slow-spreading), albeit showing a reasonable match with the across-axis first-order +curvatures in their non-rifted segments. We suggest that the MAR and SWIR topography are not +entirely controlled by the rheological setting of their sub-crustal and lower crustal mush complexes. +This mismatch indicates the possibility of tensile stress regimes to govern the axis topography where +the flow-driven stresses at the base in case of slow-spreading ridges become relatively weak due +to low-viscosity condition in the underlying mantle (e.g., Lin and Parmentier [68]). +Our model +also shows a weak match of the along-axis model topography with the Reykjanes ridge topography +where the relief is significantly higher than the model relief even in high-viscosity (µeff ∼ 1014 Pa +s) simulations (Figure 15). It is noteworthy that the 600 km long slow (2 cm/year full spreading +rate) Reykjanes ridge (57.9◦ N to 62.10◦ N) is thought to have evolved under the influence of Iceland +mantle plume, as evident from its large oblique spreading characteristics (280 from the spreading +normal) and its V shaped plan view [102, 120]. +This might be the reason for the topographic +mismatch. +Axial relief in the present model correlates positively with crystal contents, polymodality and +polydispersity in the MC that enhances the melt suspension viscosity, and in turn, the effective +viscosity of the MC. Faster spreading ridges show crystallization at shallower depths [116] and they +also undergo greater mixing of their crystal phases (olivine, plagioclase and clinopyroxene) of different +sizes and shapes at subcrustal / lower crustal MC [70]. The predominance of crystal suspensions sets +in a high-viscosity rheological condition that explains the axial highs at faster spreading segments. +In contrast, crystallization at slow spreading ridges typically occurs at greater depths [50], allowing +the melts to transport through melt channels, but loosing heavier (olivine) larger crystal in their +pathways due to slow ascent velocities [66]. In effect, slow spreading ridges are likely to produce +MCs with low crystal contents and lower polymodality and polydispersity that result in setting up +a low-viscosity setting and weak normal stress transfer in the topographic process. +Relief (km) +0.0 0.5 1.0 1.5 2.0 +0 +100 +200 +300 +400 +500 +Along axis length (km) + Density distribution +0.0 +1.0 1.5 +0.5 +-0.5 +0.0 +0.5 +1.0 +1.5 +2 +Relief (km) +Model +Ridge +(n = 249; bw= 0.113) +(n = 101; bw = 0.073) +2.0 2.5 +Ridge +Model +Relief (km) +0.0 0.5 1.0 1.5 2 +14 + µ : 1 ×10 Pa s +eff +38�W +36�W 34�W 32�W +30�W +28�W +26�W +24�W 22�W +57�N +58�N +59�N +60�N +61�N +62�N +63�N +Reykjanes ridge +Figure 15: Comparison between natural and model along-axis topography of Reykajanes Ridge +(44◦30’N to 49◦ N). A high viscosity model (µeff = 1014 Pa s) is shown in the same panel producing +Axial high topography. The box plots, scatter plots, and density distributions are also shown, along +with the natural ridge bathymetry and the evolved model ridge axis elevations. Natural data source: +GeoMapApp (http://www.geomapapp.org/)/CC BY +4.4 +Model limitations +The present FSI model is designed to study the sole role of the viscosity of melt-rich regions in +controlling the axial topography of mid-ocean ridges. However, as discussed in the Introduction, +several other factors, e.g., the diking parameter [71], can influence the ridge topography. A large +number of studies have recognized the spreading rate as a potential factor to modulate the axial +high versus valley development. The exclusion of this factor obviously imposes a limitation on our +modelling. However, this study sheds light in a new direction, showing that the viscosity changes +of sub-crustal melt regions by 101 – 102 order can alone bring a transition from flat to axial high +topography in a MOR under the same spreading rate. Secondly, MORs generally undergo extensional +26 + +Model o + Ridgeofaulting in the uppermost brittle crustal layer [12], which contributes to the development of high- +order ocean floor morphology, such ridge parallel hills at MORs. Also, extensional stresses play +a dominant role in forming axial valleys [68]. Our model excludes such tectonic stress regimes at +MOR and brittle failure in the elastic solid top layer as we focus on longer wavelength topography. +The gradient in across-axis lithospheric thickness variation might have an additional influence in the +axial topographic development. However, this factor excluded in this study to find independently the +effects of sub-crustal mush complexes (MC) on the two end-members: flat and high axial topography. +5 +Conclusions +1) One-way Fluid-Structure coupling between a sub-crustal melt accumulation zone and the overlying +solid elastic crust, has been implemented with the framework of a computational fluid dynamics +modelling of convective heat and mass transfer. The model results demonstrate that the effective +viscosity of the melt-rich zones can play a critical role in modulating the axial high versus flat +topography. 2) We claim that the mush complex (MC) dynamics involving crystallization in its melt +suspensions at the lithospheric base can largely govern the ridge axis topography. 3) A complete +description of the MOR mechanical setting demands viscosity analysis on two scales- one at magma +body/conduit scale, which is tackled by utilizing suspension theory, and mush scale, which is dealt +with a modified Arrhenius equation. 5) The effective viscosity of MC varying in the range 1012 +Pa s to 1014 Pa s produces a full spectrum of the non-rifted axial high to flat (1.27 km to - 0.06 +km) topography. Typical axial highs form in the viscosity range of 1013 Pa s - 1014 Pa s, whereas +axial lows in the viscosity range of 1012 Pa s - 5 × 1012 Pa s. 6) The onset of relative vertical +displacements in central axial regions occurs at the time of upwelling melt-bearing mushy materials +to interact with the overlying crust. The process forms a stable topography on a time scale of ∼ 6 to +7 Myr, characterized by central axial highs and off-axis depressions on their flanks. 7) This viscosity +based new model explains the following characteristics of natural MORs: a) axial high topography +in melt-rich ridge systems (e.g., EPR) and first-order axial valley in melt-poor ridges (e.g., SWIR +and MAR), b) transformation of ridge topography due to drastic changes in subcrustal magma +constituency (e.g., SEIR), c) axial seamount as a location of high upwelling rates (e.g., JdFR) and +d) stability of axial topography in large temporal and spatial settings (e.g., Southern EPR) as an +outcome of the competing factors, such as viscosity and volume percentage of crystal-bearing melt +suspensions that maintain the effective viscosity of mush complex almost at a constant level. 8) Our +FSI modelling constrains the viscosities of sub-crustal mushy regions in the following MOR systems: +1014 Pa s for JdFR, 5 × 1013 - 1014 Pa s for EPR, 1 - 5 × 1013 Pa s for western SEIR. +References +[1] Arnoux, G. 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Focused mantle upwelling beneath mid-ocean ridges: evidence from seamount +formation and isostatic compensation of topography. Earth and Planetary Science Letters 113 +(9 1992), 41–55. +[122] Zhang, C., Koepke, J., Kirchner, C., G¨otze, N., and Behrens, H. Rapid hydrother- +mal cooling above the axial melt lens at fast-spreading mid-ocean ridge. Scientific Reports 4 +(5 2015), 6342. +[123] Zhmud, B. Viscosity blending equations. Lube Mag 121 (2014), 22–27. +34 + diff --git a/dNE4T4oBgHgl3EQfQAz9/content/tmp_files/load_file.txt b/dNE4T4oBgHgl3EQfQAz9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..51ab7f0e0e522e7e7a34e633a9f5bfbdeac32b06 --- /dev/null +++ b/dNE4T4oBgHgl3EQfQAz9/content/tmp_files/load_file.txt @@ -0,0 +1,2106 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf,len=2105 +page_content='Role of mush complex viscosity in modulating axial topography in mid-oceanic ridges Joyjeet Sen∗, Shamik Sarkar†, Nibir Mandal‡ Department of Geological Sciences, Jadavpur University, Kolkata 700032, India Abstract This article exploits the interaction dynamics of the elastic oceanic crust with the underlying mush complexes (MC) to constrain the axial topography of mid-ocean ridges (MORs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The effective viscosity (µeff) of MC beneath MORs is recognized as the crucial factor in modulating their axial high versus flat topography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Based on a two-step viscosity calculation (suspension and solid-melt mixture rheology), we provide a theoretical estimate of µeff as a function of melt suspension characteristics (crystal content, polymodality, polydispersity and strain-rate), and its volume fraction in the MC region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' We then develop a numerical model to show the control of µeff on the axial topography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Using an enthalpy-porosity-based fluid-formulation of uppermost mantle the model implements a one-way fluid-structure interaction (FSI) that transmits viscous forces of the MC region to the overlying upper crust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The limiting non-rifted topographic elevations (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='06 km to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='27 km) of model MORs are found to occur in the viscosity range: µeff = 1012 to 1014 Pa s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The higher-end (1013 to 1014) Pa s of this spectrum produce axial highs, which are replaced by flat or slightly negative topography as µeff ≤ 5 × 1012 Pa s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' We discuss a number of major natural MORs to validate the model findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' 1 Introduction Many mid-ocean ridges (MORs) evolve with complex 3D axial topography, which is hard to explain with standard tectonic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Their spatially varying axial topography, such as high, flat or valley, is generally attributed to the spreading rate [106, 110], the magma availability [59, 74, 108], in a particular ridge-segment, and upper crustal faulting [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' However, these contrasting axial morphologies are often found in MORs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=', South-East Indian Ridge (SEIR), where the spreading rate shows practically no variations [16], and ultra-slow South-West Indian Ridges (SWIR) displaying typical axial valley topography, where they have large magma availability [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' A direction of MOR studies explains the rift morphology as a product of the two competing processes- tectonic and magmatic, conceived as horizontal spreading and dike opening, respectively [12, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' A non- dimensional parameter, called the M factor (a ratio of the dike intrusion to plate-spreading driven widening rates), has been used to reproduce the axial structures in numerical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' M = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=', a condition of dike intrusion rate to completely balance with the plate spreading rate, gives rise to an axial high, whose height depends on the magma density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' In contrast, M < 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=', a condition of less effective diking than the spreading rate, yields faulted axial valley [12, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Some studies have shown the axial morphology as a function of the extension rate and inherent short-wavelength seafloor heterogeneities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=', [110]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' However, their interpretation faces disagreement with the Mid- Atlantic ridge model, which proposes the magma supply as a critical factor in determining the axial morphology [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Although these models well integrate the axial morphological spectrum by a single factor- M, the modelling approach does not account for sub-crustal melt processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' It is noteworthy that many recent MOR studies demonstrated how the latter could significantly control the MOR evolution [16, 78], albeit a comprehensive model is still unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Our present article aims to bridge this gap, treating the axial morphology in the thermo-mechanical framework of an ideal three-dimensional melt upwelling system, where divergence force components act along and across the ridge axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' This modelling approach allows us to investigate the extent of magmatic control on 3D axial morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' ∗senjoyjeet@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='com †shamiksrakar@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='com ‡nibir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='mandal@jadavpuruniversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='in 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='04979v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='flu-dyn] 12 Jan 2023 While emphasizing magmatic roots, several workers considered magma buoyancy as the principal factor to elucidate the origin of axial-high topography [11, 121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' [31] provided a condition of the sub-ridge viscosity distribution required for buoyancy-driven axial high topography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' On the other hand, [82] predicted that mantle viscosities beneath the ridge must be at least two orders higher than the generally accepted values to form an axial valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' [109] also indicated viscosity as the key factor, but it is ultimately the plate velocity to regulate the sub-crustal density or viscosity that determine the axial morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Here, the most critical question is – how the plate velocity regulates the sub-crustal viscosity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' [24] showed from a 2D numerical model that the mantle viscosity at shallow depths (< 20 km) beneath the ridge should be low (∼ 1018 Pa s) to form axial high topography, but it should be high enough (∼ 1021 Pa s) to form a low axial relief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' According to their model, high- viscosity melt flows lower the hydrostatic pressure beneath the ridge, reducing the melt upwelling height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' However, none of these studies explicitly accounts for the viscosity effect of sub-crustal melt-rich zones on the axial morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' 2000 2400 2600 2800 2200 3000 B* B A A* EPR V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' = 2 C C* SEIR V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='5 B B* JDF V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' = 2 2300 2400 2500 2600 2700 2900 3000 2800 A A* C C* 2500 2300 2400 2700 2800 2900 2600 25 km 25 km 25 km Depth (m) Depth (m) Depth (m) (a) (b) } Suspension rheology + 1-2 km 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='1-1 km magma conduit melt/magma bodies Magma conduit/melt/magma bodies ~ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Mantle ~ 1 9 19 10 - 10 Pa s 10 Pa s Viscosity Values: Melt flow Oceanic upper crust Mantle Mush complex (MC) MOR axis 10 km 20 km Decompression melting stops at this level Figure 1: (a) Bathymetric profiles across the East Pacific Rise (EPR), Juan De Fuca (JDF) and South-Eastern Indian Ridge (SEIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' They show high (EPR-AA*), moderately high (JDF-BB*) and plateau dominated (SEIR-CC*) ridge-axis topography, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Data source: GeoMa- pApp (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='geomapapp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='org/)/CC BY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' (b) A conceptual cartoon diagram of the sub-ridge melt/magma settings considered for the topographic modelling in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The magma scale (Table1) covers narrow melt conduits and magma bodies containing suspended crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The mush complex (MC) represents a distinct zone consisting of melt bodies and conduits within a high- viscosity host rock matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The problem of sub-crustal melt transport mechanisms has recently rejuvenated the MOR re- search in new directions [14, 15, 32, 112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' It is now evident that melts start to localize in discrete zones during their ascent that eventually mediates for a heterogeneous magma supply to the ridge axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Earlier numerical models [74, 99] showed melt fraction as a function of spreading rates, sug- gesting that the melt fraction is substantially reduced from fast- to slow-spreading ridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Secondly, the melt upwelling processes participate in solidification at the shallow level to form isolated mushy bodies, as reported by many earlier workers [107,108,111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The crystal content in the mushy melts can largely vary depending on the degree of crystallization, and their varying relative volume ra- tios would determine the viscosity of the melt-bearing sub-ridge regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' [9] provided a depth-wise viscosity profile based on melt content (∼ 3%), dehydration, and grain boundary sliding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' This model predicts an increase in overall viscosity with height, mainly due to water extraction during partial melting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' However, later experimental studies suggested that such dehydration can hardly affect viscosity at shallower depths as partial melting generally ceases to occur at a deeper level [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The olivine rich high-fluid channels in subcrustal magma mush at a shallower depth [58] indicates crystal transport as suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' A detailed viscosity analysis of the sub-crustal regions containing 2 crystal-bearing melts beneath MORs, especially in view of the axial morphology, is yet to be fully explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' This article introduces a novel approach to model sub-crustal/lower-crustal (hereafter sub-crustal) viscosity and offers a viscosity-based explanation for the axial morphologies: highs and flat topogra- phy of MORs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=', East Pacific Rise, Juan du Fuca, and South East Indian Ridge (Figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='In the first step, we provide a series of systematic calculations of the effective viscosity of mush complex (MC) beneath the ridge axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The mush complex (MC) is defined here as a constitution of crystal- bearing-melts (with largely varying crystal contents) and host rocks [112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Our calculations consider the following parameters: the process-times, spatial magnitudes, and the constitution of sub-crustal materials (see the concept diagram, Figure 1b and Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' We then develop a three-dimensional fluid-structure interaction (FSI) approach to model the mechanical connection between the MC and the overlying crust at a mid-oceanic ridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The FSI model allows us to investigate how the viscosity of the sub-crustal mushy region can modulate the flat versus high MOR axial topography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' 2 Sub-crustal mush complexes: viscosity modelling 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='1 Mush complex in MOR settings At mid-oceanic ridges, the ascending melts produced by decompression melting − at a depth of around 40 km − focus to the ridge axis, forming large (∼ 30 km wide) melt-rich regions [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Seismic imaging and theoretical estimates indicate a wide variation in their partial melt content (∼ 10 - 70% at shallower levels, ≤ 10 km and 5 – 25% at deeper levels, ∼ 30 km) from one ridge to the other or different segments within the same ridge [7,51,80,100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' It is vital to assess how such variations in melt content can influence the mechanical strength of sub-crustal melt-rich regions (MC) at shallow depths and modulate the first-order ridge-axis topography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' In submarine systems, the temperature calculated for the critical depth of partial melting cessation constrains the amount of available melt in the subcrustal MC system [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' However, the melts ascend upward with a complex 3D pattern of their paths, determined by coupled convection-solidification processes [74, 99, 122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The volume fraction of melt-crystal aggregates goes up [43,51]as subcrustal magma bodies form at mid-oceanic ridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The plot shows a linear regression of the average melt fractions with depth (Figure 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' There can be large variations from the linear average at subcrustal regions due to significant spatial variations in the magma pool populations and their fractional crystallization beneath MORs(Figure 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Geophysical signatures, such as lower seismic velocities and high attenuation suggest the oc- currence of mushy zones beneath mid-ocean ridges [1, 121], containing suspension-rich melt bodies (super solidus) as well as subsolidus host rocks [112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Based on such sub-ridge mush-melt pat- terns reported in the literature [49, 59, 68, 106] [80] [79], we consider a mechanically distinct zone, mush complex (MC), treated as a continuum to implement the dynamic and kinematic coupling between the underlying mantle and the overlying elastic crust [15,122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' It is now a well-established fact that ascending mushy melts encounter the lithospheric base that acts as a melt barrier and forces the melts to focus into the ridge axis, forming a distinct melt-rich regime within the host rocks [49, 59, 68, 106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' From gravity anomaly data, Lin and Parmentier [68] detected a low-density zone at the base of the lithosphere at EPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Mckenzie and Bickle [80] discussed the occurrence of underlying hot sheets at the decoupling zone between the circulating mantle and the spreading plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' On the other hand, many geophysical studies found sub-crustal melt lenses, 1-2 km wide and 100 m thick, containing 30-40% crystals as suspension, in several ridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The lenses are extended horizontally up to 15-20 km with their melt content decreasing to 30%, forming spatially extensive mushy regions [13,107,113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' [107] recognized 3-4 km thick axial magma chambers at a depth of 3 km in slow-spreading, magma-rich Lucky strike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Fast and intermediate spreading ridges are reported to have magma bodies at shallower depths, ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='4 km in JdF [13] and ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='8 km in EPR [29], where their maximum thickness is ∼ 4 to 6 km [29,60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' In contrast, slow ridges generally lack such distinct magma bodies, but have mushy regions (low-velocity zones) at the crustal base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Dunn et el.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' [30] reported a 6 km thick mushy zone of sparse melt channels at a depth of 4 km at MAR (35◦N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Besides melt pockets, which are prevalent in fast-spreading ridges, discrete melt channels in the lower crust and sub-crustal axial zones [13,30] also constitute a typical feature (low-velocity mushy regions) at the crustal base, which is also considered as a part of the MC [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Based on the available reports on axial melt lenses and melt-rich bodies beneath mid-ocean ridges, the vertical extent of the mush complex (MC) was chosen in the present model in sub-crustal regions and in the uppermost mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The MC was allowed to evolve with progressively deforming overlying elastic upper crust under basal stresses that eventually decreased the axial depth and 3 Log strain rate (ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Log suspension viscosity Melt with crystals Melt (b) (a) Figure 2: (a) Variation of melt fractions with depth (plots based on available data in literature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Except Hewitt [51] and McKenzie and Bickle [80], all the data are taken from axial melt lenses, magma chambers and melt conduits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' In the plot, these data points are complemented with rocks (1% melt, [112]), marked in deep blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Red straight line shows the overall regression trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Shaded area delineates the depth range (2-8 km) of evolved MMC, where the average melt suspension fraction in MC is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='3 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' (b) The plot shows the variation of suspension viscosity as a function of strain rate and characteristics of the melt suspension (modified from [43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' increased the MC thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The MC is modelled with a triangular cross-section, describing an along-axis prismatic area in the lower crust, with a maximum thickness of 4 km beneath the ridge axis (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The upper crust (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=', solid elastic crust) at the axis is chosen 4 km thick in the initial model setting, where the MC vertically covers the lower crust and a part of the topmost mantle region (detailed in, Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The reason for choosing a larger MC depth, as compared to the available data in our initial model is that the solid crust progressively thins by elastic strains during the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' For example, the initial crustal thickness at the axis is reduced by 2 km to finally set the MC depth and thickness at 2 km and 6 km, respectively [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='2 Melt-viscosity modelling: parametric considerations The viscosity of melts and magmas at shallow depths is historically modelled within a framework of suspension rheology, following the landmark work of [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' However, natural suspensions show viscous behaviour more complex than that predicted from Einstein’s theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The complexity originates primarily from the effects of additional factors, such as packing and shapes of solid particles in suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The packing of solid components in the liquid phase is an influential factor to modify the effective viscosity of a mixture under the same solid volume fraction [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The packing vis-a-vis viscosity, depends significantly also on the solid particle size distribution in the liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' For example, a bimodal size distribution with increasing size ratios up to a threshold point lowers the effective viscosity [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Chang and Powel [20] showed that the viscosity of a suspension decreases initially with increasing smaller particle volume fraction and then increases monotonously after reaching a critical volume fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Liquid suspensions attain their maximum packing fraction in the case of multimodal size distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Such multimodal (trimodal and tetramodal) particle packing increases viscosity higher than those for bimodal and unimodal distributions [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' This study also suggests that packing with different particle sizes yields a polydispersion effect on the bulk viscosity of suspensions for the same particle volume fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Polydispersity allows smaller particles to pack more closely by forming layers between larger particles or by occupying the void spaces between larger particles [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Such 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='0 Depth from seafloor (km) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='15 20 Hewitt(2010) 25 Arnulfetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' (2018) Mainprice（1997) Xuetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' (2014) McKenzie and Bickle (1988) Gonnerman andManga (2007) 30 Rock 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='0 Melt fractionpolymodal and polydisperse particle (heterogeneous packing) distributions can thus significantly enhance the bulk viscosity of melt suspensions in a mushy region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Petrological and compositional data indicate mineral phases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=', olivine, plagioclase and clinopy- roxene can crystallize in partially molten zones at successive stages during the melt ascent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' However, their depth correlation becomes weak with decreasing spreading rate [62,67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Volcanic studies sug- gest that the polydispersity characteristics of mushy melts and their variations hold a connection with the spreading rates at MORs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Mushy melts can strongly differ in their crystal and bubble contents depending upon the spreading rates [32, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Slow-spreading ridges generally show larger compositional variations in erupting lavas than the fast-spreading ridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Higher degrees of composi- tional homogenization in the fast-spreading ridges are commonly attributed to the magma chamber processes [88], in contrast to slow-spreading ridges, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=', the Mid Atlantic Ridge (MAR), which are devoid of large, stable magma chambers and involves fractional melting throughout the whole melting regime to produce melts with a strong compositional variability [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' It is noteworthy that magma chamber processes act as potential sites for hot mafic magma replenishment into the cold silicic resident magmas (Figures 3a and 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Several workers have reported mafic enclaves from volcanic rocks as an evidence of the magma replenishing process (Figure 3a) [26,77,83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Cold resident magma Hot repleneshing magma Solid crystal t� t� T (°C) T (°C) T (°C) T (°C) t� t� (a) (b) Figure 3: A cartoon presentation of two different replenishment dynamics in the subcrustal magma- chambers (adapted from [77]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' (a) Volumetrically small and slow magma replenishment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' In this case a thin layer of groundmass crystallization forms, and the thermal gradient between the hot emplaced magma and the host mafic magma is dissipated and reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' (b) Fast and large magma replenishment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' In this case, the hot magma disperses into a magma chamber like fountains before crystallizing to form enclaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Moreover, a large volume of replenishing magma causes a thicker hot layer, which sustains the thermal gradient for a longer period with convective churning and multimodal crystal enclaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The effects of replenishment dynamics on magma chamber processes depend mainly on the magma flow conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Turbulent fountain-type injections produce dispersed spherical enclaves of crystals, whereas a non-turbulent slow influx of hot magma (Re ≤ 400) (Figure 3) results in ponding of groundmass crystals at the magma-chamber base [26,77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' [77] also suggested that the more replenishing magma volume flux, the larger olivine phenocrysts would form, resulting in 5 convection-driven churning within the magma chamber due to a longstanding steeper temperature gradient (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' However, recent geochemical studies focus on the general mushy environment as a host of crystallization and magma mixing [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Geochemical investigations of crystal-bearing enclaves within erupted lavas in volcanic settings allow us to recognize several factors controlling shallow magmatic environments beneath mid-oceanic ridges: a) occurrence of magma chamber, b) melt-upwelling velocity, and c) volume in the mushy regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' These factors determine whether magma-dominated, volatile environments can significantly influence the dispersion and multimodal- ity characteristics of solid suspensions in hot magmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' In contrast, magma-poor mushy regions, consisting of small and unstable non-convecting magma chambers, can form dominantly monodis- persed microlith ponds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Petrogenetic analysis of a recent East Pacific Rise (EPR) eruption suggests the presence of sub- ridge crystal-bearing mushy regions, which are thought to be a product of consistent replenishment (Figures 3a and 3b) by evolved magmas from a deeper level source [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Rubin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=', [98] reported homogeneity in lava basalts from fast ridges [Heterogeneity Index (HI) ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='6 for spreading rate > 10 cm/yr], but strong heterogeneity from slower ones [HI ∼ 3 for spreading rate < 4 cm/yr].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' They accounted for the relative thermal stability to explain the higher degree of homogeneity in the fast ridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Phyric basalts containing mushy zone crystals suggest the injection of primitive magmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' However, the petrological estimates of mid-ocean ridge basalt (MORB) at MAR point to larger volumes of aphyric basalt, representing a collection of aphyric magmas within a mushy zone at shallow depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The magmas originated from convection-assisted melt segregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' [66] calculated mush viscosity [41] as a function of plagioclase phenocryst content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' According to their estimate, the viscosity of melt suspensions can increase by eight times with an increase in small-size (∼ max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' 10 mm) plagioclase phenocryst fraction, where the crystallinity is 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Their analysis predicts the maximum size of olivine phenocrysts in erupting plagioclase-ultraphyric-basalts (PUB) in the range of 1 to 3 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' They also explain the presence of PUB selectively in slow and intermediate ridges as a consequence of olivine phenocryst segregation in conduits during the magma ascent rather than in the magma chambers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' [66] hypothesized that during the ascent of melt-crystal aggregates through conduits, the melt to crystal ratio is high, and their bulk viscosity is thereby low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' However, it is hard for low-viscosity magmas to transport crystals without segregation in the conduit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Going by these arguments for the petrogenesis of plagioclase-phyric basalts, we infer that the bulk viscosity of magmas at the time of their ascent through conduits must be low in cases of slow and moderately spreading ridges, allowing extensive crystal segregation to produce monomodal crystal packing with little or no polydispersity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' In contrast, the melt suspension viscosity in fast-spreading ridges, or ridges with prominent magma chambers, should be high due to greater polydispersity and polymodality even the melt volume fraction is relatively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The polydispersity variation in sub-ridge magmatic processes is thus an influential factor to modulate the melt suspension viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Based on the preceding discussions of melt characteristics, we thus consider the packing factors in the viscosity analysis of crystal-bearing melts in mushy regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Table 1: Model Parameters used in determining the viscosity scale Domain Properties Lava Scale Conduit length (lc) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='1 km;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Conduit diameter (dc) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='1 km;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Transmitted volume (vc) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='01 km3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Strain rate ( ˙ε)= 10−5s−1 Transmitting time (tc) = 5 hrs Magma Scale Conduit length (lc) = 1 km;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Conduit diameter (dc) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='1 km;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Transmitted volume (vc) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='01 km3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Strain rate ( ˙ε)= 10−9s−1 Transmitting time (tc) = 13 yrs Lava Scale Conduit length (lc) = 10 km;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Conduit diameter (dc) = 1 km;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Transmitted volume (vc) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='01 km3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Strain rate ( ˙ε)= 10−14s−1 Transmitting time (tc) = 100 hrs 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='3 Mush complex (MC) viscosity: a two-step calculation We calculate the viscosity of mush complex (MC) in two steps: 1) viscosity calculation of crystal- bearing melts, based on the theory of suspension rheology, called melt suspension, and 2) viscosity of MC (host rock + melt suspensions), based on the theory of two-phase fluid mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' To calculate the viscosity of a crystal-melt aggregate, we assume the solid (crystals) component as a suspension in the liquid matrix, as described in the preceding section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Again, the solid part in suspension is treated as rigid particles and the liquid part as a continuous, viscous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The calculation is carried out using the equations and calculations found in the literature for erupted lavas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Figure 1b provides a cartoon diagram to show its conceptual framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='Melt fraction (wt%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='Effective viscosity of MC ( log µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=') (Pa s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='eff ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='Melt fraction (wt%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='Viscosity of melt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='(log µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=') (Pa s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='melt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='Effective viscosity of MC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='( log µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=') (Pa s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='eff ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='Melt fraction (wt%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='(d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='Effective viscosity of MC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='( log µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=') (Pa s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='eff ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='Viscosity of melt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='(log µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=') (Pa s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='melt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='Effective viscosity of MC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='( log µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=') (Pa s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='eff ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='Viscosity of melt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='(log µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=') (Pa s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='melt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='Effective viscosity of MC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='( log µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=') (Pa s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='eff ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='Viscosity of melt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='(log µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=') (Pa s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='melt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='Melt fraction (wt%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='Figure 4: Three-dimensional plots of the effective viscosity of MC (µeff) as a function of melt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='suspension viscosity (µM) and molar volume fraction (φ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' obtained from the two-stage viscosity calculations for increasing values of the cohesion parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' (a) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='6, characterized by convex surface plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' (b) α = 1 (an ideal situation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' (c) and (d) α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='4 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Note the transformation of convex to concave curvature of the surface plot with increasing α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Insets show the full-length plots, highlighting the effective viscosity range of 1010 to 1018 Pa s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Two limiting effective viscosity (µeff) values, 1012 Pa s and 1014 Pa s are shown to constrain the viscosity range in our FSI model to reproduce the spectrum of axial high to flat topography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Host rock viscosity is chosen 1019 Pa s in these calculations, emulating mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' 7 α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='6 α= 1 18 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' 17 17 ~ 16 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Effective viscosity of (s ed) (n Bo) sn mush (log Hefr) (Pa s) Effective viscosity of 15y 14 13 12 、 12 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' 11 10岁 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='5 g viscosity Ato 7 5 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='1 0 magma fraction magma fraction α= 2 α= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='4 53 1、 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='5， 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' 18 17 、 17 16、 16 ~ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Effective viscosity Effective viscosity mush (log Hef) (Pa s 14 13 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' 12、 11↓ 10 10 7 5 5 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='9 1 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='7 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='3 0 magma fraction magma fraction 16 18 2 6 8 10 12 14 Effective viscosity (log μueff) (Pa s)18 16 14 12 10 8 6 4 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='1 Melt suspension viscosity We introduce a scale of the magmatic process with characteristic lengths, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='01 to 1 km, as applicable to shallow level magma conduit dimensions (diameter and length) in the sub-ridge region [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Lava eruption episodes determine the characteristic time through the conduits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' According to the Volcanic Explosivity Index (VEI) [85] study, covering more than 75% of the documented Holocene eruptions, almost 50% of them record continuous blast duration of less than 6 hrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Furthermore, 63% of the eruptions yield eruptive volumes ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='001 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='1 km3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Considering the median of time intervals, successive eruptions of a volcano is found to occur in a time-frequency of 13 years [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' We use these data to calculate the strain rates associated with magma flows in shallow conduits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' To simplify the calculation, we consider 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='001- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='1 km3 magma undergoing eruption through a cylindrical conduit of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='01-1 km diameter and on a time duration of 13 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' This wider range of diameters and conduit lengths is chosen because eruptions generally occur through multiple magma conduits of varying lengths, and a cumulative effect of the aforesaid parameters is required in our present analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' For further simplification, we deal with a representative set of their values, where the magma conduit length (lc) and diameter (dc) are 1 km and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='1 km, respectively, and the erupted magma volume (vc) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='01 km3, which passes through the conduit on a time scale (tc) of 13 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' This set of values yields an average characteristic strain rate (ε ′) of shallow-conduit upwelling of magma, ε ′ = � vc/(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='25πdc 2lctc) � = 10−9s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The characteristic strain rate ε ′ can range from 10−6 to 10−12s−1 if the parametric values were varied to cover the entire range discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' It is noteworthy that this strain rate scale supports suspended crystals to move passively within the melt phase [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' It is necessary to treat lava eruption on a different time scale (designated as lava scale), which represents the duration of a continuous flow event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' These events usually take place in a duration of 1 to 12 hours [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' We thus choose an average value of 5 hours to represent the lava scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Considering the lava conduit length and cumulative diameter as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='1 km and the erupted magma volume as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='01 km3, we obtain the characteristic strain rate 10−3s−1 (Table1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The suspension parameters, polymodality, and polydispersity can increase the viscosity of melt suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Now, we calculate the degree of viscosity increase possible in melt suspensions under a sub-crustal environment at MORs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Several studies have provided empirical relations to express the melt viscosity as a function of suspension properties [25,27,34,64,75,81,104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Consider first the effect of crystal content in melts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Costa [27] enumerated crystal-free melt viscosity, µl = 105 Pa s at a temperature of 800◦C and a pressure of 300 MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The effective viscosity of melts increases with increasing solid volume fraction (φs) in the suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' [76] suggested that melts erupt as lava with a maximum viscosity, µM = 107Pas corresponding to φ(s,c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='55, called a critical solid fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' However, [114] suggested that the critical solid fraction can be further large, φ(s,c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='6 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='7 at the time of lava eruption, implying crystal-bearing melt viscosity, µM = 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='5 Pa s, which means the enhancement of suspension melt viscosity by an order: I1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='5 [27,43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Polydispersity (δ) is a measure of the size variation of suspended solid particles in magma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' For packing with particle distribution on radii, P(R), the parameter can be expressed as, δ = �� ∆R2� ⟨R⟩ (1) where δR = R − ⟨R⟩, and the moments of R is defined by Rn > � RnP(R)dR [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' It is noteworthy that an increase in δ allows the suspension to increase the maximum limit of critical solid fraction � φ(s,c) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The polydispersity, in turn, multiplies the suspension viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The maximum packing ratio of mono-dispersed spheres accommodates a maximum solid fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='64, which can increase to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='75 for suspensions with a polydispersity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='65 [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Roscoe [96] derived a couple of equations using experimental results [33,117], showing that various size distribution of rigid spheres influences the viscosity of suspensions less than a uniform size distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' However, we consider here the theoretical work of Klein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' [63], who showed that polydispersity would steeply increase the maximum packing fraction after a threshold limit for monomodal size distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' This packing effect results in an exponential increase of the suspension viscosity and multiplies its magnitude 40 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Moreover, the experimental study suggested that an increase in crystal polydispersity might augment volcanic lava viscosity up to 3 orders of magnitude at a higher deformation rate [81, 92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' We thus consider the maximum viscosity enhancement in the order, I2 = 3, corresponding to the polydispersity of crystal-bearing melts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Strain rate is another factor in our viscosity calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Experimental studies suggest Newtonian melt rheology prevails at strain rates lower than 10−5s−1 [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' But, at higher strain rates, the melts develop shear thinning behaviour [118], which reduces the viscosity by more than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='5 orders in case of larger solid fraction � φ(s,c) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='8 � [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Considering the strain rates in the order of 10−6 to 10−12 8 s−1 on the magma scale and 10−3 s−1 on the lava scale, we choose a maximum viscosity enhancement in the order, I3 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='5, solely due to the decreasing strain rate, leaving out other variables [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' To summarize, we use a suspension factor (I), taking into account the cumulative effects of solid crystal fraction (I1), size distribution (polydispersity) (I2), and strain rate (I3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Considering pure melt viscosity in the order of 105 Pa s, as an example, the suspension viscosity (µM) can be enhanced to a maximum extent of 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='5 Pa s for a limiting solid fraction (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='6 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='7), implying that I1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='5 [27,43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='On the other hand, an increase in crystal polydispersity can multiply µM by an order of 103 at a higher deformation rate [81,92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' We thus consider I2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Finally, for the strain rate effects, µM can multiply by a factor of 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='5 depending on the variation of strain rates in the range 10−3 to 10−12 s−1, as applicable to the MC in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' That means, I3 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Taking their net effects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=', I1 + I2 + I3), we obtain I = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='2 Viscosity of mush complex We are now estimating the viscosity (µeff) of mush complexes (MC), using the theory of mixture rheology within a framework of continuum mechanics [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Consider a mixture of host rock (µR = 1019 Pa s) and melt suspensions (µM = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='5 − 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='5 Pa s), the effective viscosity of MC (µeff) can be expressed by the Lederer-Roegiers equation for a two-phase liquid system as, ln µ12 = x1 x1 + ax2 ln (µ1) + (ax2) x1 + ax2 ln (µ2) (2) where α is a constant used to represent the difference in intermolecular cohesive energy between the participating two components, 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' xi and µi (i = 1, 2) are the mole fraction and the viscosity of ith component in the mixture, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The Lederer-Roegiers equation provides an accurate viscosity calculation of multi-phase fluids with contrasting component viscosities [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Equation (2) is close to the Arhenius equation, which can be demonstrated from Roegiers and Zhmud’s [93] approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Fluidity (inverse of viscosity) of a fluid phase depends on the molar flow activation energy, ∆E (a measure of intermolecular cohesion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The Arrhenius relation describes the fluidity in the framework of Eyring’s rate process theory ( [42]) as, 1 µ = K �h exp �∆E RT � (3) which leads to, ln µi = C1 + ∆Ei RT (4) where C1 is a constant, �h is Planck’s Constant, T is absolute temperature, R is the universal gas constant, and K is the ratio of molar volume and Avogadro number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Subscript i refers to the fluid component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' For a two-phase liquid system, we consider an additive principle to find the net activation energy of the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' According to Eyering’s Rate Process theory of viscosity [42], the relative motion of one fluid layer over the other demands a molecule to overcome a potential-energy barrier, called flow activation energy per molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The total flow activation energy is obtained by taking a product of this quantity with the number of molecules in the system, neglecting any energy dissipation during the molecular transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Based on this assumption, the total flow activation energy follows, ∆E12 = x1∆E1 + x2∆E2 (5) Using equations (4) and (5), we arrive at the Arrhenius equation for the binary mixture viscosity, ln µ12 = x1 ln µ1 + x2 ln µ2 (6) Equation (5) can be generalized with an asymmetric mixing rule (Roegiers and Zhumd 2011) for the flow activation energy: ∆E12 = (1 − γ)x1 (1 − γ)x1 + γx2 ∆E1 + γx2 (1 − γ)x1 + γx2 ∆E2 (7) where 0 < γ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' For γ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='5, the contribution of component 1 to the flow activation energy is greater than that of component 2, and vice-versa for γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Using equations (4) and (7), we obtain the Roegiers equation (2) by replacing α = γ/(1 − γ) in equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' α = 1 implies an equal contribution of flow activation energy by the components, whereas α ̸= 1 indicates their unequal 9 contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' For asymmetric two-liquid mixtures, Roegiers and Roegiers [95], and Roegiers [94] considered α as the ratio of the specific intermolecular attraction energies of the components to derive Equation (2), where α was held constant for an ideal binary system at a particular temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The equation, validated experimentally by Roegiers [94], and later tackled analytically by Zhmud [123] yields α as the ratio ln(µ12/µ1)/ ln(µ2/µ12) for a two-phase system with equal mole fraction of the participating components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' In the foregoing analysis we use equation (2) with µ1 = µR and µ2 = µM, x2 = φ (molar volume fraction of melt suspension, and µ12 = µeff (MC viscosity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' A set of 3D graphical plots presents the calculated µeff as a function of φ and µM for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='4 and 2 (Figures 4a-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' All of them show an inverse relation of the MC viscosity (µeff) with melt volume fraction (φ) and suspension viscosity (µM), as widely reported in the literature [27,43,81], for the entire range of α values considered in the present calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' µeff is reduced by two orders (1014 to 1012 Pa s) depending on the φ and µM variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' Our model calculations suggest that µeff can increase with suspension melt fraction in specific conditions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=', suspensions with large volume fractions of crystals, as observed in magmatically robust ridge settings at fast spreading ridges where magmas are extremely enriched with crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' This model provides the MC viscosity estimates also in opposite environments in slow spreading ridges, characterized by magma poor and low in crystal content, where crystals readily settle down in the course of magma ascent [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' 3 Axial topography: fluid-structure interaction (FSI) mod- elling 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='1 Numerical methods This model couples the three-dimensional convective melt upwelling in the melt-bearing mantle part (fluid region) with the overlying elastic layer (oceanic crust) (Figure 5) in the framework of a Fluid-Structure Interaction (FSI) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The fluid sub-problem is tackled using the finite volume computational dynamics code Fluent®, where the CFD model idealizes the mechanical setting as a two-layer system: basal layer (uppermost mantle part), thermomechanically coupled with an overlying high-viscosity layer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=', elastic solid crust) (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The model base is subjected to thermal perturbations to simulate thermo-chemical convection with synchronous Darcy’s (porous melt flows) and crystallization (phase transition) [74, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' We then take an average of the three- dimensional velocity data, calculated at the interface above the MC region, and use as the fluid structure interface velocity to set a mechanical (FSI) coupling of the fluid domain with the top elastic crust in the finite element (FE) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' This FSI coupling principally aims to reproduce finite deformations in the crustal layer, which otherwise cannot be implemented through the control- volume based fluid simulations used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' It is noteworthy that this combined fluid-solid (CFD-FSI-FE) modelling approach (Figure 5) geophysically conceptualizes the MC (prismatic sub- axial zone) as a control volume that conserves mass and momentum by a combination of material influx from below and solidification / recycling / eruption, and partly outflux across the top surface [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The melts in the fluid domain ascend with a complex heterogeneous pattern due to the 3D convection structures (Figure 6b), and consequently form ellipsoidal magma pockets with circular plan views, as seen in the FE results (see, Figure 8a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The mechanical properties of MC are allowed to evolve with the convection in the overall fluid domain, but maintaining both the mass (continuity) and momentum conservations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The theoretical framework of convection in sub-ridge fluid domains is developed on the following conservation equations: continuity, momentum and energy equations, where we introduce a number of source terms: Darcy and buoyancy source terms in the momentum equation, and an enthalpy source term in the energy equation [10,115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The continuity, momentum and energy equations are finally expressed as follows, ∇ · v = 0 (8) ρ ∂ ∂tv + ρv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='∇v = −∇p + µfd∇2v + Sg + SD (9) ρ ∂ ∂t(ρh) + ∇ · (ρvh) = ∇a∇h − Sh (10) where p, ρ and µfd denote pressure, density and viscosity of the fluid domain, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' T, h and a represent temperature, enthalpy, and thermal diffusivity (a = k/ρc, k and c are the thermal conductivity and specific heat, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The fluid velocity, v, is chosen to vary linearly with the 10 melt fraction, φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' In this single-phase idealization, the domain viscosity µfd is varied as a power-law function of temperature [22,99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' In the momentum equation 9 SD regulates the dominance of Darcy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=', porous) flow, whereas Sg implements the buoyancy factor through Boussinesque approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' In equation10 acts as an enthalpy factor to incorporate the energy involved in the phase (solid-melt) transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The mathematical expressions of these source terms are, SD = −C (1 − φ)2 (φ3 + ε)v (11) Sg = ρgθ∆T (12) Sh = ∂ ∂t(ρ∆H) + ∇ · (ρv∆H) (13) C and ε in equation11 are constants, whose values are taken as 1e5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content='001 respectively, after [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' In equation12, ∆T represents temperature fluctuations with respect to the reference temperature, and θ is the co-efficient of thermal expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' In equation13, ∆H is the mean latent heat content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' The fluid subdomain base is subjected to a random thermal perturbation (RTP) condition, which aims to initiate convective flows in the sub-crustal region ( [99]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE4T4oBgHgl3EQfQAz9/content/2301.04979v1.pdf'} +page_content=' In this random thermal condition partial melting occurs in the domains of high temperatures (>solidus), whereas solidification in the domains of low temperatures ( 1 be a terminal stage. +For stage k = +0, 1, . . . , N − 1, consider an investor forming a portfolio consisting of m ≥ 2 assets and assume +that at least one asset is riskless with a rate of return rf ≥ 0. That is, if an asset is riskless, +its return is deterministic and is treated as a degenerate random variable with value X(k) = rf +for all k with probability one.2 Alternatively, if Asset i is a risky asset whose price at time k +is Si(k) > 0, then its per-period return is given by Xi(k) = Si(k+1)−Si(k) +Si(k) +. In the sequel, for +risky assets, we assume that the return vectors X(k) := [X1(k) X2(k) · · · Xm(k)]T have a known +distribution and have components Xi(·) which can be arbitrarily correlated.3 We also assume +that these vectors are i.i.d. with components satisfying Xmin,i ≤ Xi(k) ≤ Xmax,i with known +bounds above and with Xmax,i being finite and Xmin,i > −1. The latter constraint on Xmin,i +means that the loss per time step is limited to less than 100% and the price of a stock cannot +drop to zero. +2.1 +Linear Policy and Unit Simplex Constraint +Consistent with the literature, e.g., Barmish and Primbs (2015); Hsieh et al. (2020, 2018a,b); +Zhang (2001); Primbs (2007), we consider a linear policy with a weight vector K ∈ Rm. Let +V (k) be the investor’s account value at stage k and the weight for Asset i is given by 0 ≤ Ki ≤ 1 +represents the fraction of the account allocated to the ith asset for i = 1, . . . , m. Said another +way, the policy for the ith asset is of a linear form ui(k) := KiV (k). Note that the number of +shares invested on the ith asset is ui(k)/Si(k). Since Ki ≥ 0, the investor is going long. In view +2In practice, the actual distribution of returns may not be available to the investor, but one can always estimate +it and work with the empirical surrogate. +3Again, if the ith asset is riskless, then we put Xi(k) = rf ≥ 0 with probability one. If an investor maintains +cash in its portfolio, then this corresponds to the case rf = 0. +3 + +of this, and given that there is at least one riskless asset available, we consider the unit simplex +constraint +K ∈ K := +� +K ∈ Rm : Ki ≥ 0 for all i = 1, . . . , m, +m +� +i=1 +Ki = 1 +� +(1) +which is classical constraint in finance; e.g., see Cvitanic and Zapatero (2004); Cover and Thomas +(2006); Luenberger (2013); Cuchiero et al. (2019); Hsieh (2021). With K ∈ K, we guarantee +that 100% of the account is invested. +Remark 2.1. In the finance literature, it is known that long-only constraints like (1) can be +used to mitigate the overconcentration of weight. These constraints can also assist in containing +volatility and trading performance; see Jagannathan and Ma (2003). +2.2 +Frequency Dependent Account Value Dynamics with Transaction +Costs +Letting n ≥ 1 be the number of steps between rebalancings, at time k = 0, the investor be- +gins with initial investments u(0) = �m +i=1 ui(0) with ui(0) := KiV (0) with Ki being the ith +component of the portfolio weight K satisfying constraint (1). It is worth mentioning that the +investment level ui(0) can be converted to the number of shares by dividing it by the price Si(0); +i.e., ui(0)/Si(0). The investor then waits n steps in the spirit of buy and hold. When k = n, the +investment control is updated to be u(n) = �m +i=1 KiV (n). Continuing in this manner, a waiting +period of n stages is enforced between each rebalance. +To incorporate the transaction costs into the frequency-dependent framework, let ci ∈ [0, cmax] +be a percentage transaction costs imposed on Asset i where cmax ∈ (0, 1) is a predetermined +maximum transaction cost.4 That is, at stage k = 0, if one invests ui(0) at Asset i, then the +associated transaction costs in dollar is ui(0)ci ≥ 0. Then the dynamics of account value at +stage n ≥ 1 is characterized by the following stochastic recursive equation:5 +V (n) = V (0) + +m +� +i=1 +ui(0) +Si(0)(Si(n) − Si(0)) − +m +� +i=1 +ui(0)ci. +(2) +In the sequel, we may sometimes write VK(n) instead of V (n) to emphasize the dependence on +portfolio weight K. +Remark 2.2 (Transaction Costs). If there are no transaction costs; i.e., ci := 0 for all i = +1, 2, . . . , m, then the account value dynamics (2) reduces to the existing formulation in Rujeer- +apaiboon et al. (2018); Hsieh et al. (2018b); see also the frictionless market setting discussed +in Merton (1992). +4Nowadays, while some online brokerage services offer fee-free trades for certain exchange-traded funds (ETFs) +in the United States, transaction costs are typically required. For example, trading on the Taiwan Stock Exchange +typically incurs a transaction cost of α · 0.1425% of the trade value for some α ∈ (0, 1). As a second example, +using professional broker services such as Interactive Brokers Pro., may incur a fee of $0.005 per share, with a +minimum fee of $1 dollar and a maximum fee of 1% of the trade value. +5At stage ℓ ≥ 0 and rebalancing period n ≥ 1, the stochastic recursion of account value becomes +V (n(ℓ + 1)) = V (nℓ) + +m +� +i=1 +ui(nℓ) +Si(nℓ)(Si(n(ℓ + 1)) − Si(nℓ)) − +m +� +i=1 +ui(nℓ)ci. +4 + +2.3 +Frequency-Dependent Optimization Problem +Following previous research in Hsieh et al. (2018b); Hsieh (2021), to study the performance +which is dependent on rebalancing frequency, for i = 1, 2, . . . , m, we work with the n-period +compound returns for each asset i, call it Xn,i, defined as +Xn,i = Si(n) − Si(0) +Si(0) +. +It is readily verified that Xn,i = �n−1 +k=0(1 + Xi(k)) − 1 and −1 < Xmin,i ≤ Xn,i ≤ Xmax,i where +Xmax,i := (1 + Xmax,i)n − 1 and Xmin,i := (1 + Xmin,i)n − 1 > −1 for all n ≥ 1. In the sequel, we +work with the random vector Xn having ith component Xn,i. +Now for any rebalancing period n ≥ 1, we define the expected logarithmic growth (ELG) +gn(K) := 1 +nE +� +log VK(n) +V (0) +� +. +Our goal is to solve the following frequency-dependent stochastic maximization problem: +sup {gn(K) : K ∈ K} +(3) +s.t. +V (n) = V (0) + +m +� +i=1 +ui(0) +Si(0)(Si(n) − Si(0)) − +m +� +i=1 +ui(0)ci +where K is the unit simplex defined previously in Equation (1). The following lemma shows +that maximizing the frequency-dependent ELG with nonzero costs is indeed solving a concave +program. +Lemma 2.1 (ELG Optimization as a Concave Program). Fix n ≥ 1 and ci ∈ (0, 1). +The +frequency-dependent ELG optimization problem (3) is equivalent to +max +� 1 +nE +� +log(1 + KT � +Xn) +� +: K ∈ K +� +. +(4) +where � +Xn is a vector with the ith component given by � +Xn,i := Xn,i −ci. Additionally, Problem (4) +is a concave program. +Proof. We begin by observing that the account value dynamics +V (n) = V (0) + +m +� +i=1 +ui(0) +Si(0)(Si(n) − Si(0)) − +m +� +i=1 +ui(0)ci += V (0) + +m +� +i=1 +KiV (0) +�Si(n) − Si(0) +Si(0) +� +− +m +� +i=1 +ui(0)ci += V (0) + +m +� +i=1 +KiV (0)Xn,i − +m +� +i=1 +ui(0)ci += (1 + KT Xn)V (0) − +m +� +i=1 +ui(0)ci += (1 + KT � +Xn)V (0) +(5) +5 + +where � +Xn is a vector with the ith component given by � +Xn,i := Xn,i − ci. Hence, it follows that +gn(K) = 1 +nE +� +log VK(n) +V (0) +� += 1 +nE +� +log(1 + KT � +Xn) +� +. +Therefore, the original Problem (3) reduces to max +� +gn(K) = 1 +nE +� +log(1 + KT � +Xn) +� +: K ∈ K +� +. +The supremum operator is replaced by the maximum since gn(K) is continuous in K over a +compact domain K. Hence, the Weierstrass extremum theorem; see Rudin (1976), guarantees +that the maximum is attained. To complete the proof, it remains to show that Problem (4) is a +concave program. This is accomplished by a standard convexity argument. Since 1 + KT � +Xn is +affine in K, taking the logarithm function yields a concave function. Moreover, taking the expec- +tation and multiplying a scaling factor 1/n preserve the concavity; see Boyd and Vandenberghe +(2004). Therefore, the objective function 1 +nE +� +log(1 + KT � +Xn) +� +is a concave function in K. On +the other hand, K is a unit simplex which is a convex compact set. Therefore, the maximization +considered in Problem (4) is a concave function over a convex compact set, hence, is a concave +program. +Henceforth, we denote g∗ +n as the optimal expected logarithmic growth associated with the +given rebalancing period of length n. A vector K∗ ∈ K ⊂ Rm satisfying gn(K∗) = g∗ +n is called +a log-optimal weight. The portfolio that uses the log-optimal fraction vector is called frequency- +dependent log-optimal portfolio. +2.4 +Dominance Lemma with Costs +In this section, a version of the dominance lemma with costs is stated below. +Lemma 2.2 (Dominance). Given a collection of m ≥ 2 assets, if Asset j satisfying +E +� +1 + � +Xn,i +1 + � +Xn,j +� +≤ 1, +for all i ̸= j with i, j ∈ {1, 2, . . ., m}, then, for all n ≥ 1, gn(K) is maximized by K∗ = ej +where ej is the unit vector in the jth coordinate direction. +Proof. To prove K∗ = ej, it suffices to show that gn(K) ≤ gn(ej) for K ∈ K. For notational +convenience, we work with the random vector �Rn := � +Xn + 1 where 1 := [1 1 · · · 1]T ∈ Rm. +Since KT 1 = 1 for K ∈ K, it follows that gn(K) = 1 +nE[log KT �Rn]. Hence, by applying Jensen’s +6 + +inequality to the concave logarithmic function, we obtain +gn(K) − gn(ej) = 1 +nE +� +log KT �Rn +�Rn,j +� +≤ 1 +n log E +� +KT �Rn +�Rn,j +� += 1 +n log +� m +� +i=1 +KiE +� �Rn,i +�Rn,j +�� += 1 +n log +� m +� +i=1 +KiE +� +1 + � +Xn,i +1 + � +Xn,j +�� +≤ 1 +n log +� m +� +i=1 +Ki · 1 +� +≤ 1 +n log 1 = 0 +where the second last inequality holds since E +� +1+ � +Xn,i +1+ � +Xn,j +� +≤ 1 and the last inequality holds +since �m +i=1 Ki = 1. Therefore, gn(K) ≤ gn(ej). +Remark 2.3. Lemma 2.2 indicates that, under certain conditions, an optimal log-optimal in- +vestor must invest all available funds in a specific asset when transaction costs are present. This +result can be viewed as an extension of the Dominant Asset Theorem in Hsieh (2021) to include +transaction costs. To see this, consider the case where there are no costs; i.e., ci = 0 for all i, +then � +Xn,i = Xn,i. This implies that the ratio +E +� +1 + � +Xn,i +1 + � +Xn,j +� += E +�n−1 +� +k=0 +1 + Xi(k) +1 + Xj(k) +� += +� +E +� 1 + Xi(0) +1 + Xj(0) +��n +, +where the last equality holds since Xi(k) are i.i.d. in k. Thus, the condition E +� +1+ � +Xn,i +1+ � +Xn,j +� +≤ 1 +reduces to a much simpler condition E +� +1+Xi(0) +1+Xj(0) +� +≤ 1, which is consistent with the Dominant +Asset Theorem proved in Hsieh (2021). +3 +An Approximate Log-Optimal Portfolio Problem with +Costs +When the transaction costs are present, the corresponding fee-adjusted return is given by � +Xn,i = +Xn,i − ci. The next lemma provides a sufficient condition for ensuring that the trades survive up +to stage n. +Lemma 3.1 (Probability of Having Survival Trades under Transaction Costs). Fix n ≥ 1. If +Xmin,i > c1/n +i +− 1 for all i = 1, 2, . . ., m, then the probability P(V (n) > 0) = 1. +7 + +Proof. Let n ≥ 1 be given. Observe that +P(V (n) > 0) = P((1 + KT � +Xn)V (0) > 0) += P(1 + KT � +Xn > 0) += P +� m +� +i=1 +Ki(1 + � +Xn,i) > 0 +� += P +� m +� +i=1 +Ki +� +1 + +n−1 +� +k=0 +(1 + Xi(k)) − 1 − ci +� +> 0 +� += P +� m +� +i=1 +Ki +�n−1 +� +k=0 +(1 + Xi(k)) − ci +� +> 0 +� +. +(6) +where the third equality holds by invoking the fact that �m +i=1 Ki = 1. Now note that the +event +��m +i=1 Ki +��n−1 +k=0(1 + Xi(k)) − ci +� +> 0 +� +⊇ +��n−1 +k=0(1 + Xi(k)) > ci +� +. With the aids of +monotonicity of probability measure, Equality (6) becomes +P(V (n) > 0) ≥ P +�n−1 +� +k=0 +(1 + Xi(k)) > ci +� +. +Since �n−1 +k=0(1 + Xi(k)) ≥ (1 + Xmin,i)n for all i = 1, 2, . . . , m and Xmin,i > c1/n +i +− 1 for all +i = 1, 2, . . . , m, it follows that �n−1 +k=0(1+Xi(k)) > ci for all i. Therefore, we have P(V (n) > 0) = +1. +Remark 3.1. (i) To assure a survival trade, Lemma 3.1 indicates that the worst returns must +be large enough. Specifically, for n = 1, it requires Xmin,i > ci − 1 for all i. On the other hand, +if n → ∞, which corresponds to buy and hold, then we must have Xmin,i > 0 for all i. (ii) On +the converse of the Lemma 3.1, it is readily verified that if mini ci > 0 and P(V (n) > 0) = 1 +for n ≥ 1, then �m +i=1 Ki((1 + µi)n − ci) ≥ 0 where µi := E[Xi(k)]; see Lemma A.1. This reveals +a gap in obtaining a necessary condition for survival trades in Lemma 3.1. +Lemma 3.1 implies that for a fixed c∗ := maxi ci ∈ (0, 1), there exists Xmin,i > −1 such +that V (n) ≤ 0 with positive probability. Said another way, the investor’s account may experience +a “survival issue” when the rebalancing frequency and costs are taken into consideration. In +addition, this survival issue may cause the gn(K) to become ill-defined. To address this issue, +we use a Taylor-based quadratic approximation of gn(K) around K = 0; see Casella and Berger +(2001) and write +gn(K) ≈ 1 +n +� +KT E +� +� +Xn +� +− 1 +2KTE +� +� +Xn � +X T +n +� +K +� +:= �gn(K). +(7) +It is well-known that such a quadratic approximation is accurate for small returns; see Pulley +(1983).6 +Hence, in the sequel, we consider an approximate frequency-dependent log-optimal +portfolio problem with costs as follows: +max {�gn(K) : K ∈ K} . +(8) +6Without loss of generality, set ci := 0 for all i. +Then the Taylor expansion of E �log(1 + KT Xn)� = +E +��∞ +d=1(−1)d+1 (KT Xn)d +d +� +converges for all K ∈ K if |KT Xn| ≤ 1 with probability one. +8 + +Remark 3.2. It is readily verified that the approximate problem (8) described above is a concave +quadratic program, which enables us to solve it in an efficient manner; e.g., see Diamond and +Boyd (2016). +3.1 +Optimality Conditions +In this section, we investigate the optimality conditions for the approximate frequency-dependent +log-optimal problem (8). +Lemma 3.2 (Necessity and Sufficiency). Fix n ≥ 1. Given a percentage costs ci ∈ (0, 1) for i = +1, 2, . . . , m, the portfolio weight �K∗ ∈ K is optimal to the approximate frequency-dependent log- +optimal problem (8), if and only if +E +� +� +Xn,i +� +− +m +� +j=1 +�K∗ +j E +� +� +Xn,i � +Xn,j +� += �K∗T E +� +� +Xn +� +− �K∗T E +� +� +Xn � +X T +n +� +�K∗, if �K∗ +i > 0 +(9) +E +� +� +Xn,i +� +− +m +� +j=1 +�K∗ +j E +� +� +Xn,i � +Xn,j +� +≤ �K∗T E +� +� +Xn +� +− �K∗T E +� +� +Xn � +X T +n +� +�K∗, if �K∗ +i = 0 +(10) +Proof. Let n ≥ 1 and ci ∈ (0, 1) for all i be given. +We begin by considering an equivalent +constrained stochastic minimization problem described as follows: +min +K −KTE +� +� +Xn +� ++ 1 +2KT E +� +� +Xn � +X T +n +� +K +s.t. KT 1 − 1 = 0; +− KTei ≤ 0, i = 1, 2, . . . , m +where ei ∈ Rm is unit vector having one at the ith component and zeros on the other components. +Consider the Lagrangian +L(K, λ, µ) := −KT E +� +� +Xn +� ++ 1 +2KTE +� +� +Xn � +X T +n +� +K + λ(KT 1 − 1) − µT K. +By the Karush-Kuhn-Tucker (KKT) conditions; e.g., see (Boyd and Vandenberghe, 2004, Chap- +ter 5), if �K∗ is a local maximum then there is a scalar λ ∈ R1 and a vector µ ∈ Rm with +component µj ≥ 0 such that, for i = 1, 2, . . ., m, +− E +� +� +Xn,i +� ++ +m +� +j=1 +�K∗ +j E +� +� +Xn,i � +Xn,j +� ++ λ − µi = 0 +(11) +�K∗T 1 − 1 = 0 +(12) +µi �K∗ +i = 0. +(13) +From Equation (11), we obtain, for i = 1, . . . , m, +µi = −E +� +� +Xn,i +� ++ +m +� +j=1 +�K∗ +j E +� +� +Xn,i � +Xn,j +� ++ λ. +(14) +Since µi �K∗ +i = 0 for all i, we take weighted sum of Equation (14); i.e., +m +� +i=1 +µi �K∗ +i = − �K∗TE +� +� +Xn +� ++ �K∗T E +� +� +Xn � +X T +n +� +�K∗ + λ = 0. +(15) +9 + +This implies that λ = �K∗TE +� +� +Xn +� +− �K∗T E +� +� +Xn � +X T +n +� +�K∗. Substituting this into Equation (14), we +have, for i = 1, 2, . . ., m, +µi = −E +� +� +Xn,i +� ++ +m +� +j=1 +�K∗ +j E +� +� +Xn,i � +Xn,j +� ++ �K∗T E +� +� +Xn +� +− �K∗T E +� +� +Xn � +X T +n +� +�K∗. +(16) +From Equation (16) and the fact that µi �K∗ +i = 0, it follows that for i = 1, 2, . . ., m, if �K∗ +i > 0, +then µi = 0 and +E +� +� +Xn,i +� +− +m +� +j=1 +�K∗ +j E +� +� +Xn,i � +Xn,j +� += �K∗T E +� +� +Xn +� +− �K∗T E +� +� +Xn � +X T +n +� +�K∗. +On the other hand, if �K∗ +i = 0, then µi ≥ 0 and +E +� +� +Xn,i +� +− +m +� +j=1 +�K∗ +j E +� +� +Xn,i � +Xn,j +� +≤ �K∗T E +� +� +Xn +� +− �K∗T E +� +� +Xn � +X T +n +� +�K∗. +To prove sufficiency, let �K∗ ∈ K and satisfies the conditions (9) and (10). Then it follows +that there exists λ ∈ R and µj > 0 such that the KKT conditions (11) to (13) hold at �K∗. Since +the constrained minimization problem is a convex optimization problem, it follows that the KKT +conditions are also sufficient for optimality. Hence, �K∗ is optimal; see Boyd et al. (2017). +Remark 3.3. Let �K∗ be the optimum obtained by solving the approximate frequency-dependent +log-optimal portfolio problem (8) and K∗ be the true log-optimum. Using Jensen’s inequality, +we have +0 ≤ g(K∗) − g( �K∗) = E +� +log 1 + K∗T � +Xn +1 + �K∗T � +Xn +� +≤ log E +� +1 + K∗T � +Xn +1 + �K∗T � +Xn +� +. +The right-hand side is approximately zero when K∗ ≈ �K∗. As we will see later in this paper, +this is typically the case. More interestingly, Lemma 3.2 serves to compliment Lemma 2.2 by +characterizing the log-optimal weights; see Example 3.1 below. +Example 3.1 (Two-Asset Toy Example). To demonstrate the application of Lemmas 2.2 and 3.2, +we first consider a high-frequency investor who rebalances her portfolio at every period; i.e., +n := 1. Specifically, consider a two-asset portfolio including a risk-free cash asset with zero +interest rate; i.e. X1(k) := rf = 0 with probability one and a risky asset with a binomial return +X2(k) ∈ {− 1 +2, 1 +2} with probability P +� +X2(k) = 1 +2 +� +:= p ∈ +� 1 +2 + c2, 1 +� +. The transaction costs are +c1 = 0 for cash and c2 < 1/2 for the risky asset. If �K∗ +2 > 0, by Lemma 3.2, we have +(1 − �K∗ +2) +� +p − 1 +2 − c2 +� +− �K∗ +2(1 − �K∗ +2) +�1 +4 − 2c2 +� +p − 1 +2 + c2 +2 +�� += 0. +This implies that �K∗ +2 = +−(4c2−4p+2) +4c2 +2+4c2−8c2p+1. Incorporating with Lemma 2.2, we conclude +K∗ +2 := + + + +�K∗ +2 +if p ∈ +� +1 +2 + c2, 4c2 +2+8c2+3 +4+8c2 +� +1 +if p ∈ +� +4c2 +2+8c2+3 +4+8c2 +, 1 +� +(17) +10 + +and K∗ +1 = 1 − K∗ +2. Note that if c2 = 0, then K∗ +2 = 2(2p − 1) for p ∈ +� 1 +2, 3 +4 +� +or K∗ +2 = 1 for +p ∈ +� 3 +4, 1 +� +, which reduces to the classical ELG result in gambling; see Kelly jr (1956); Hsieh et al. +(2018a). +To see the effect of rebalancing period n > 1, we consider a second example with n = 2; i.e., +one rebalances the portfolio for every two periods. For c2 ∈ +� +0, 1 +4 +� +, applying Lemmas 2.2 and 3.2 +yield +K∗ +2 := + + + +�K∗ +2, +if p ∈ +� +1 +2 + c2, − 4c2−9 +8c2+6 − 1 +4C +� +1, +if p ∈ +� +− 4c2−9 +8c2+6 − 1 +4C, 1 +� +, +(18) +where �K∗ +2 = +16p2+16p−16c2−12 +16c2 +2+24c2+32p2−16p−32p2c2−32pc2+9, and C := +√ +−256c4 +2+384c3 +2+640c2 +2−504c2+81 +4c2+3 +and +K∗ +1 = 1 − K∗ +2. If c2 = 0, we have K∗ +2 = 16p2+16p−12 +32p2−16p+9 for p ∈ +� 1 +2, 3 +4 +� +and K∗ +2 = 1 for p ∈ +� 3 +4, 1 +� +. +4 +Feasible Region and Efficient Frontier +Similar to how the performance of a portfolio can be characterized by its expected return and +variance in the celebrated Markowitz framework, the performance of log-optimal portfolios can +be characterized by the expected logarithmic growth and variance of the logarithmic growth and +plotted on a two-dimensional diagram; see Luenberger (2013). The region mapped out by all +possible portfolios defines the feasible region. That is, for any fixed n ≥ 1, we consider +K �→ +� +E +� +log VK(n) +V (0) +� +, var +� +log VK(n) +V (0) +�� +⊂ R2. +As demonstrated later in Example 4.1, the feasible region is convex to the left. This means that +if we take any two points within the region, the straight line connecting them does not cross +the left boundary of the feasible region. A similar idea about analyzing the efficient frontier +analytically can be found in Merton (1972). +4.1 +A Version of The Two-Fund Theorem +In the approximate log-optimal portfolio problem, as defined in (8), the upper left-hand portion +at the boundary of the feasible region is referred to as the approximate efficient frontier. This +frontier is considered efficient in terms of expected logarithmic growth rate and its variance; see +also (Luenberger, 2013, Chapter 14). Then, with the aid of Lemma 3.2, we can obtain a version +of the two-fund theorem, which states that any convex combination of two optimal weights from +the optimality conditions is still optimal. +Theorem 4.1 (A Version of Two-Fund Theorem). Let K′, K′′ ∈ K be two weights satisfying the +optimality conditions stated in Lemma 3.2. Define a convex combination Kα := αK′+(1−α)K′′ +with α ∈ [0, 1]. Then Kα also satisfies the optimality conditions. +Proof. Take K′ and K′′ be two weights satisfying Equations (11) to (13), for all α ∈ [0, 1], we +must show that the convex combination of the two weights K′ and K′′, Kα := αK′ + (1 − α)K′′, +with the jth component Kα,j, also satisfies the same optimality equations. In particular, we +begin by proving that Kα satisfies Equation (12). Indeed, we observe that +(αK′ + (1 − α)K′′)T 1 − 1 = αK′T 1 + (1 − α)K′′T 1 − 1 +(19) +where 1 := [1 1 · · · 1]T ∈ Rm. Since K′, K′′ satisfy Equation (12), it follows that K′T 1 = 1 +and K′′T 1 = 1. Therefore, Equation (19) becomes (αK′ + (1 − α)K′′)T 1 − 1 = α + (1 − α) = 1 +11 + +which proves that the convex combination Kα satisfies Equation (12). To see it also satisfies +µi �K∗ +i = 0 for i = 1, . . . , m, we observe that +µi(αK′ +i + (1 − α)K′′ +i ) = αµiK′ +i + (1 − α)µiK′′ +i += α · 0 + (1 − α) · 0 = 0. +To complete the proof, we show that Kα satisfies Equation (11). It suffices to show that for i = +1, . . . , m, −E +� +� +Xn,i +� ++ �m +j=1 Kα,jE +� +� +Xn,i � +Xn,j +� ++ λ = µi. Note that the left-hand side using Kα +yields +− (α + (1 − α))E +� +� +Xn,i +� ++ +m +� +j=1 +(αK′ +j + (1 − α)K′′ +j )E +� +� +Xn,i � +Xn,j +� ++ (α + (1 − α))λ += α + +−E +� +� +Xn,i +� ++ +m +� +j=1 +K′ +jE +� +� +Xn,i � +Xn,j +� ++ λ + + + (1 − α) + +−E +� +� +Xn,i +� ++ +m +� +j=1 +K′′ +j E +� +� +Xn,i � +Xn,j +� ++ λ + + += αµi + (1 − α)µi = µi +which completes the proof. +Example 4.1 (Five-Asset Portfolio with Intraday Minute-by-Minute Data). This example il- +lustrates the feasible region, efficient frontier, and Two-Fund Theorem 4.1 using a five-asset +portfolio consisting of a bank account, Vanguard Total Stock Market Index Fund ETF (Ticker: +VTI), Vanguard Total Bond Market Index Fund ETF (Ticker: BND), Vanguard Emerging Mar- +kets Stock Index Fund ETF (Ticker: VWO), and Bitcoin to the USD exchange rate (Ticker: +XBTUSD). The portfolio is well-diversified, covering the large US-Euro stock market, the global +bond market, and cryptocurrency. +Here, transaction costs ci = 0.001% are imposed on the +ETFs (i.e., i ∈ {VTI, BND, VWO}) and costs cXBTUSD = 0.1% on the XBTUSD.7 Besides, in- +vestors receive interest at a (per-minute) rate rf = 0.0001% if they keep their funds in the bank +account. The data used in this example spans from 09 : 30 : 00 AM to 15 : 59 : 00 PM on +December 3, 2021, where the associated price trajectories for the four risky assets are shown +in Figure 1.8 To derive the approximate log-optimal portfolio and examine its trading perfor- +mance, we split the entire data set into two parts: The first portion from 09 : 30 : 00 AM +to 12 : 29 : 00 PM is for the in-sample optimization, and the second portion 12 : 30 : 00 PM +to 15 : 59 : 00 PM is for the out-of-sample testing.9 +Fix n ≥ 1. We define the approximate feasible region H := +�� +�gn(K), var +� +log VK(n) +V (0) +�� +: K ∈ K +� +. +Figures 2 and 3 show the points in H and the approximate efficient frontier for different rebal- +ancing periods n = 1 and n = 5, respectively. +As predicted by Theorem 4.1, any convex +combination of two optimal weights K′ and K′′ satisfying optimality conditions 3.2, denoted +as Kα = αK′ + (1 − α)K′′ with α ∈ [0, 1], satisfies the optimality conditions. Interestingly, it +also lies on the approximate efficient frontier due to the small scale of the minute-by-minute price +data10; see Figures 2 and 3 for an example with α = 0.5. Similar findings also hold for other +rebalancing periods n > 5. +7According to the platform Binance binance.com/en, regular users are charged a transaction cost of 0.1% for +Bitcoin trades. +8The price data for the four underlying risky assets (VTI, BND, VWO, XBTUSD) are retrieved from the +Bloomberg terminal (accessed on November 17, 2022). +9This will be demonstrated later in Example 5.2 in the next section. +10This phenomenon disappears when using daily data; see also Remark 4.1 for more information. +12 + +10:00 +12:00 +14:00 +16:00 +Dec 03, 2021 +226 +228 +230 +232 +VTI +10:00 +12:00 +14:00 +16:00 +Dec 03, 2021 +83.2 +83.4 +83.6 +83.8 +BND +10:00 +12:00 +14:00 +16:00 +Dec 03, 2021 +47.4 +47.6 +47.8 +48 +VWO +10:00 +12:00 +14:00 +16:00 +Dec 03, 2021 +5.2 +5.3 +5.4 +5.5 +5.6 +104 +XBTUSD +Figure 1: Intraday Minute-by-Minute Prices for VTI, BND, VWO, and XBTUSD. +Remark 4.1. While not pursued further in this paper, the optimality conditions derived in +Lemma 3.2 only consider the approximate logarithmic growth function �gn(K) without taking +into account the log-variance var(log Vn(K)/V (0)). As a result, to ensure that any convex com- +bination of two points on the approximate efficient frontier is still on the frontier, the log-variance +must be included in the optimization problem (8). This topic presents a promising research di- +rection. +5 +Illustrative Examples +This section presents empirical examples to demonstrate the validity of our theory. In the first +two examples, we use the same intraday data set as Example 4.1 to compare the log-optimal +and approximate log-optimal results. We evaluate the impact of different rebalancing periods +and levels of costs on trading performance. The third example examines the capability of our +theory to handle the mid-sized portfolio case by considering a portfolio of thirty-two assets (with +a Bank account, Dow-30 stocks, and cryptocurrency) using daily historical price data. +Example 5.1 (Five-Asset Portfolio Revisited). This example demonstrates that the approxi- +mate optimal weights �K∗ from Lemma 3.2 is sufficiently close to the optimal weights K∗. To +demonstrate this, we choose the weights K∗ on the efficient frontier that satisfy the logarith- +mic variance condition: var +� +log VK∗(n) +V (0) +� +≡ var +� +log +V� +K∗(n) +V (0) +� +. Figures 4 and 5 show the portfolio +weights of the two trading strategies: the approximate log-optimal weights �K∗, and the true +13 + +Figure 2: +An illustration of Feasible Set, Efficient Frontier, and Two-Fund Theorem (Kα +with α = 0.5) using Rebalancing Period n = 1 (Minute). +log-optimal weights K∗ with different rebalancing periods n = 1 and n = 5. The results show +that the weights of the two strategies are nearly identical, i.e., �K∗ +i ≈ K∗ +i , for all i = 1, 2, . . ., 5. +This suggests that the approximate optimal weights �K∗ are a good approximation of the true +optimal weights K∗. While not showing here, it is also worth mentioning that if the transaction +costs are sufficiently large, then both of the optima K∗ and �K∗ will tend to fully invest in the +bank account, meaning that K∗ +Bank account ≈ �K∗ +Bank account ≈ 1. +Example 5.2 (Trading Performance with Different Rebalancing Periods and Costs). This exam- +ple illustrates the in-sample and out-of-sample trading performances using the solutions obtained +in previous Example 5.1. Specifically, let V (N) be the account value at the terminal stage N. +The portfolio realized return in period k is Rp(k) := +V (k+1)−V (k) +V (k) +. With the aid of this real- +ized return, we consider the following metrics to study the trading performance: The realized +cumulative rate of return +V (N)−V (0) +V (0) +, realized log-return log V (N) +V (0) , volatility σ := std(Rp(k)), +maximum percentage drawdown d∗ := max0≤k≤N +Vmax(k)−V (k) +Vmax(k) +with Vmax(k) := max0≤i≤k V (i), +and the N-period Sharpe ratio +√ +N · SR with SR being the per-period realized Sharpe ratio.11 +Starting with initial account V (0) = $1, Figures 6 and 7 reveal the in-sample and out-of- +sample values of the trading account using the three trading strategies: The log-optimal portfolio +11Given a sequence of the realized portfolio per-period returns {Rp(k) : k = 0, 1, . . . , N − 1}, the per-period +Sharpe ratio is SR := +Rp−rf +s +where R +p := +1 +N +�N−1 +k=0 Rp(k) is the sample mean return, rf is the per-period +risk-free rate, and s := +� +1 +N−1 +�N−1 +k=0 (Rp(k) − R +p)2 is the sample standard deviation of portfolio returns. A +detailed discussion of this topic can be found in Lo (2002). +14 + +×10-5 +Approximate feasible region (n=1) +6 +Approximate feasible points +Approximate efficient frontier +Points on the approximate efficient frontier +4 +2 +E[log-growth] +0 +2 +-4 +-6 +0 +1 +2 +3 +4 +5 +6 +7 +8 +Var(log-growth) +×10-8Figure 3: +An Illustration of Feasible Set, Efficient Frontier, and Two-Fund Theorem (Kα +with α = 0.5) using Rebalancing Period n = 5 (Minutes). +with weight K∗, the approximate log-optimum �K∗, and buy-and-hold with equal weight K = +1/m, for the same five-asset portfolio considered in Example 4.1. Note that there are nonzero +transaction costs of 0.001% for trading ETFs and a cost of 0.1% for trading cryptocurrency. +From the figures, we see that the account value trajectory obtained using �K∗ is similar to that +obtained using K∗. Moreover, both of the portfolios outperform the equally-weighted buy-and- +hold strategy. +To see clearly the effect of transaction costs on trading performance, we consider an additional +scenario with zero costs for trading both ETFs and cryptocurrency; see Figures 8 and 9 for the in- +sample and out-of-sample account value trajectories under rebalancing period n = 1 and n = 5, +with zero costs. Both figures demonstrate that the account values are improved when there are +no costs. +Tables 1 and 2 provide an overview of the out-of-sample trading performance metrics of the +three trading strategies for different rebalancing periods n = 1 and n = 5, respectively. For +the case of n = 1, i.e., the portfolio is rebalanced every minute, we find that the zero costs +lead to better performance of the log-optimal portfolio in terms of the Sharpe ratio. +When +nonzero transaction costs are imposed, the Sharpe ratios for K∗ and �K∗ become negative. This +suggests that transaction costs have a negative impact on trading performance especially when +rebalancing occurs frequently. On the other hand, for the case of n = 5, where the portfolio is +rebalanced every five minutes, the Sharpe ratios are positive and generally higher than those for +n = 1. This indicates that a longer rebalancing period incurs fewer costs and may lead to better +trading performance. +15 + +×10-5 +Approximatefeasibleregion(n=5) +5 +Approximate feasible points +Approximate eficient frontier +4 +Points on the approximate efficient frontier +3 +2 +E[log-growth] +0 +.1 +-2 +-3 +-4 +-5 +0 +1 +2 +3 +4 +5 +6 +7 +8 +Var(log-growth) +×10-7Weights for each trading strategy (n=1) +BND +Bank Account +VTI +VWO +XBTUSD +Asset +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Weight +Figure 4: Portfolio Weights K∗ versus �K∗ with Rebalancing Period n = 1 (Minute). +Weights for each trading strategy (n=5) +BND +Bank Account +VTI +VWO +XBTUSD +Asset +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Weight +Figure 5: Portfolio Weights K∗ versus �K∗ with Rebalancing Period n = 5 (Minutes). +16 + +09:30 +10:00 +10:30 +11:00 +11:30 +12:00 +12:30 +Time +Dec 03, 2021 +0.99 +0.995 +1 +Account value +In-sample trading performance (n=1) +12:30 +13:00 +13:30 +14:00 +14:30 +15:00 +15:30 +16:00 +Time +Dec 03, 2021 +0.992 +0.994 +0.996 +0.998 +1 +1.002 +Account value +Out-of-sample trading performance (n=1) +Figure 6: Account Value Trajectories under Three Trading Strategies (Optimal K∗, �K∗, and +Equally-Weighted K = 1/m) with Rebalancing Period n = 1 (Minute) and Nonzero Costs. +Table 1: Out-of-Sample Trading Performance Metrics with Different Transaction Costs with +Rebalancing Period n = 1 (Minute) +Costs of 0% for ETFs and cryptocurrency +K∗ +�K∗ +Buy and hold +Cumulative rate of return V (N)−V (0) +V (0) +(%) +0.13 +0.13 +−0.38 +Realized log-growth log V (N) +V (0) (%) +0.13 +0.13 +−0.38 +Volatility σ (%) +0.01 +0.01 +0.05 +Maximum percentage drawdown d∗ (%) +0.16 +0.16 +1.20 +Sharpe ratio +√ +NSR +0.74 +0.75 +−0.58 +Costs of 0.001% for ETFs and 0.1% for cryptocurrency +K∗ +�K∗ +Buy and hold +Cumulative rate of return V (N)−V (0) +V (0) +(%) +−0.05 +−0.06 +−0.42 +Realized log-growth log V (N) +V (0) (%) +−0.05 +−0.06 +−0.43 +Volatility σ (%) +0.01 +0.01 +0.05 +Maximum percentage drawdown d∗ (%) +0.18 +0.19 +1.20 +Sharpe ratio +√ +NSR +−0.58 +−0.65 +−0.64 +Example 5.3 (Mid-Sized Portfolio: Thirty-Two Assets with Daily Price Data). Our theory is +readily applied to a mid-sized (or large-sized) portfolio. As an example, we consider a portfolio +consisting of 32 assets involving a bank account, Dow 30 Stocks,12 and the Bitcoin-to-USD +exchange rate (Ticker: BTC-USD) over a one-year horizon from November 20, 2021 to November +12Dow 30 Stocks consist of the thirty stocks that make up the Dow Jones Industrial Average. +17 + +09:30 +10:00 +10:30 +11:00 +11:30 +12:00 +12:30 +Time +Dec 03, 2021 +0.99 +0.995 +1 +Account value +In-sample trading performance (n=5) +12:30 +13:00 +13:30 +14:00 +14:30 +15:00 +15:30 +16:00 +Time +Dec 03, 2021 +0.992 +0.994 +0.996 +0.998 +1 +1.002 +Account value +Out-of-sample trading performance (n=5) +Figure 7: Account Value Trajectories under Three Trading Strategies (Optimal K∗, �K∗, and +Equally-Weighted K = 1/m) with Rebalancing Period n = 5 (Minutes) and Nonzero Costs. +Table 2: Out-of-Sample Trading Performance Metrics with Different Transaction Costs with +Rebalancing Period n = 5 (Minutes) +Costs of 0% for ETFs and cryptocurrency +K∗ +�K∗ +Buy and hold +Cumulative rate of return V (N)−V (0) +V (0) +(%) +0.14 +0.14 +−0.38 +Realized log-growth log V (N) +V (0) (%) +0.14 +0.14 +−0.38 +Volatility σ (%) +0.02 +0.02 +0.05 +Maximum percentage drawdown d∗ (%) +0.16 +0.16 +1.20 +Sharpe ratio +√ +NSR +0.88 +0.88 +−0.58 +Costs of 0.001% for ETFs and 0.1% for cryptocurrency +K∗ +�K∗ +Buy and hold +Cumulative rate of return V (N)−V (0) +V (0) +(%) +0.10 +0.10 +−0.42 +Realized log-growth log V (N) +V (0) (%) +0.10 +0.10 +−0.43 +Volatility σ (%) +0.02 +0.02 +0.05 +Maximum percentage drawdown d∗ (%) +0.17 +0.17 +1.20 +Sharpe ratio +√ +NSR +0.61 +0.61 +−0.64 +20, 2022.13 +The one-year data is divided into two parts: The first 90 days are used for in-sample optimiza- +13The data considered in this example is retrieved from Yahoo Finance. +It is worth noting that the time +period considered for this example is significant because the third-largest cryptocurrency exchange, FTX, declared +bankruptcy on November 11, 2022, which had a significant impact on cryptocurrency markets. +18 + +09:30 +10:00 +10:30 +11:00 +11:30 +12:00 +12:30 +Time +Dec 03, 2021 +0.99 +0.995 +1 +Account value +In-sample trading performance (n=1) +12:30 +13:00 +13:30 +14:00 +14:30 +15:00 +15:30 +16:00 +Time +Dec 03, 2021 +0.992 +0.994 +0.996 +0.998 +1 +1.002 +Account value +Out-of-sample trading performance (n=1) +Figure 8: Account Value Trajectories under Three Trading Strategies (Optimal K∗, �K∗, and +Equally-Weighted K = 1/m) with Rebalancing Period n = 1 (Minute) and Zero Costs. +tion and the remainder is used for out-of-sample testing. Here, we consider different scenarios +for the transaction costs: zero costs, 0.01%, 0.1%, and 0.5% for trading stocks, and zero costs +and 0.1% fees for trading cryptocurrency. If investors retain their capital in the bank account, +they earn daily interest with a rate rf := 1%/365. +Fix n = 1, i.e., the portfolio is rebalanced on a daily basis. When costs for trading stocks +are 0%, 0.01% and 0.1%, we find that K∗ +CV X ≈ �K∗ +CV X ≈ 1.14 However, when the proportional +cost is 0.5%, the approximate optimum becomes K∗ +Bank account ≈ �K∗ +Bank account ≈ 1, indicating +that it is optimal to hold all capital in the bank account. Table 3 summarizes the performance of +the three trading strategies under different levels of costs for trading stocks and cryptocurrency. +As expected, higher costs result in a significant decrease in investor revenue. The corresponding +account value trajectories are plotted in Figure 10. +Subsequently, we examine the effects of different rebalancing periods by setting the rebal- +ancing period to every five days, i.e., n = 5. In this case, we find that K∗ +CV X ≈ �K∗ +CV X ≈ 1 +for all four different levels of costs (0%, 0.01%, 0.1%, and 0.5%) for trading stocks. This is in +contrast to the case with n = 1, where the optimal weights dictated K∗ +bank account ≈ 1 when the +proportional cost was 0.5%. Figure 11 shows that the associated trading performance using K∗ +and �K∗ are similar and outperforms the buy-and-hold strategy with equal weights 1/m over the +given time period. Table 4 provides a summary of the performance metrics under four different +levels of costs with rebalancing periods n = 5. +14Note that, in this example, Chevron Corporation (Ticker: CVX) is the dominant asset since the estimated +dominance condition max1≤i≤32, i̸=CV X +1 +N +�N +k=1 +1+Xi(k) +1+XCV X (k) = 0.998 < 1 for all the assets in the portfolio +except for CVX. Hence, according to Lemma 2.2, a log-optimal investor should invest all the available capital in +this asset. +19 + +09:30 +10:00 +10:30 +11:00 +11:30 +12:00 +12:30 +Time +Dec 03, 2021 +0.99 +0.995 +1 +Account value +In-sample trading performance (n=5) +12:30 +13:00 +13:30 +14:00 +14:30 +15:00 +15:30 +16:00 +Time +Dec 03, 2021 +0.992 +0.994 +0.996 +0.998 +1 +1.002 +Account value +Out-of-sample trading performance (n=5) +Figure 9: Account Value Trajectories under Three Trading Strategies (Optimal K∗, �K∗, and +Equally-Weighted K = 1/m) with Rebalancing Period n = 5 (Minutes) and Zero Costs. +For an even longer rebalancing period, say n = 10 and n = 30, the optimal weight K∗ +CV X = 1 +remains under proportional cost for stocks being 0.5%. +6 +Online Trading with Sliding Window Approach +In previous sections, optimal weights K∗ and its approximation counterpart �K∗ were obtained as +fixed values based on the empirical distributions of returns, rather than true distribution, which +is typically unknown to the investor in practice. Moreover, these fixed weights cannot adapt to +the constantly changing information in a dynamic market. To address this issue, we apply a +data-driven sliding window approach that generates time-varying log-optimal weights online; see +also Wang and Hsieh (2022) for a similar idea for online trading. +The idea of the sliding window approach is as follows. For k = 0, 1, . . . , the investor first +declares a fixed window size M ≥ 1. With k = 0, 1, . . . , M −1, one solves the log-optimal portfolio +problem (4) to obtain K∗ or the approximation counterpart (8) to obtain �K∗. These optimum +weights are then applied in the next stage. Having done that, one re-solves the log-optimal +portfolio problem again using the data from k = 1, 2, . . ., M. Repeating this procedure until the +end, one obtains a time-varying optimum K∗(k) or �K∗(k). This approach has a computational +advantage because it solves a sequence of concave optimization problems rather than a stochastic +dynamic programming problem. The details of this approach can be found in Algorithm 1 below. +Example 6.1 (Mid-Sized Portfolio Revisited: Online Trading via the Sliding Window Ap- +proach). To illustrate the sliding window approach, we conduct additional empirical studies +20 + +Nov 2021 +Dec 2021 +Jan 2022 +Feb 2022 +Mar 2022 +Apr 2022 +Time +1 +1.2 +1.4 +Account value +In-sample trading performance (n=1) +Apr +May +Jun +Jul +Aug +Sep +Oct +Nov +Dec +Time +2022 +0.9 +1 +1.1 +Account value +Out-of-sample trading performance (n=1) +Figure 10: Account Value Trajectories under Three Trading Strategies (Optimal K∗, �K∗, and +Equally-Weighted K = 1/m) with Rebalancing Period n = 1 (Day) and Costs of 0.01% for +Stocks and 0.1% for Cryptocurrency. +using the daily price data considered in Example 5.3 with the costs of 0.01% for stocks and +0.1% for cryptocurrency. Here, we first fix the rebalancing period n = 1 day and consider three +different window sizes: M = 10, 20, 30 days. By solving the log-optimal and approximate log- +optimal portfolio problems, we obtain the resulting time-varying optimal weights K∗(k) and +the approximate log-optimum �K∗(k) for k = 1, 2, . . . , see Figure 12 for an illustration. The +associated account value trajectories of three portfolios with different weights ( �K∗, K∗, and an +equally-weights K = 1/m) are depicted in Figure 13. See also Table 5 for a summary of the +trading performance metrics under three different window sizes M. It is interesting to note that +the portfolios with weights K∗ and �K∗ using the window size M = 30 outperform the buy-and- +hold strategy in terms of Sharpe ratio. This observation suggests that the window size M may +be an important factor in determining the overall trading performance. While this point is not +pursued further in this paper, it is worth considering in future work when implementing the +sliding window approach in practice. +Likewise, we also study the performance with different rebalancing periods n = 5 and n = 10 +and with different window sizes M = 10, 20, and 30. These results are summarized in Tables 6 +and 7. Similar to the n = 1 case, we see that for both n = 5 and n = 10, the best performance +is obtained with M = 30 and M = 20, respectively in this example. +7 +Concluding Remarks +This paper focuses on incorporating rebalancing frequency and transaction costs into the log- +optimal portfolio formulation, which aims to maximize the expected logarithmic growth rate of +21 + +Table 3: Out-of-Sample Trading Performance with Zero Costs and Different Nonzero Costs for +Stocks and Cryptocurrency with Rebalancing Period n = 1 (Day). +Costs of 0% for stocks and cryptocurrency +K∗ +�K∗ +Buy and hold +Cumulative rate of return V (N)−V (0) +V (0) +(%) +14.20 +14.18 +−6.47 +Realized log-growth log V (N) +V (0) (%) +13.28 +13.26 +−6.68 +Volatility σ (%) +2.20 +2.20 +1.37 +Maximum percentage drawdown d∗ (%) +24.88 +24.89 +20.42 +Sharpe ratio +√ +NSR +0.60 +0.60 +−0.33 +Costs of 0.01% for stocks and 0.1% for cryptocurrency +K∗ +�K∗ +Buy and hold +Cumulative rate of return V (N)−V (0) +V (0) +(%) +12.39 +12.37 +−6.49 +Realized log-growth log V (N) +V (0) (%) +11.68 +11.66 +−6.71 +Volatility σ (%) +2.20 +2.20 +1.37 +Maximum percentage drawdown d∗ (%) +25.06 +25.07 +20.42 +Sharpe ratio +√ +NSR +0.54 +0.54 +−0.33 +Costs of 0.1% for stocks and 0.1% for cryptocurrency +K∗ +�K∗ +Buy and hold +Cumulative rate of return V (N)−V (0) +V (0) +(%) +−2.68 +−2.7 +−6.65 +Realized log-growth log V (N) +V (0) (%) +−2.72 +−2.73 +−6.88 +Volatility σ (%) +2.20 +2.20 +1.37 +Maximum percentage drawdown d∗ (%) +27.15 +27.17 +20.42 +Sharpe ratio +√ +NSR +0.03 +0.02 +−0.34 +Costs of 0.5% for stocks and 0.1% for cryptocurrency +K∗ +�K∗ +Buy and hold +Cumulative rate of return V (N)−V (0) +V (0) +(%) +0.22 +0.17 +−7.34 +Realized log-growth log V (N) +V (0) (%) +0.22 +0.17 +−7.63 +Volatility σ (%) +3.9 × 10−5 +4.7 × 10−5 +1.37 +Maximum percentage drawdown d∗ (%) +0.02 +0.03 +20.42 +Sharpe ratio +√ +NSR +−4.45 +−4.55 +−0.38 +an investor’s wealth. We demonstrate that solving a frequency-dependent optimization problem +with costs is equivalent to solving a concave program. Conditions under which a log-optimal +investor would invest all available funds in a specific asset are provided. We also consider the +issue of bankruptcy that can arise due to transaction costs in the frequency-dependent formu- +lation and propose an approximate solution using a quadratic concave program. Additionally, +a version of the two-fund theorem is proven, demonstrating that a convex combination of two +optimal weights is still optimal. We present various empirical studies to explore the effect of +considering percentage transaction cost and rebalancing periods from the small to mid-sized +portfolio optimization problems. Lastly, we extend our empirical studies to an online trading +scenario by implementing a sliding window approach, which allows us to solve a sequence of +concave programs rather than a complex stochastic dynamic programming problem. +Regarding further research, one possible continuation is to consider additional practical trad- +ing issues; e.g., allowing to short an asset, i.e., Ki < 0 for some i and/or modeling the impact +of dividend/taxes. Another feasible direction is to incorporate an risk term into the objective +function for the ELG maximization problem, which would mitigate the situation when the op- +22 + +Nov 2021 +Dec 2021 +Jan 2022 +Feb 2022 +Mar 2022 +Apr 2022 +Time +1 +1.2 +1.4 +Account value +In-sample trading performance (n=5) +Apr +May +Jun +Jul +Aug +Sep +Oct +Nov +Dec +Time +2022 +0.9 +1 +1.1 +Account value +Out-of-sample trading performance (n=5) +Figure 11: Account Value Trajectories under Three Trading Strategies (Optimal K∗, �K∗, and +Equally-Weighted K = 1/m) with Rebalancing Period n = 5 (Days) and Costs of 0.01% for +Stocks and 0.1% for Cryptocurrency.. +timum suggests betting all capital on a specific asset; e.g., see Davis and Lleo (2008). Another +important consideration is the potential for estimation error in the distribution of returns, which +is often unknown and must be estimated in practice. In this case, it may be useful to study +the robust counterpart of the problem considered in this paper. +That is, instead of solving +supK E[log VK(N) +V (0) ], one seeks to solve a data-driven distributional robust log-optimal portfolio +problem +sup +K∈K +inf +P ∈P EP +� +log VK(N) +V (0) +� +where P is the ambiguity set of probability distribution; e.g., see Mohajerin Esfahani and Kuhn +(2018); Wu et al. (2022) for an approach using Wasserstein metric to characterize the ambigu- +ity set. +References +Algoet, P. H. and Cover, T. M. (1988). Asymptotic Optimality and Asymptotic Equipartition +Properties of Log-Optimum Investment. The Annals of Probability, 16(2):876–898. +Almgren, R. and Chriss, N. (2001). Optimal Execution of Portfolio Transactions. Journal of +Risk, 3:5–40. +Barmish, B. R. and Primbs, J. A. (2015). On a New Paradigm for Stock Trading via a Model-Free +Feedback Controller. IEEE Transactions on Automatic Control, 61(3):662–676. +23 + +Table 4: Out-of-Sample Trading Performance with Zero Costs and Different Nonzero Costs for +Stocks and Cryptocurrency with Rebalancing Period n = 5 (Days). +Costs of 0% for stocks and cryptocurrency +K∗ +�K∗ +Buy and hold +Cumulative rate of return V (N)−V (0) +V (0) +(%) +14.25 +14.23 +−6.47 +Realized log-growth log V (N) +V (0) (%) +13.32 +13.31 +−6.68 +Volatility σ (%) +5.18 +5.18 +1.37 +Maximum percentage drawdown d∗ (%) +24.55 +24.55 +20.42 +Sharpe ratio +√ +NSR +0.61 +0.60 +−0.33 +Costs of 0.01% for stocks and 0.1% for cryptocurrency +K∗ +�K∗ +Buy and hold +Cumulative rate of return V (N)−V (0) +V (0) +(%) +13.88 +13.87 +−6.49 +Realized log-growth log V (N) +V (0) (%) +13.00 +12.99 +−6.71 +Volatility σ (%) +5.18 +5.18 +1.37 +Maximum percentage drawdown d∗ (%) +24.59 +24.59 +20.42 +Sharpe ratio +√ +NSR +0.59 +0.59 +−0.33 +Costs of 0.1% for stocks and 0.1% for cryptocurrency +K∗ +�K∗ +Buy and hold +Cumulative rate of return V (N)−V (0) +V (0) +(%) +10.66 +10.64 +−6.65 +Realized log-growth log V (N) +V (0) (%) +10.13 +10.12 +−6.88 +Volatility σ (%) +5.18 +5.18 +1.37 +Maximum percentage drawdown d∗ (%) +24.95 +24.95 +20.42 +Sharpe ratio +√ +NSR +0.49 +0.49 +−0.34 +Costs of 0.5% for stocks and 0.1% for cryptocurrency +K∗ +�K∗ +Buy and hold +Cumulative rate of return V (N)−V (0) +V (0) +(%) +−2.64 +−2.65 +−7.34 +Realized log-growth log V (N) +V (0) (%) +−2.67 +−2.69 +−7.63 +Volatility σ (%) +5.18 +5.18 +1.37 +Maximum percentage drawdown d∗ (%) +26.53 +26.53 +20.42 +Sharpe ratio +√ +NSR +0.05 +0.05 +−0.38 +Bertsimas, D. and Lo, A. W. (1998). Optimal Control of Execution Costs. Journal of financial +markets, 1(1):1–50. +Bogle, J. C. (2017). The Little Book of Common Sense Investing: The Only Way to Guarantee +Your Fair Share of Stock Market Returns. John Wiley & Sons. +Boyd, S., Busseti, E., Diamond, S., Kahn, R. N., Koh, K., Nystrup, P., and Speth, J. (2017). +Multi-Period Trading via Convex Optimization. arXiv preprint arXiv:1705.00109. +Boyd, S. and Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press. +Breiman, L. (1961). Optimal Gambling Systems for Favorable Games. In Proceedings of the +Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contribu- +tions to the Theory of Statistics. The Regents of the University of California. +Browne, S. and Whitt, W. (1996). Portfolio Choice and the Bayesian Kelly Criterion. Advances +in Applied Probability, 28(4):1145–1176. +24 + +Algorithm 1 Online Trading via Sliding Window Approach +Require: Consider m ≥ 2 assets, realized returns {Xi(k) : k ≥ 0} and transaction cost ci for +i = 1, 2, · · · , m, and sliding window size M ≥ 1. +Ensure: Optimal portfolio weight K∗ or �K∗ for each stage. +1: Compute compound returns { � +Xn,i(s)} for each asset i in the portfolio with +� +Xn,i(s) := +�ns−1 +k=n(s−1)(1 + Xi(k)) − 1 − ci. +2: if k ≥ M then +3: +Solve the maximization Problem (4) to obtain K∗ (or solve Problem (8) to obtain �K∗). +4: +Having obtained optimal K∗(k) := K∗ (or K∗(k) := �K∗(k)), we apply it at stage k + 1. +Set k := k + 1 then back to Step 2. +5: end if +Table 5: Online Trading Performance Using Sliding Window Approach with Transaction Costs +of 0.01% for Stocks and 0.1% for Cryptocurrency with rebalancing period n = 1 (Day). +M = 10 +K∗ +�K∗ +Buy and hold +Cumulative rate of return V (N)−V (0) +V (0) +(%) +−22.88 +−23.11 +−5.78 +Realized log-growth log V (N) +V (0) (%) +−25.99 +−26.28 +−5.95 +Volatility σ (%) +2.16 +2.16 +1.24 +Maximum percentage drawdown d∗ (%) +42.19 +42.35 +22.08 +Sharpe ratio +√ +NSR +−0.63 +−0.62 +−0.25 +M = 20 +K∗ +�K∗ +Buy and hold +Cumulative rate of return V (N)−V (0) +V (0) +(%) +−11.74 +−11.65 +−5.89 +Realized log-growth log V (N) +V (0) (%) +−12.48 +−12.38 +−6.07 +Volatility σ (%) +1.86 +1.86 +1.26 +Maximum percentage drawdown d∗ (%) +37.85 +37.80 +22.08 +Sharpe ratio +√ +NSR +−0.33 +−0.30 +−0.26 +M = 30 +K∗ +�K∗ +Buy and hold +Cumulative rate of return V (N)−V (0) +V (0) +(%) +14.12 +14.04 +−8.42 +Realized log-growth log V (N) +V (0) (%) +13.21 +13.14 +−8.79 +Volatility σ (%) +1.71 +1.71 +1.28 +Maximum percentage drawdown d∗ (%) +17.33 +17.33 +21.80 +Sharpe ratio +√ +NSR +0.62 +0.64 +−0.40 +Casella, G. and Berger, R. L. (2001). Statistical Inference. Cengage Learning. +Cornuejols, G. and T¨ut¨unc¨u, R. (2006). Optimization Methods in Finance. Cambridge University +Press. +Cover, T. M. (1984). An Algorithm for Maximizing Expected Log Investment Return. IEEE +Transactions on Information Theory, 30(2):369–373. +Cover, T. M. and Thomas, J. A. (2006). Elements of Information Theory. Wiley-Interscience. +Cuchiero, C., Schachermayer, W., and Wong, T.-K. L. (2019). +Cover’s Universal Portfolio, +25 + +Table 6: Online Trading Performance Using Sliding Window Approach with Transaction Costs +of 0.01% for Stocks and 0.1% for Cryptocurrency with Rebalancing Period n = 5 (Days). +M = 10 +K∗ +�K∗ +Buy and hold +Cumulative rate of return V (N)−V (0) +V (0) +(%) +−2.09 +−2.28 +−7.45 +Realized log-growth log V (N) +V (0) (%) +−2.11 +−2.31 +−7.75 +Volatility σ (%) +4.62 +4.61 +1.33 +Maximum percentage drawdown d∗ (%) +30.90 +31.09 +21.03 +Sharpe ratio +√ +NSR +0.07 +0.06 +−0.35 +M = 20 +K∗ +�K∗ +Buy and hold +Cumulative rate of return V (N)−V (0) +V (0) +(%) +−9.55 +−9.56 +−3.97 +Realized log-growth log V (N) +V (0) (%) +−10.03 +−10.05 +−4.05 +Volatility σ (%) +4.55 +4.55 +1.40 +Maximum percentage drawdown d∗ (%) +27.58 +27.59 +18.17 +Sharpe ratio +√ +NSR +−0.29 +−0.29 +−0.18 +M = 30 +K∗ +�K∗ +Buy and hold +Cumulative rate of return V (N)−V (0) +V (0) +(%) +3.50 +3.64 +6.96 +Realized log-growth log V (N) +V (0) (%) +3.44 +3.57 +6.73 +Volatility σ (%) +3.42 +3.42 +1.33 +Maximum percentage drawdown d∗ (%) +10.54 +10.52 +16.39 +Sharpe ratio +√ +NSR +0.30 +0.31 +0.56 +Table 7: Online Trading Performance Using Sliding Window Approach with Transaction Costs +of 0.01% for Stocks and 0.1% for Cryptocurrency with Rebalancing Period n = 10 (Days). +M = 10 +K∗ +�K∗ +Buy and hold +Cumulative rate of return V (N)−V (0) +V (0) +(%) +−9.43 +−9.12 +−1.15 +Realized log-growth log V (N) +V (0) (%) +−9.91 +−9.56 +−1.16 +Volatility σ (%) +6.31 +6.34 +1.39 +Maximum percentage drawdown d∗ (%) +27.56 +27.56 +18.15 +Sharpe ratio +√ +NSR +−0.30 +−0.28 +−0.01 +M = 20 +K∗ +�K∗ +Buy and hold +Cumulative rate of return V (N)−V (0) +V (0) +(%) +11.32 +11.19 +11.20 +Realized log-growth log V (N) +V (0) (%) +10.73 +10.60 +10.61 +Volatility σ (%) +7.26 +7.24 +1.53 +Maximum percentage drawdown d∗ (%) +9.51 +9.51 +4.74 +Sharpe ratio +√ +NSR +0.80 +0.79 +1.13 +Stochastic Portfolio Theory, and the Num´eraire Portfolio. 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(2013). +Portfolio Rebalancing with an +Investment Horizon and Transaction Costs. Omega, 41(2):406–420. +Wu, M.-E., Tsai, H.-H., Chung, W.-H., and Chen, C.-M. (2020). Analysis of Kelly Betting on +Finite Repeated Games. Applied Mathematics and Computation, 373:125028. +Wu, Q., Li, J. Y.-M., and Mao, T. (2022). On Generalization and Regularization via Wasserstein +Distributionally Robust Optimization. arXiv preprint arXiv:2212.05716. +Zhang, Q. (2001). Stock Trading: An Optimal Selling Rule. SIAM Journal on Control and +Optimization, 40(1):64–87. +A +Technical Lemma on Survival Trades +Lemma A.1. Let mini ci > 0. If P(V (n) > 0) = 1, then �m +i=1 Ki((1 + µi)n − ci) ≥ 0. +Proof. Fix mini ci > 0 and assume that P(V (n) > 0) = 1. Then +1 = P(V (n) > 0) = P +� m +� +i=1 +Ki +�n−1 +� +k=0 +(1 + Xi(k)) +� +> +m +� +i=1 +Kici +� +. +Since �m +i=1 Kici > 0 and �m +i=1 Ki +��n−1 +k=0(1 + Xi(k)) +� +≥ 0, applying Markov inequality yields +1 = P +� m +� +i=1 +Ki +�n−1 +� +k=0 +(1 + Xi(k)) +� +> +m +� +i=1 +Kici +� +≤ +E +��m +i=1 Ki +��n−1 +k=0(1 + Xi(k)) +�� +�m +i=1 Kici += +�m +i=1 Ki(1 + µi)n +�m +i=1 Kici +. +where the last equality holds since the returns {Xi(k) : k ≥ 0} are i.i.d. Hence, by rearranging +the inequality above, we obtain �m +i=1 Ki((1 + µi)n − ci) ≥ 0, which is desired. +29 + +0 +100 +200 +0 +0.5 +1 +Bank Account +0 +100 +200 +0 +0.5 +1 +AXP +0 +100 +200 +0 +0.5 +1 +AMGN +0 +100 +200 +0 +0.5 +1 +AAPL +0 +100 +200 +0 +0.5 +1 +BA +0 +100 +200 +0 +0.5 +1 +CAT +0 +100 +200 +0 +5 +10-4CSCO +0 +100 +200 +0 +0.5 +1 +CVX +0 +100 +200 +0 +2 +4 +10-4 GS +0 +100 +200 +0 +1 +2 +10-4 HD +0 +100 +200 +0 +0.5 +1 +10-3 HON +0 +100 +200 +0 +0.5 +1 +IBM +0 +100 +200 +0 +5 +10-4 INTC +0 +100 +200 +0 +0.5 +1 +JNJ +0 +100 +200 +0 +0.5 +1 +KO +0 +100 +200 +0 +1 +2 +10-3 JPM +0 +100 +200 +0 +0.5 +MCD +0 +100 +200 +0 +1 +2 +10-4 MMM +0 +100 +200 +0 +0.5 +1 +MRK +0 +100 +200 +0 +1 +2 +10-4MSFT +0 +100 +200 +0 +1 +2 +10-4 NKE +0 +100 +200 +0 +5 +10-3 +PG +0 +100 +200 +0 +0.5 +1 +TRV +0 +100 +200 +0 +0.5 +UNH +0 +100 +200 +0 +0.5 +1 +CRM +0 +100 +200 +0 +0.5 +1 +VZ +0 +100 +200 +0 +2 +4 +10-4 +V +0 +100 +200 +0 +0.5 +1 +WBA +0 +100 +200 +0 +0.5 +1 +WMT +0 +100 +200 +0 +0.5 +1 +DIS +0 +100 +200 +0 +0.5 +1 +Dow +0 +100 +200 +0 +5 +10-4 BTC-USD +Figure 12: Time-Varying Portfolio Weight K∗(k) (red dash line) and �K∗(k) (blue solid line) with +Window Size M = 30 (Days) and Rebalancing Period n = 1 (Day). +30 + +Jan +Feb +Mar +Apr +May +Jun +Jul +Aug +Sep +Oct +Nov +Dec +Time +2022 +0.75 +0.8 +0.85 +0.9 +0.95 +1 +1.05 +1.1 +1.15 +1.2 +Account Value +Figure 13: Equally Weighted Portfolio Versus Sliding Window Approach with Window Size +M = 30 (Days) and Rebalancing Period n = 1 (Day). +31 + diff --git a/etE0T4oBgHgl3EQf5wIr/content/tmp_files/load_file.txt b/etE0T4oBgHgl3EQf5wIr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..29172aeda198ab302fcdba431574b79e133124ec --- /dev/null +++ b/etE0T4oBgHgl3EQf5wIr/content/tmp_files/load_file.txt @@ -0,0 +1,1448 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf,len=1447 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='02754v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='PM] 7 Jan 2023 On Frequency Dependent Log-Optimal Portfolio with Transaction Costs Chung-Han Hsieh∗†∗and Yi-Shan Wong†† Department of Quantitative Finance, National Tsing Hua University, Hsinchu 300044, Taiwan R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The aim of this paper is to investigate the impact of rebalancing frequency and transaction costs on the log-optimal portfolio, which is a portfolio that maximizes the expected logarithmic growth rate of an investor’s wealth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' We prove that the frequency-dependent log- optimal portfolio problem with costs is equivalent to a concave program and provide a version of the dominance theorem with costs to determine when an investor should invest all available funds in a particular asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Then, we show that transaction costs may cause a bankruptcy issue for the frequency-dependent log-optimal portfolio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' To address this issue, we approximate the problem to obtain a quadratic concave program and derive necessary and sufficient optimality conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Additionally, we prove a version of the two-fund theorem, which states that any convex combination of two optimal weights from the optimality conditions is still optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' We test our proposed methods using both intraday and daily price data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Finally, we extend our empirical studies to an online trading scenario by implementing a sliding window approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' This approach enables us to solve a sequence of concave programs rather than a potentially computational complex stochastic dynamic programming problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Keywords: Portfolio Optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Transaction Costs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Control and Optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Log- Optimal Portfolio;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Rebalancing Frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Classcodes: G11, G14, C61 1 Introduction The takeoff point of this paper is to study the celebrated log-optimal portfolio, which calls for maximizing the Expected Logarithmic Growth (ELG) of an investor’s wealth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' This ELG maximization idea was introduced by Kelly jr (1956) and is also known as the Kelly Criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 Some earlier works related to ELG maximization and its possible applications in gambling and trading can be found in Breiman (1961);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Thorp (1975);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Cover (1984);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Algoet and Cover (1988);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Rotando and Thorp (1992);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Browne and Whitt (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Many subsequent papers contributed to the ELG problem and its various ramifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' For example, Thorp (2006) studied ELG problems in blackjack, sports betting, and the stock market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Maclean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' MacLean ∗∗†Corresponding author: Chung-Han Hsieh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Email: ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='hsieh@mx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='nthu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' ††Email: yishan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='wong13@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' This paper was supported in part by the Ministry of Science and Tech- nology, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Taiwan, under Grants: MOST110–2222–E–007–005– and MOST111–2221–E–007–124–.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 1See Poundstone (2010) for a storytelling book that brought the Kelly criterion to the attention of practical investors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 1 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2011) summarized the good and bad properties of maximizing ELG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' MacLean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2016) studied the risky short-run properties of the ELG criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Cover and Thomas (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Luenberger (2013) are textbooks that contain an introductory chapter on the ELG problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Kim and Shin (2017) demonstrated the superior return of the log-optimal portfolio compared to a traditional mean-variance portfolio in the Korean stock market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Lo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2018) connected ELG results to population genetics and discussed testable findings using experimental evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' More recently, Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2020) analyzed Kelly betting in finite repeated games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' MacLean and Zhao (2022) studied the ELG problem in a regime-switching market framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Wang and Hsieh (2022) proposed a data-driven log-optimal portfolio via a sliding window approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' However, among all of these papers, the effects of rebalancing frequency have not been extensively considered in previous literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 Rebalancing Frequency Considerations There are some existing results regarding rebalancing frequency in a log-optimal portfolio, as found in Kuhn and Luenberger (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Das and Goyal (2015), and Hsieh (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Specifically, Kuhn and Luenberger (2010) considered a portfolio optimization with re- turns following a continuous geometric Brownian motion, but only focused on two extreme cases: High-frequency trading and buy and hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' On the other hand, Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2014) and Das and Goyal (2015) studied log-optimal portfolio with the constant weight K selected without regard for the frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' However, when the same weight K is used to find an optimal rebalancing pe- riod, the resulting ELG levels are arguably suboptimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Lastly, in our prior work Hsieh (2021), we formulated a discrete-time frequency-dependent log-optimal portfolio problem and derived various optimality conditions, but we did not consider the effects of the transaction costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 Transaction Costs Considerations Transaction costs are known to significantly impact trading performance in practice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' see Cornue- jols and T¨ut¨unc¨u (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Bogle (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' These costs may include execution commissions, bid-ask spreads, latency costs, and the price impact of trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' As a result, an optimal trading policy may no longer be optimal when the transaction costs are not zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' see Cvitanic and Zapatero (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Hsieh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Consequently, many previous papers have focused on addressing the effect of transaction costs on a portfolio optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' One of the earliest models for expected execution costs was developed by Bertsimas and Lo (1998), where dynamic programming techniques are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Almgren and Chriss (2001) studied an optimal tradeoff between expected cost and risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' In the meantime, Magill and Constantinides (1976);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Shreve and Soner (1994);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Muthuraman and Kumar (2006) have all studied proportional transaction costs in continuous-time portfolio optimization problems, but none of these studies considered the effects of rebalancing frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Other transaction cost models can be found in the literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', Lobo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2007) considered fixed transaction costs in a portfolio optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Woodside-Oriakhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2013) studied fixed, and V-shaped variable transaction costs in a mean-variance model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' A recent empirical study;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' see Ruf and Xie (2020), analyzed portfolios’ performance in the presence of proportional transaction costs under various discrete rebalancing frequencies, constituent list size, and renewing frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' While it compared trading performance under various classical portfolios arising in stochastic portfolio theory, it does not take portfolio optimization into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Following the frequency-dependent portfolio optimization framework described by Hsieh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2018b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Hsieh (2021), this paper extends the formulation to incorporate proportional transaction costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' When there are nonzero proportional transaction costs, the frequency-dependent log- 2 optimal portfolio problem would be intractable since there may be no (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', because bankruptcy might occur) or only a trivial solution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', zero investments) for such a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' To address this issue, we propose an approximation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' We also derive optimality conditions for the approximate problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' In addition, we prove a version of the Dominance Theorem involving proportional transaction costs, which shows under what circumstances a log-optimal investor would invest all available funds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='3 Contributions of the Paper In Section 2, we formulate the frequency-dependent log-optimal portfolio problem involving pro- portional transaction costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' We prove that the problem is equivalent to a concave program (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1) and show a version of the Dominance Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 with cost considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' We also state a sufficient condition to invest all available funds in a single asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Then, in Section 3, we investigate bankruptcy issues when there are nonzero costs, see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Then by approximat- ing the ordinary optimization problem with a concave quadratic program, we provide necessary and sufficient optimality conditions for an approximate log-optimal portfolio;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' We further prove a version of the Two-Fund Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1, demonstrating that a combination of two optimal weights is still optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Finally, in Section 6, we extend our theory to online trading via a sliding window approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 2 Problem Formulation This section provides some background information and the formulation for the frequency- dependent log-optimal problem with costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Let N > 1 be a terminal stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' For stage k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' , N − 1, consider an investor forming a portfolio consisting of m ≥ 2 assets and assume that at least one asset is riskless with a rate of return rf ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' That is, if an asset is riskless, its return is deterministic and is treated as a degenerate random variable with value X(k) = rf for all k with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 Alternatively, if Asset i is a risky asset whose price at time k is Si(k) > 0, then its per-period return is given by Xi(k) = Si(k+1)−Si(k) Si(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' In the sequel, for risky assets, we assume that the return vectors X(k) := [X1(k) X2(k) · · · Xm(k)]T have a known distribution and have components Xi(·) which can be arbitrarily correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='3 We also assume that these vectors are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' with components satisfying Xmin,i ≤ Xi(k) ≤ Xmax,i with known bounds above and with Xmax,i being finite and Xmin,i > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The latter constraint on Xmin,i means that the loss per time step is limited to less than 100% and the price of a stock cannot drop to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 Linear Policy and Unit Simplex Constraint Consistent with the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', Barmish and Primbs (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Hsieh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2020, 2018a,b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Zhang (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Primbs (2007), we consider a linear policy with a weight vector K ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Let V (k) be the investor’s account value at stage k and the weight for Asset i is given by 0 ≤ Ki ≤ 1 represents the fraction of the account allocated to the ith asset for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Said another way, the policy for the ith asset is of a linear form ui(k) := KiV (k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Note that the number of shares invested on the ith asset is ui(k)/Si(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Since Ki ≥ 0, the investor is going long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' In view 2In practice, the actual distribution of returns may not be available to the investor, but one can always estimate it and work with the empirical surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 3Again, if the ith asset is riskless, then we put Xi(k) = rf ≥ 0 with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' If an investor maintains cash in its portfolio, then this corresponds to the case rf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 3 of this, and given that there is at least one riskless asset available, we consider the unit simplex constraint K ∈ K := � K ∈ Rm : Ki ≥ 0 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' , m, m � i=1 Ki = 1 � (1) which is classical constraint in finance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', see Cvitanic and Zapatero (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Cover and Thomas (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Luenberger (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Hsieh (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' With K ∈ K, we guarantee that 100% of the account is invested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' In the finance literature, it is known that long-only constraints like (1) can be used to mitigate the overconcentration of weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' These constraints can also assist in containing volatility and trading performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' see Jagannathan and Ma (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 Frequency Dependent Account Value Dynamics with Transaction Costs Letting n ≥ 1 be the number of steps between rebalancings, at time k = 0, the investor be- gins with initial investments u(0) = �m i=1 ui(0) with ui(0) := KiV (0) with Ki being the ith component of the portfolio weight K satisfying constraint (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' It is worth mentioning that the investment level ui(0) can be converted to the number of shares by dividing it by the price Si(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', ui(0)/Si(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The investor then waits n steps in the spirit of buy and hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' When k = n, the investment control is updated to be u(n) = �m i=1 KiV (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Continuing in this manner, a waiting period of n stages is enforced between each rebalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' To incorporate the transaction costs into the frequency-dependent framework, let ci ∈ [0, cmax] be a percentage transaction costs imposed on Asset i where cmax ∈ (0, 1) is a predetermined maximum transaction cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='4 That is, at stage k = 0, if one invests ui(0) at Asset i, then the associated transaction costs in dollar is ui(0)ci ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Then the dynamics of account value at stage n ≥ 1 is characterized by the following stochastic recursive equation:5 V (n) = V (0) + m � i=1 ui(0) Si(0)(Si(n) − Si(0)) − m � i=1 ui(0)ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2) In the sequel, we may sometimes write VK(n) instead of V (n) to emphasize the dependence on portfolio weight K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 (Transaction Costs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' If there are no transaction costs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', ci := 0 for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' , m, then the account value dynamics (2) reduces to the existing formulation in Rujeer- apaiboon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Hsieh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2018b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' see also the frictionless market setting discussed in Merton (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 4Nowadays, while some online brokerage services offer fee-free trades for certain exchange-traded funds (ETFs) in the United States, transaction costs are typically required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' For example, trading on the Taiwan Stock Exchange typically incurs a transaction cost of α · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1425% of the trade value for some α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' As a second example, using professional broker services such as Interactive Brokers Pro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', may incur a fee of $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='005 per share, with a minimum fee of $1 dollar and a maximum fee of 1% of the trade value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 5At stage ℓ ≥ 0 and rebalancing period n ≥ 1, the stochastic recursion of account value becomes V (n(ℓ + 1)) = V (nℓ) + m � i=1 ui(nℓ) Si(nℓ)(Si(n(ℓ + 1)) − Si(nℓ)) − m � i=1 ui(nℓ)ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='3 Frequency-Dependent Optimization Problem Following previous research in Hsieh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2018b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Hsieh (2021), to study the performance which is dependent on rebalancing frequency, for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' , m, we work with the n-period compound returns for each asset i, call it Xn,i, defined as Xn,i = Si(n) − Si(0) Si(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' It is readily verified that Xn,i = �n−1 k=0(1 + Xi(k)) − 1 and −1 < Xmin,i ≤ Xn,i ≤ Xmax,i where Xmax,i := (1 + Xmax,i)n − 1 and Xmin,i := (1 + Xmin,i)n − 1 > −1 for all n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' In the sequel, we work with the random vector Xn having ith component Xn,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Now for any rebalancing period n ≥ 1, we define the expected logarithmic growth (ELG) gn(K) := 1 nE � log VK(n) V (0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Our goal is to solve the following frequency-dependent stochastic maximization problem: sup {gn(K) : K ∈ K} (3) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' V (n) = V (0) + m � i=1 ui(0) Si(0)(Si(n) − Si(0)) − m � i=1 ui(0)ci where K is the unit simplex defined previously in Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The following lemma shows that maximizing the frequency-dependent ELG with nonzero costs is indeed solving a concave program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 (ELG Optimization as a Concave Program).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Fix n ≥ 1 and ci ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The frequency-dependent ELG optimization problem (3) is equivalent to max � 1 nE � log(1 + KT � Xn) � : K ∈ K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (4) where � Xn is a vector with the ith component given by � Xn,i := Xn,i −ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Additionally, Problem (4) is a concave program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' We begin by observing that the account value dynamics V (n) = V (0) + m � i=1 ui(0) Si(0)(Si(n) − Si(0)) − m � i=1 ui(0)ci = V (0) + m � i=1 KiV (0) �Si(n) − Si(0) Si(0) � − m � i=1 ui(0)ci = V (0) + m � i=1 KiV (0)Xn,i − m � i=1 ui(0)ci = (1 + KT Xn)V (0) − m � i=1 ui(0)ci = (1 + KT � Xn)V (0) (5) 5 where � Xn is a vector with the ith component given by � Xn,i := Xn,i − ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Hence, it follows that gn(K) = 1 nE � log VK(n) V (0) � = 1 nE � log(1 + KT � Xn) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Therefore, the original Problem (3) reduces to max � gn(K) = 1 nE � log(1 + KT � Xn) � : K ∈ K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The supremum operator is replaced by the maximum since gn(K) is continuous in K over a compact domain K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Hence, the Weierstrass extremum theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' see Rudin (1976), guarantees that the maximum is attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' To complete the proof, it remains to show that Problem (4) is a concave program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' This is accomplished by a standard convexity argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Since 1 + KT � Xn is affine in K, taking the logarithm function yields a concave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Moreover, taking the expec- tation and multiplying a scaling factor 1/n preserve the concavity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' see Boyd and Vandenberghe (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Therefore, the objective function 1 nE � log(1 + KT � Xn) � is a concave function in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' On the other hand, K is a unit simplex which is a convex compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Therefore, the maximization considered in Problem (4) is a concave function over a convex compact set, hence, is a concave program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Henceforth, we denote g∗ n as the optimal expected logarithmic growth associated with the given rebalancing period of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' A vector K∗ ∈ K ⊂ Rm satisfying gn(K∗) = g∗ n is called a log-optimal weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The portfolio that uses the log-optimal fraction vector is called frequency- dependent log-optimal portfolio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='4 Dominance Lemma with Costs In this section, a version of the dominance lemma with costs is stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 (Dominance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Given a collection of m ≥ 2 assets, if Asset j satisfying E � 1 + � Xn,i 1 + � Xn,j � ≤ 1, for all i ̸= j with i, j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', m}, then, for all n ≥ 1, gn(K) is maximized by K∗ = ej where ej is the unit vector in the jth coordinate direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' To prove K∗ = ej, it suffices to show that gn(K) ≤ gn(ej) for K ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' For notational convenience, we work with the random vector �Rn := � Xn + 1 where 1 := [1 1 · · · 1]T ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Since KT 1 = 1 for K ∈ K, it follows that gn(K) = 1 nE[log KT �Rn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Hence, by applying Jensen’s 6 inequality to the concave logarithmic function, we obtain gn(K) − gn(ej) = 1 nE � log KT �Rn �Rn,j � ≤ 1 n log E � KT �Rn �Rn,j � = 1 n log � m � i=1 KiE � �Rn,i �Rn,j �� = 1 n log � m � i=1 KiE � 1 + � Xn,i 1 + � Xn,j �� ≤ 1 n log � m � i=1 Ki · 1 � ≤ 1 n log 1 = 0 where the second last inequality holds since E � 1+ � Xn,i 1+ � Xn,j � ≤ 1 and the last inequality holds since �m i=1 Ki = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Therefore, gn(K) ≤ gn(ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 indicates that, under certain conditions, an optimal log-optimal in- vestor must invest all available funds in a specific asset when transaction costs are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' This result can be viewed as an extension of the Dominant Asset Theorem in Hsieh (2021) to include transaction costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' To see this, consider the case where there are no costs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', ci = 0 for all i, then � Xn,i = Xn,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' This implies that the ratio E � 1 + � Xn,i 1 + � Xn,j � = E �n−1 � k=0 1 + Xi(k) 1 + Xj(k) � = � E � 1 + Xi(0) 1 + Xj(0) ��n , where the last equality holds since Xi(k) are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Thus, the condition E � 1+ � Xn,i 1+ � Xn,j � ≤ 1 reduces to a much simpler condition E � 1+Xi(0) 1+Xj(0) � ≤ 1, which is consistent with the Dominant Asset Theorem proved in Hsieh (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 3 An Approximate Log-Optimal Portfolio Problem with Costs When the transaction costs are present, the corresponding fee-adjusted return is given by � Xn,i = Xn,i − ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The next lemma provides a sufficient condition for ensuring that the trades survive up to stage n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 (Probability of Having Survival Trades under Transaction Costs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Fix n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' If Xmin,i > c1/n i − 1 for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', m, then the probability P(V (n) > 0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Let n ≥ 1 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Observe that P(V (n) > 0) = P((1 + KT � Xn)V (0) > 0) = P(1 + KT � Xn > 0) = P � m � i=1 Ki(1 + � Xn,i) > 0 � = P � m � i=1 Ki � 1 + n−1 � k=0 (1 + Xi(k)) − 1 − ci � > 0 � = P � m � i=1 Ki �n−1 � k=0 (1 + Xi(k)) − ci � > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (6) where the third equality holds by invoking the fact that �m i=1 Ki = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Now note that the event ��m i=1 Ki ��n−1 k=0(1 + Xi(k)) − ci � > 0 � ⊇ ��n−1 k=0(1 + Xi(k)) > ci � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' With the aids of monotonicity of probability measure, Equality (6) becomes P(V (n) > 0) ≥ P �n−1 � k=0 (1 + Xi(k)) > ci � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Since �n−1 k=0(1 + Xi(k)) ≥ (1 + Xmin,i)n for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' , m and Xmin,i > c1/n i − 1 for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' , m, it follows that �n−1 k=0(1+Xi(k)) > ci for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Therefore, we have P(V (n) > 0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (i) To assure a survival trade, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 indicates that the worst returns must be large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Specifically, for n = 1, it requires Xmin,i > ci − 1 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' On the other hand, if n → ∞, which corresponds to buy and hold, then we must have Xmin,i > 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (ii) On the converse of the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1, it is readily verified that if mini ci > 0 and P(V (n) > 0) = 1 for n ≥ 1, then �m i=1 Ki((1 + µi)n − ci) ≥ 0 where µi := E[Xi(k)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' see Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' This reveals a gap in obtaining a necessary condition for survival trades in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 implies that for a fixed c∗ := maxi ci ∈ (0, 1), there exists Xmin,i > −1 such that V (n) ≤ 0 with positive probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Said another way, the investor’s account may experience a “survival issue” when the rebalancing frequency and costs are taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' In addition, this survival issue may cause the gn(K) to become ill-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' To address this issue, we use a Taylor-based quadratic approximation of gn(K) around K = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' see Casella and Berger (2001) and write gn(K) ≈ 1 n � KT E � � Xn � − 1 2KTE � � Xn � X T n � K � := �gn(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (7) It is well-known that such a quadratic approximation is accurate for small returns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' see Pulley (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='6 Hence, in the sequel, we consider an approximate frequency-dependent log-optimal portfolio problem with costs as follows: max {�gn(K) : K ∈ K} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (8) 6Without loss of generality, set ci := 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Then the Taylor expansion of E �log(1 + KT Xn)� = E ��∞ d=1(−1)d+1 (KT Xn)d d � converges for all K ∈ K if |KT Xn| ≤ 1 with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 8 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' It is readily verified that the approximate problem (8) described above is a concave quadratic program, which enables us to solve it in an efficient manner;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', see Diamond and Boyd (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 Optimality Conditions In this section, we investigate the optimality conditions for the approximate frequency-dependent log-optimal problem (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 (Necessity and Sufficiency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Fix n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Given a percentage costs ci ∈ (0, 1) for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' , m, the portfolio weight �K∗ ∈ K is optimal to the approximate frequency-dependent log- optimal problem (8), if and only if E � � Xn,i � − m � j=1 �K∗ j E � � Xn,i � Xn,j � = �K∗T E � � Xn � − �K∗T E � � Xn � X T n � �K∗, if �K∗ i > 0 (9) E � � Xn,i � − m � j=1 �K∗ j E � � Xn,i � Xn,j � ≤ �K∗T E � � Xn � − �K∗T E � � Xn � X T n � �K∗, if �K∗ i = 0 (10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Let n ≥ 1 and ci ∈ (0, 1) for all i be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' We begin by considering an equivalent constrained stochastic minimization problem described as follows: min K −KTE � � Xn � + 1 2KT E � � Xn � X T n � K s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' KT 1 − 1 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' − KTei ≤ 0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' , m where ei ∈ Rm is unit vector having one at the ith component and zeros on the other components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Consider the Lagrangian L(K, λ, µ) := −KT E � � Xn � + 1 2KTE � � Xn � X T n � K + λ(KT 1 − 1) − µT K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' By the Karush-Kuhn-Tucker (KKT) conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', see (Boyd and Vandenberghe, 2004, Chap- ter 5), if �K∗ is a local maximum then there is a scalar λ ∈ R1 and a vector µ ∈ Rm with component µj ≥ 0 such that, for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', m, − E � � Xn,i � + m � j=1 �K∗ j E � � Xn,i � Xn,j � + λ − µi = 0 (11) �K∗T 1 − 1 = 0 (12) µi �K∗ i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (13) From Equation (11), we obtain, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' , m, µi = −E � � Xn,i � + m � j=1 �K∗ j E � � Xn,i � Xn,j � + λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (14) Since µi �K∗ i = 0 for all i, we take weighted sum of Equation (14);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', m � i=1 µi �K∗ i = − �K∗TE � � Xn � + �K∗T E � � Xn � X T n � �K∗ + λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (15) 9 This implies that λ = �K∗TE � � Xn � − �K∗T E � � Xn � X T n � �K∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Substituting this into Equation (14), we have, for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', m, µi = −E � � Xn,i � + m � j=1 �K∗ j E � � Xn,i � Xn,j � + �K∗T E � � Xn � − �K∗T E � � Xn � X T n � �K∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (16) From Equation (16) and the fact that µi �K∗ i = 0, it follows that for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', m, if �K∗ i > 0, then µi = 0 and E � � Xn,i � − m � j=1 �K∗ j E � � Xn,i � Xn,j � = �K∗T E � � Xn � − �K∗T E � � Xn � X T n � �K∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' On the other hand, if �K∗ i = 0, then µi ≥ 0 and E � � Xn,i � − m � j=1 �K∗ j E � � Xn,i � Xn,j � ≤ �K∗T E � � Xn � − �K∗T E � � Xn � X T n � �K∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' To prove sufficiency, let �K∗ ∈ K and satisfies the conditions (9) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Then it follows that there exists λ ∈ R and µj > 0 such that the KKT conditions (11) to (13) hold at �K∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Since the constrained minimization problem is a convex optimization problem, it follows that the KKT conditions are also sufficient for optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Hence, �K∗ is optimal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' see Boyd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Let �K∗ be the optimum obtained by solving the approximate frequency-dependent log-optimal portfolio problem (8) and K∗ be the true log-optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Using Jensen’s inequality, we have 0 ≤ g(K∗) − g( �K∗) = E � log 1 + K∗T � Xn 1 + �K∗T � Xn � ≤ log E � 1 + K∗T � Xn 1 + �K∗T � Xn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The right-hand side is approximately zero when K∗ ≈ �K∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' As we will see later in this paper, this is typically the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' More interestingly, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 serves to compliment Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 by characterizing the log-optimal weights;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' see Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 (Two-Asset Toy Example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' To demonstrate the application of Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2, we first consider a high-frequency investor who rebalances her portfolio at every period;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', n := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Specifically, consider a two-asset portfolio including a risk-free cash asset with zero interest rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' X1(k) := rf = 0 with probability one and a risky asset with a binomial return X2(k) ∈ {− 1 2, 1 2} with probability P � X2(k) = 1 2 � := p ∈ � 1 2 + c2, 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The transaction costs are c1 = 0 for cash and c2 < 1/2 for the risky asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' If �K∗ 2 > 0, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2, we have (1 − �K∗ 2) � p − 1 2 − c2 � − �K∗ 2(1 − �K∗ 2) �1 4 − 2c2 � p − 1 2 + c2 2 �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' This implies that �K∗ 2 = −(4c2−4p+2) 4c2 2+4c2−8c2p+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Incorporating with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2, we conclude K∗ 2 := \uf8f1 \uf8f2 \uf8f3 �K∗ 2 if p ∈ � 1 2 + c2, 4c2 2+8c2+3 4+8c2 � 1 if p ∈ � 4c2 2+8c2+3 4+8c2 , 1 � (17) 10 and K∗ 1 = 1 − K∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Note that if c2 = 0, then K∗ 2 = 2(2p − 1) for p ∈ � 1 2, 3 4 � or K∗ 2 = 1 for p ∈ � 3 4, 1 � , which reduces to the classical ELG result in gambling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' see Kelly jr (1956);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Hsieh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' To see the effect of rebalancing period n > 1, we consider a second example with n = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', one rebalances the portfolio for every two periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' For c2 ∈ � 0, 1 4 � , applying Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 yield K∗ 2 := \uf8f1 \uf8f2 \uf8f3 �K∗ 2, if p ∈ � 1 2 + c2, − 4c2−9 8c2+6 − 1 4C � 1, if p ∈ � − 4c2−9 8c2+6 − 1 4C, 1 � , (18) where �K∗ 2 = 16p2+16p−16c2−12 16c2 2+24c2+32p2−16p−32p2c2−32pc2+9, and C := √ −256c4 2+384c3 2+640c2 2−504c2+81 4c2+3 and K∗ 1 = 1 − K∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' If c2 = 0, we have K∗ 2 = 16p2+16p−12 32p2−16p+9 for p ∈ � 1 2, 3 4 � and K∗ 2 = 1 for p ∈ � 3 4, 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 4 Feasible Region and Efficient Frontier Similar to how the performance of a portfolio can be characterized by its expected return and variance in the celebrated Markowitz framework, the performance of log-optimal portfolios can be characterized by the expected logarithmic growth and variance of the logarithmic growth and plotted on a two-dimensional diagram;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' see Luenberger (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The region mapped out by all possible portfolios defines the feasible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' That is, for any fixed n ≥ 1, we consider K �→ � E � log VK(n) V (0) � , var � log VK(n) V (0) �� ⊂ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' As demonstrated later in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1, the feasible region is convex to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' This means that if we take any two points within the region, the straight line connecting them does not cross the left boundary of the feasible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' A similar idea about analyzing the efficient frontier analytically can be found in Merton (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 A Version of The Two-Fund Theorem In the approximate log-optimal portfolio problem, as defined in (8), the upper left-hand portion at the boundary of the feasible region is referred to as the approximate efficient frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' This frontier is considered efficient in terms of expected logarithmic growth rate and its variance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' see also (Luenberger, 2013, Chapter 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Then, with the aid of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2, we can obtain a version of the two-fund theorem, which states that any convex combination of two optimal weights from the optimality conditions is still optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 (A Version of Two-Fund Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Let K′, K′′ ∈ K be two weights satisfying the optimality conditions stated in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Define a convex combination Kα := αK′+(1−α)K′′ with α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Then Kα also satisfies the optimality conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Take K′ and K′′ be two weights satisfying Equations (11) to (13), for all α ∈ [0, 1], we must show that the convex combination of the two weights K′ and K′′, Kα := αK′ + (1 − α)K′′, with the jth component Kα,j, also satisfies the same optimality equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' In particular, we begin by proving that Kα satisfies Equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Indeed, we observe that (αK′ + (1 − α)K′′)T 1 − 1 = αK′T 1 + (1 − α)K′′T 1 − 1 (19) where 1 := [1 1 · · · 1]T ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Since K′, K′′ satisfy Equation (12), it follows that K′T 1 = 1 and K′′T 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Therefore, Equation (19) becomes (αK′ + (1 − α)K′′)T 1 − 1 = α + (1 − α) = 1 11 which proves that the convex combination Kα satisfies Equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' To see it also satisfies µi �K∗ i = 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' , m, we observe that µi(αK′ i + (1 − α)K′′ i ) = αµiK′ i + (1 − α)µiK′′ i = α · 0 + (1 − α) · 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' To complete the proof, we show that Kα satisfies Equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' It suffices to show that for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' , m, −E � � Xn,i � + �m j=1 Kα,jE � � Xn,i � Xn,j � + λ = µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Note that the left-hand side using Kα yields − (α + (1 − α))E � � Xn,i � + m � j=1 (αK′ j + (1 − α)K′′ j )E � � Xn,i � Xn,j � + (α + (1 − α))λ = α \uf8eb \uf8ed−E � � Xn,i � + m � j=1 K′ jE � � Xn,i � Xn,j � + λ \uf8f6 \uf8f8 + (1 − α) \uf8eb \uf8ed−E � � Xn,i � + m � j=1 K′′ j E � � Xn,i � Xn,j � + λ \uf8f6 \uf8f8 = αµi + (1 − α)µi = µi which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 (Five-Asset Portfolio with Intraday Minute-by-Minute Data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' This example il- lustrates the feasible region, efficient frontier, and Two-Fund Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 using a five-asset portfolio consisting of a bank account, Vanguard Total Stock Market Index Fund ETF (Ticker: VTI), Vanguard Total Bond Market Index Fund ETF (Ticker: BND), Vanguard Emerging Mar- kets Stock Index Fund ETF (Ticker: VWO), and Bitcoin to the USD exchange rate (Ticker: XBTUSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The portfolio is well-diversified, covering the large US-Euro stock market, the global bond market, and cryptocurrency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Here, transaction costs ci = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='001% are imposed on the ETFs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', i ∈ {VTI, BND, VWO}) and costs cXBTUSD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1% on the XBTUSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='7 Besides, in- vestors receive interest at a (per-minute) rate rf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='0001% if they keep their funds in the bank account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The data used in this example spans from 09 : 30 : 00 AM to 15 : 59 : 00 PM on December 3, 2021, where the associated price trajectories for the four risky assets are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='8 To derive the approximate log-optimal portfolio and examine its trading perfor- mance, we split the entire data set into two parts: The first portion from 09 : 30 : 00 AM to 12 : 29 : 00 PM is for the in-sample optimization, and the second portion 12 : 30 : 00 PM to 15 : 59 : 00 PM is for the out-of-sample testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='9 Fix n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' We define the approximate feasible region H := �� �gn(K), var � log VK(n) V (0) �� : K ∈ K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Figures 2 and 3 show the points in H and the approximate efficient frontier for different rebal- ancing periods n = 1 and n = 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' As predicted by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1, any convex combination of two optimal weights K′ and K′′ satisfying optimality conditions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2, denoted as Kα = αK′ + (1 − α)K′′ with α ∈ [0, 1], satisfies the optimality conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Interestingly, it also lies on the approximate efficient frontier due to the small scale of the minute-by-minute price data10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' see Figures 2 and 3 for an example with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Similar findings also hold for other rebalancing periods n > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 7According to the platform Binance binance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='com/en, regular users are charged a transaction cost of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1% for Bitcoin trades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 8The price data for the four underlying risky assets (VTI, BND, VWO, XBTUSD) are retrieved from the Bloomberg terminal (accessed on November 17, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 9This will be demonstrated later in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 10This phenomenon disappears when using daily data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' see also Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 12 10:00 12:00 14:00 16:00 Dec 03, 2021 226 228 230 232 VTI 10:00 12:00 14:00 16:00 Dec 03, 2021 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='4 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='8 BND 10:00 12:00 14:00 16:00 Dec 03, 2021 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='4 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='6 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='8 48 VWO 10:00 12:00 14:00 16:00 Dec 03, 2021 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='6 104 XBTUSD Figure 1: Intraday Minute-by-Minute Prices for VTI, BND, VWO, and XBTUSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' While not pursued further in this paper, the optimality conditions derived in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 only consider the approximate logarithmic growth function �gn(K) without taking into account the log-variance var(log Vn(K)/V (0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' As a result, to ensure that any convex com- bination of two points on the approximate efficient frontier is still on the frontier, the log-variance must be included in the optimization problem (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' This topic presents a promising research di- rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 5 Illustrative Examples This section presents empirical examples to demonstrate the validity of our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' In the first two examples, we use the same intraday data set as Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 to compare the log-optimal and approximate log-optimal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' We evaluate the impact of different rebalancing periods and levels of costs on trading performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The third example examines the capability of our theory to handle the mid-sized portfolio case by considering a portfolio of thirty-two assets (with a Bank account, Dow-30 stocks, and cryptocurrency) using daily historical price data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 (Five-Asset Portfolio Revisited).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' This example demonstrates that the approxi- mate optimal weights �K∗ from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 is sufficiently close to the optimal weights K∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' To demonstrate this, we choose the weights K∗ on the efficient frontier that satisfy the logarith- mic variance condition: var � log VK∗(n) V (0) � ≡ var � log V� K∗(n) V (0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Figures 4 and 5 show the portfolio weights of the two trading strategies: the approximate log-optimal weights �K∗, and the true 13 Figure 2: An illustration of Feasible Set, Efficient Frontier, and Two-Fund Theorem (Kα with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5) using Rebalancing Period n = 1 (Minute).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' log-optimal weights K∗ with different rebalancing periods n = 1 and n = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The results show that the weights of the two strategies are nearly identical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', �K∗ i ≈ K∗ i , for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' This suggests that the approximate optimal weights �K∗ are a good approximation of the true optimal weights K∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' While not showing here, it is also worth mentioning that if the transaction costs are sufficiently large, then both of the optima K∗ and �K∗ will tend to fully invest in the bank account, meaning that K∗ Bank account ≈ �K∗ Bank account ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 (Trading Performance with Different Rebalancing Periods and Costs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' This exam- ple illustrates the in-sample and out-of-sample trading performances using the solutions obtained in previous Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Specifically, let V (N) be the account value at the terminal stage N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The portfolio realized return in period k is Rp(k) := V (k+1)−V (k) V (k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' With the aid of this real- ized return, we consider the following metrics to study the trading performance: The realized cumulative rate of return V (N)−V (0) V (0) , realized log-return log V (N) V (0) , volatility σ := std(Rp(k)), maximum percentage drawdown d∗ := max0≤k≤N Vmax(k)−V (k) Vmax(k) with Vmax(k) := max0≤i≤k V (i), and the N-period Sharpe ratio √ N · SR with SR being the per-period realized Sharpe ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='11 Starting with initial account V (0) = $1, Figures 6 and 7 reveal the in-sample and out-of- sample values of the trading account using the three trading strategies: The log-optimal portfolio 11Given a sequence of the realized portfolio per-period returns {Rp(k) : k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' , N − 1}, the per-period Sharpe ratio is SR := Rp−rf s where R p := 1 N �N−1 k=0 Rp(k) is the sample mean return, rf is the per-period risk-free rate, and s := � 1 N−1 �N−1 k=0 (Rp(k) − R p)2 is the sample standard deviation of portfolio returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' A detailed discussion of this topic can be found in Lo (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 14 ×10-5 Approximate feasible region (n=1) 6 Approximate feasible points Approximate efficient frontier Points on the approximate efficient frontier 4 2 E[log-growth] 0 2 4 6 0 1 2 3 4 5 6 7 8 Var(log-growth) ×10-8Figure 3: An Illustration of Feasible Set, Efficient Frontier, and Two-Fund Theorem (Kα with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5) using Rebalancing Period n = 5 (Minutes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' with weight K∗, the approximate log-optimum �K∗, and buy-and-hold with equal weight K = 1/m, for the same five-asset portfolio considered in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Note that there are nonzero transaction costs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='001% for trading ETFs and a cost of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1% for trading cryptocurrency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' From the figures, we see that the account value trajectory obtained using �K∗ is similar to that obtained using K∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Moreover, both of the portfolios outperform the equally-weighted buy-and- hold strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' To see clearly the effect of transaction costs on trading performance, we consider an additional scenario with zero costs for trading both ETFs and cryptocurrency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' see Figures 8 and 9 for the in- sample and out-of-sample account value trajectories under rebalancing period n = 1 and n = 5, with zero costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Both figures demonstrate that the account values are improved when there are no costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Tables 1 and 2 provide an overview of the out-of-sample trading performance metrics of the three trading strategies for different rebalancing periods n = 1 and n = 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' For the case of n = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', the portfolio is rebalanced every minute, we find that the zero costs lead to better performance of the log-optimal portfolio in terms of the Sharpe ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' When nonzero transaction costs are imposed, the Sharpe ratios for K∗ and �K∗ become negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' This suggests that transaction costs have a negative impact on trading performance especially when rebalancing occurs frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' On the other hand, for the case of n = 5, where the portfolio is rebalanced every five minutes, the Sharpe ratios are positive and generally higher than those for n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' This indicates that a longer rebalancing period incurs fewer costs and may lead to better trading performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 15 ×10-5 Approximatefeasibleregion(n=5) 5 Approximate feasible points Approximate eficient frontier 4 Points on the approximate efficient frontier 3 2 E[log-growth] 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 2 3 4 5 0 1 2 3 4 5 6 7 8 Var(log-growth) ×10-7Weights for each trading strategy (n=1) BND Bank Account VTI VWO XBTUSD Asset 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='8 Weight Figure 4: Portfolio Weights K∗ versus �K∗ with Rebalancing Period n = 1 (Minute).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Weights for each trading strategy (n=5) BND Bank Account VTI VWO XBTUSD Asset 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='9 1 Weight Figure 5: Portfolio Weights K∗ versus �K∗ with Rebalancing Period n = 5 (Minutes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 16 09:30 10:00 10:30 11:00 11:30 12:00 12:30 Time Dec 03, 2021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='995 1 Account value In-sample trading performance (n=1) 12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00 Time Dec 03, 2021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='998 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='002 Account value Out-of-sample trading performance (n=1) Figure 6: Account Value Trajectories under Three Trading Strategies (Optimal K∗, �K∗, and Equally-Weighted K = 1/m) with Rebalancing Period n = 1 (Minute) and Nonzero Costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Table 1: Out-of-Sample Trading Performance Metrics with Different Transaction Costs with Rebalancing Period n = 1 (Minute) Costs of 0% for ETFs and cryptocurrency K∗ �K∗ Buy and hold Cumulative rate of return V (N)−V (0) V (0) (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='38 Realized log-growth log V (N) V (0) (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='38 Volatility σ (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='05 Maximum percentage drawdown d∗ (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='20 Sharpe ratio √ NSR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='58 Costs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='001% for ETFs and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1% for cryptocurrency K∗ �K∗ Buy and hold Cumulative rate of return V (N)−V (0) V (0) (%) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='42 Realized log-growth log V (N) V (0) (%) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='43 Volatility σ (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='05 Maximum percentage drawdown d∗ (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='20 Sharpe ratio √ NSR −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='58 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='64 Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='3 (Mid-Sized Portfolio: Thirty-Two Assets with Daily Price Data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Our theory is readily applied to a mid-sized (or large-sized) portfolio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' As an example, we consider a portfolio consisting of 32 assets involving a bank account, Dow 30 Stocks,12 and the Bitcoin-to-USD exchange rate (Ticker: BTC-USD) over a one-year horizon from November 20, 2021 to November 12Dow 30 Stocks consist of the thirty stocks that make up the Dow Jones Industrial Average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 17 09:30 10:00 10:30 11:00 11:30 12:00 12:30 Time Dec 03, 2021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='995 1 Account value In-sample trading performance (n=5) 12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00 Time Dec 03, 2021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='998 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='002 Account value Out-of-sample trading performance (n=5) Figure 7: Account Value Trajectories under Three Trading Strategies (Optimal K∗, �K∗, and Equally-Weighted K = 1/m) with Rebalancing Period n = 5 (Minutes) and Nonzero Costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Table 2: Out-of-Sample Trading Performance Metrics with Different Transaction Costs with Rebalancing Period n = 5 (Minutes) Costs of 0% for ETFs and cryptocurrency K∗ �K∗ Buy and hold Cumulative rate of return V (N)−V (0) V (0) (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='38 Realized log-growth log V (N) V (0) (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='38 Volatility σ (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='05 Maximum percentage drawdown d∗ (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='20 Sharpe ratio √ NSR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='88 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='58 Costs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='001% for ETFs and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1% for cryptocurrency K∗ �K∗ Buy and hold Cumulative rate of return V (N)−V (0) V (0) (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='42 Realized log-growth log V (N) V (0) (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='43 Volatility σ (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='05 Maximum percentage drawdown d∗ (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='20 Sharpe ratio √ NSR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='61 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='64 20, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='13 The one-year data is divided into two parts: The first 90 days are used for in-sample optimiza- 13The data considered in this example is retrieved from Yahoo Finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' It is worth noting that the time period considered for this example is significant because the third-largest cryptocurrency exchange, FTX, declared bankruptcy on November 11, 2022, which had a significant impact on cryptocurrency markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 18 09:30 10:00 10:30 11:00 11:30 12:00 12:30 Time Dec 03, 2021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='995 1 Account value In-sample trading performance (n=1) 12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00 Time Dec 03, 2021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='998 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='002 Account value Out-of-sample trading performance (n=1) Figure 8: Account Value Trajectories under Three Trading Strategies (Optimal K∗, �K∗, and Equally-Weighted K = 1/m) with Rebalancing Period n = 1 (Minute) and Zero Costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' tion and the remainder is used for out-of-sample testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Here, we consider different scenarios for the transaction costs: zero costs, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='01%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1%, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5% for trading stocks, and zero costs and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1% fees for trading cryptocurrency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' If investors retain their capital in the bank account, they earn daily interest with a rate rf := 1%/365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Fix n = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', the portfolio is rebalanced on a daily basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' When costs for trading stocks are 0%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='01% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1%, we find that K∗ CV X ≈ �K∗ CV X ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='14 However, when the proportional cost is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5%, the approximate optimum becomes K∗ Bank account ≈ �K∗ Bank account ≈ 1, indicating that it is optimal to hold all capital in the bank account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Table 3 summarizes the performance of the three trading strategies under different levels of costs for trading stocks and cryptocurrency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' As expected, higher costs result in a significant decrease in investor revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The corresponding account value trajectories are plotted in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Subsequently, we examine the effects of different rebalancing periods by setting the rebal- ancing period to every five days, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', n = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' In this case, we find that K∗ CV X ≈ �K∗ CV X ≈ 1 for all four different levels of costs (0%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='01%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1%, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5%) for trading stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' This is in contrast to the case with n = 1, where the optimal weights dictated K∗ bank account ≈ 1 when the proportional cost was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Figure 11 shows that the associated trading performance using K∗ and �K∗ are similar and outperforms the buy-and-hold strategy with equal weights 1/m over the given time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Table 4 provides a summary of the performance metrics under four different levels of costs with rebalancing periods n = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 14Note that, in this example, Chevron Corporation (Ticker: CVX) is the dominant asset since the estimated dominance condition max1≤i≤32, i̸=CV X 1 N �N k=1 1+Xi(k) 1+XCV X (k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='998 < 1 for all the assets in the portfolio except for CVX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Hence, according to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2, a log-optimal investor should invest all the available capital in this asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 19 09:30 10:00 10:30 11:00 11:30 12:00 12:30 Time Dec 03, 2021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='995 1 Account value In-sample trading performance (n=5) 12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00 Time Dec 03, 2021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='998 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='002 Account value Out-of-sample trading performance (n=5) Figure 9: Account Value Trajectories under Three Trading Strategies (Optimal K∗, �K∗, and Equally-Weighted K = 1/m) with Rebalancing Period n = 5 (Minutes) and Zero Costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' For an even longer rebalancing period, say n = 10 and n = 30, the optimal weight K∗ CV X = 1 remains under proportional cost for stocks being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 6 Online Trading with Sliding Window Approach In previous sections, optimal weights K∗ and its approximation counterpart �K∗ were obtained as fixed values based on the empirical distributions of returns, rather than true distribution, which is typically unknown to the investor in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Moreover, these fixed weights cannot adapt to the constantly changing information in a dynamic market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' To address this issue, we apply a data-driven sliding window approach that generates time-varying log-optimal weights online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' see also Wang and Hsieh (2022) for a similar idea for online trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The idea of the sliding window approach is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' For k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' , the investor first declares a fixed window size M ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' With k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' , M −1, one solves the log-optimal portfolio problem (4) to obtain K∗ or the approximation counterpart (8) to obtain �K∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' These optimum weights are then applied in the next stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Having done that, one re-solves the log-optimal portfolio problem again using the data from k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Repeating this procedure until the end, one obtains a time-varying optimum K∗(k) or �K∗(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' This approach has a computational advantage because it solves a sequence of concave optimization problems rather than a stochastic dynamic programming problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The details of this approach can be found in Algorithm 1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 (Mid-Sized Portfolio Revisited: Online Trading via the Sliding Window Ap- proach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' To illustrate the sliding window approach, we conduct additional empirical studies 20 Nov 2021 Dec 2021 Jan 2022 Feb 2022 Mar 2022 Apr 2022 Time 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='4 Account value In-sample trading performance (n=1) Apr May Jun Jul Aug Sep Oct Nov Dec Time 2022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 Account value Out-of-sample trading performance (n=1) Figure 10: Account Value Trajectories under Three Trading Strategies (Optimal K∗, �K∗, and Equally-Weighted K = 1/m) with Rebalancing Period n = 1 (Day) and Costs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='01% for Stocks and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1% for Cryptocurrency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' using the daily price data considered in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='3 with the costs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='01% for stocks and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1% for cryptocurrency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Here, we first fix the rebalancing period n = 1 day and consider three different window sizes: M = 10, 20, 30 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' By solving the log-optimal and approximate log- optimal portfolio problems, we obtain the resulting time-varying optimal weights K∗(k) and the approximate log-optimum �K∗(k) for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' , see Figure 12 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The associated account value trajectories of three portfolios with different weights ( �K∗, K∗, and an equally-weights K = 1/m) are depicted in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' See also Table 5 for a summary of the trading performance metrics under three different window sizes M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' It is interesting to note that the portfolios with weights K∗ and �K∗ using the window size M = 30 outperform the buy-and- hold strategy in terms of Sharpe ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' This observation suggests that the window size M may be an important factor in determining the overall trading performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' While this point is not pursued further in this paper, it is worth considering in future work when implementing the sliding window approach in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Likewise, we also study the performance with different rebalancing periods n = 5 and n = 10 and with different window sizes M = 10, 20, and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' These results are summarized in Tables 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Similar to the n = 1 case, we see that for both n = 5 and n = 10, the best performance is obtained with M = 30 and M = 20, respectively in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 7 Concluding Remarks This paper focuses on incorporating rebalancing frequency and transaction costs into the log- optimal portfolio formulation, which aims to maximize the expected logarithmic growth rate of 21 Table 3: Out-of-Sample Trading Performance with Zero Costs and Different Nonzero Costs for Stocks and Cryptocurrency with Rebalancing Period n = 1 (Day).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Costs of 0% for stocks and cryptocurrency K∗ �K∗ Buy and hold Cumulative rate of return V (N)−V (0) V (0) (%) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='20 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='18 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='47 Realized log-growth log V (N) V (0) (%) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='28 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='26 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='68 Volatility σ (%) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='37 Maximum percentage drawdown d∗ (%) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='88 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='89 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='42 Sharpe ratio √ NSR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='33 Costs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='01% for stocks and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1% for cryptocurrency K∗ �K∗ Buy and hold Cumulative rate of return V (N)−V (0) V (0) (%) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='39 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='37 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='49 Realized log-growth log V (N) V (0) (%) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='68 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='66 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='71 Volatility σ (%) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='37 Maximum percentage drawdown d∗ (%) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='06 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='07 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='42 Sharpe ratio √ NSR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='33 Costs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1% for stocks and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1% for cryptocurrency K∗ �K∗ Buy and hold Cumulative rate of return V (N)−V (0) V (0) (%) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='68 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='7 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='65 Realized log-growth log V (N) V (0) (%) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='72 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='73 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='88 Volatility σ (%) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='37 Maximum percentage drawdown d∗ (%) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='15 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='17 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='42 Sharpe ratio √ NSR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='34 Costs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5% for stocks and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1% for cryptocurrency K∗ �K∗ Buy and hold Cumulative rate of return V (N)−V (0) V (0) (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='17 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='34 Realized log-growth log V (N) V (0) (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='17 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='63 Volatility σ (%) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='9 × 10−5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='7 × 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='37 Maximum percentage drawdown d∗ (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='03 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='42 Sharpe ratio √ NSR −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='45 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='38 an investor’s wealth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' We demonstrate that solving a frequency-dependent optimization problem with costs is equivalent to solving a concave program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Conditions under which a log-optimal investor would invest all available funds in a specific asset are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' We also consider the issue of bankruptcy that can arise due to transaction costs in the frequency-dependent formu- lation and propose an approximate solution using a quadratic concave program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Additionally, a version of the two-fund theorem is proven, demonstrating that a convex combination of two optimal weights is still optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' We present various empirical studies to explore the effect of considering percentage transaction cost and rebalancing periods from the small to mid-sized portfolio optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Lastly, we extend our empirical studies to an online trading scenario by implementing a sliding window approach, which allows us to solve a sequence of concave programs rather than a complex stochastic dynamic programming problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Regarding further research, one possible continuation is to consider additional practical trad- ing issues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', allowing to short an asset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', Ki < 0 for some i and/or modeling the impact of dividend/taxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Another feasible direction is to incorporate an risk term into the objective function for the ELG maximization problem, which would mitigate the situation when the op- 22 Nov 2021 Dec 2021 Jan 2022 Feb 2022 Mar 2022 Apr 2022 Time 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='4 Account value In-sample trading performance (n=5) Apr May Jun Jul Aug Sep Oct Nov Dec Time 2022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 Account value Out-of-sample trading performance (n=5) Figure 11: Account Value Trajectories under Three Trading Strategies (Optimal K∗, �K∗, and Equally-Weighted K = 1/m) with Rebalancing Period n = 5 (Days) and Costs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='01% for Stocks and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1% for Cryptocurrency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='. timum suggests betting all capital on a specific asset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', see Davis and Lleo (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Another important consideration is the potential for estimation error in the distribution of returns, which is often unknown and must be estimated in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' In this case, it may be useful to study the robust counterpart of the problem considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' That is, instead of solving supK E[log VK(N) V (0) ], one seeks to solve a data-driven distributional robust log-optimal portfolio problem sup K∈K inf P ∈P EP � log VK(N) V (0) � where P is the ambiguity set of probability distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', see Mohajerin Esfahani and Kuhn (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2022) for an approach using Wasserstein metric to characterize the ambigu- ity set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' References Algoet, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' and Cover, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Asymptotic Optimality and Asymptotic Equipartition Properties of Log-Optimum Investment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The Annals of Probability, 16(2):876–898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Almgren, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' and Chriss, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Optimal Execution of Portfolio Transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Journal of Risk, 3:5–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Barmish, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' and Primbs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' On a New Paradigm for Stock Trading via a Model-Free Feedback Controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' IEEE Transactions on Automatic Control, 61(3):662–676.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 23 Table 4: Out-of-Sample Trading Performance with Zero Costs and Different Nonzero Costs for Stocks and Cryptocurrency with Rebalancing Period n = 5 (Days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Costs of 0% for stocks and cryptocurrency K∗ �K∗ Buy and hold Cumulative rate of return V (N)−V (0) V (0) (%) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='25 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='23 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='47 Realized log-growth log V (N) V (0) (%) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='32 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='31 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='68 Volatility σ (%) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='37 Maximum percentage drawdown d∗ (%) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='55 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='55 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='42 Sharpe ratio √ NSR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='33 Costs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='01% for stocks and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1% for cryptocurrency K∗ �K∗ Buy and hold Cumulative rate of return V (N)−V (0) V (0) (%) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='88 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='87 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='49 Realized log-growth log V (N) V (0) (%) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='00 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='99 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='71 Volatility σ (%) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='37 Maximum percentage drawdown d∗ (%) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='59 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='59 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='42 Sharpe ratio √ NSR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='33 Costs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1% for stocks and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1% for cryptocurrency K∗ �K∗ Buy and hold Cumulative rate of return V (N)−V (0) V (0) (%) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='66 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='64 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='65 Realized log-growth log V (N) V (0) (%) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='13 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='12 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='88 Volatility σ (%) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='37 Maximum percentage drawdown d∗ (%) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='95 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='95 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='42 Sharpe ratio √ NSR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='49 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='34 Costs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5% for stocks and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1% for cryptocurrency K∗ �K∗ Buy and hold Cumulative rate of return V (N)−V (0) V (0) (%) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='64 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='65 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='34 Realized log-growth log V (N) V (0) (%) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='67 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='69 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='63 Volatility σ (%) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='37 Maximum percentage drawdown d∗ (%) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='53 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='53 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='42 Sharpe ratio √ NSR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='38 Bertsimas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' and Lo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Optimal Control of Execution Costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Journal of financial markets, 1(1):1–50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Bogle, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The Little Book of Common Sense Investing: The Only Way to Guarantee Your Fair Share of Stock Market Returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' John Wiley & Sons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Boyd, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', Busseti, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', Diamond, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', Kahn, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', Koh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', Nystrup, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', and Speth, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Multi-Period Trading via Convex Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' arXiv preprint arXiv:1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='00109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Boyd, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' and Vandenberghe, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Convex Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Breiman, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Optimal Gambling Systems for Favorable Games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contribu- tions to the Theory of Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' The Regents of the University of California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Browne, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' and Whitt, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Portfolio Choice and the Bayesian Kelly Criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Advances in Applied Probability, 28(4):1145–1176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 24 Algorithm 1 Online Trading via Sliding Window Approach Require: Consider m ≥ 2 assets, realized returns {Xi(k) : k ≥ 0} and transaction cost ci for i = 1, 2, · · · , m, and sliding window size M ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Ensure: Optimal portfolio weight K∗ or �K∗ for each stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 1: Compute compound returns { � Xn,i(s)} for each asset i in the portfolio with � Xn,i(s) := �ns−1 k=n(s−1)(1 + Xi(k)) − 1 − ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 2: if k ≥ M then 3: Solve the maximization Problem (4) to obtain K∗ (or solve Problem (8) to obtain �K∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 4: Having obtained optimal K∗(k) := K∗ (or K∗(k) := �K∗(k)), we apply it at stage k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Set k := k + 1 then back to Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 5: end if Table 5: Online Trading Performance Using Sliding Window Approach with Transaction Costs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='01% for Stocks and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1% for Cryptocurrency with rebalancing period n = 1 (Day).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' M = 10 K∗ �K∗ Buy and hold Cumulative rate of return V (N)−V (0) V (0) (%) −22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='88 −23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='11 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='78 Realized log-growth log V (N) V (0) (%) −25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='99 −26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='28 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='95 Volatility σ (%) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='24 Maximum percentage drawdown d∗ (%) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='19 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='35 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='08 Sharpe ratio √ NSR −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='25 M = 20 K∗ �K∗ Buy and hold Cumulative rate of return V (N)−V (0) V (0) (%) −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='74 −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='65 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='89 Realized log-growth log V (N) V (0) (%) −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='48 −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='38 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='07 Volatility σ (%) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='26 Maximum percentage drawdown d∗ (%) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='85 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='80 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='08 Sharpe ratio √ NSR −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='26 M = 30 K∗ �K∗ Buy and hold Cumulative rate of return V (N)−V (0) V (0) (%) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='12 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='04 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='42 Realized log-growth log V (N) V (0) (%) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='21 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='14 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='79 Volatility σ (%) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='28 Maximum percentage drawdown d∗ (%) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='33 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='33 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='80 Sharpe ratio √ NSR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='40 Casella, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' and Berger, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Statistical Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Cengage Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Cornuejols, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' and T¨ut¨unc¨u, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Optimization Methods in Finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Cover, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' An Algorithm for Maximizing Expected Log Investment Return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' IEEE Transactions on Information Theory, 30(2):369–373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Cover, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' and Thomas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Elements of Information Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Wiley-Interscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Cuchiero, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', Schachermayer, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', and Wong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Cover’s Universal Portfolio, 25 Table 6: Online Trading Performance Using Sliding Window Approach with Transaction Costs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='01% for Stocks and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1% for Cryptocurrency with Rebalancing Period n = 5 (Days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' M = 10 K∗ �K∗ Buy and hold Cumulative rate of return V (N)−V (0) V (0) (%) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='09 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='28 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='45 Realized log-growth log V (N) V (0) (%) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='11 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='31 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='75 Volatility σ (%) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='62 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='61 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='33 Maximum percentage drawdown d∗ (%) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='90 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='09 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='03 Sharpe ratio √ NSR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='35 M = 20 K∗ �K∗ Buy and hold Cumulative rate of return V (N)−V (0) V (0) (%) −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='55 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='56 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='97 Realized log-growth log V (N) V (0) (%) −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='03 −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='05 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='05 Volatility σ (%) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='55 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='40 Maximum percentage drawdown d∗ (%) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='58 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='59 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='17 Sharpe ratio √ NSR −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='18 M = 30 K∗ �K∗ Buy and hold Cumulative rate of return V (N)−V (0) V (0) (%) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='64 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='96 Realized log-growth log V (N) V (0) (%) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='44 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='57 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='73 Volatility σ (%) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='42 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='33 Maximum percentage drawdown d∗ (%) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='54 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='52 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='39 Sharpe ratio √ NSR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='56 Table 7: Online Trading Performance Using Sliding Window Approach with Transaction Costs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='01% for Stocks and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1% for Cryptocurrency with Rebalancing Period n = 10 (Days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' M = 10 K∗ �K∗ Buy and hold Cumulative rate of return V (N)−V (0) V (0) (%) −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='43 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='12 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='15 Realized log-growth log V (N) V (0) (%) −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='91 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='56 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='16 Volatility σ (%) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='31 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='34 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='39 Maximum percentage drawdown d∗ (%) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='56 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='56 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='15 Sharpe ratio √ NSR −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='01 M = 20 K∗ �K∗ Buy and hold Cumulative rate of return V (N)−V (0) V (0) (%) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='32 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='19 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='20 Realized log-growth log V (N) V (0) (%) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='73 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='60 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='61 Volatility σ (%) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='26 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='53 Maximum percentage drawdown d∗ (%) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='51 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='51 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='74 Sharpe ratio √ NSR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='79 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='13 Stochastic Portfolio Theory, and the Num´eraire Portfolio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Mathematical Finance, 29(3):773– 803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Cvitanic, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' and Zapatero, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Introduction to the Economics and Mathematics of Finan- cial Markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' MIT press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 26 Das, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' and Goyal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Computing Optimal Rebalance Frequency for Log-Optimal Portfolios in Linear Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Quantitative Finance, 15(7):1191–1204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Das, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', Kaznachey, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', and Goyal, M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Analysis of Kelly Betting on Finite Repeated Games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Applied Mathematics and Computation, 373:125028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Wu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=', Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Y.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Zhang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Stock Trading: An Optimal Selling Rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' SIAM Journal on Control and Optimization, 40(1):64–87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' A Technical Lemma on Survival Trades Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Let mini ci > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' If P(V (n) > 0) = 1, then �m i=1 Ki((1 + µi)n − ci) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Fix mini ci > 0 and assume that P(V (n) > 0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Then 1 = P(V (n) > 0) = P � m � i=1 Ki �n−1 � k=0 (1 + Xi(k)) � > m � i=1 Kici � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Since �m i=1 Kici > 0 and �m i=1 Ki ��n−1 k=0(1 + Xi(k)) � ≥ 0, applying Markov inequality yields 1 = P � m � i=1 Ki �n−1 � k=0 (1 + Xi(k)) � > m � i=1 Kici � ≤ E ��m i=1 Ki ��n−1 k=0(1 + Xi(k)) �� �m i=1 Kici = �m i=1 Ki(1 + µi)n �m i=1 Kici .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' where the last equality holds since the returns {Xi(k) : k ≥ 0} are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' Hence, by rearranging the inequality above, we obtain �m i=1 Ki((1 + µi)n − ci) ≥ 0, which is desired.' metadata={'source': 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100 200 0 1 2 10-3 JPM 0 100 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5 MCD 0 100 200 0 1 2 10-4 MMM 0 100 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5 1 MRK 0 100 200 0 1 2 10-4MSFT 0 100 200 0 1 2 10-4 NKE 0 100 200 0 5 10-3 PG 0 100 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5 1 TRV 0 100 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5 UNH 0 100 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5 1 CRM 0 100 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5 1 VZ 0 100 200 0 2 4 10-4 V 0 100 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5 1 WBA 0 100 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5 1 WMT 0 100 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5 1 DIS 0 100 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='5 1 Dow 0 100 200 0 5 10-4 BTC-USD Figure 12: Time-Varying Portfolio Weight K∗(k) (red dash line) and �K∗(k) (blue solid line) with Window Size M = 30 (Days) and Rebalancing Period n = 1 (Day).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 30 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Time 2022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content='2 Account Value Figure 13: Equally Weighted Portfolio Versus Sliding Window Approach with Window Size M = 30 (Days) and Rebalancing Period n = 1 (Day).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} +page_content=' 31' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQf5wIr/content/2301.02754v1.pdf'} diff --git a/fNAzT4oBgHgl3EQfoP3V/content/2301.01595v1.pdf b/fNAzT4oBgHgl3EQfoP3V/content/2301.01595v1.pdf new file mode 100644 index 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sha256:cf58e9d3b0a9e4adcba47458bdedbda2b23f40b0b81f32543e3205c573ba76d7 +size 168056 diff --git a/g9AzT4oBgHgl3EQfa_wH/content/tmp_files/2301.01376v1.pdf.txt b/g9AzT4oBgHgl3EQfa_wH/content/tmp_files/2301.01376v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b09fa12aaaab0fba8ef61143aa4796eacecd508d --- /dev/null +++ b/g9AzT4oBgHgl3EQfa_wH/content/tmp_files/2301.01376v1.pdf.txt @@ -0,0 +1,1604 @@ +ON abc TRIPLES OF THE FORM (1, c − 1, c) +ELISE ALVAREZ-SALAZAR, ALEXANDER J. BARRIOS, CALVIN HENAKU, AND SUMMER SOLLER +Abstract. By an abc triple, we mean a triple (a, b, c) of relatively prime positive integers a, b, and c +such that a+b = c and rad(abc) < c, where rad(n) denotes the product of the distinct prime factors +of n. The necessity of the ϵ in the abc conjecture is demonstrated by the existence of infinitely many +abc triples. For instance, +� +1, 9k − 1, 9k� +is an abc triple for each positive integer k. In this article, +we study abc triples of the form (1, c − 1, c) and deduce two general results that allow us to recover +existing sequences in the literature of abc triples with a = 1. +1. Introduction +In 1985, Masser and Oesterl´e proposed the abc conjecture [Oes88, Mas17], which states: +The abc conjecture. For every ϵ > 0, there are finitely many relatively prime positive integers +a, b, and c with a + b = c such that +rad(abc)1+ϵ < c, +where rad(n) denotes the product of the distinct prime factors of a positive integer n. +Due to its profound implications, this simple-to-state conjecture is one of the most important +open questions in number theory. For instance, some consequences of the abc conjecture include an +asymptotic version of Fermat’s Last Theorem, Faltings’s Theorem, Roth’s Theorem, and Szpiro’s +Conjecture [Elk91, Lan90, Oes88]. For further information on the abc conjecture, see the excellent +survey article [MM16]. +The statement of the abc conjecture naturally leads us to ask if the ϵ is necessary? This leads +us to the “simplistic abc conjecture,” which asks if there are finitely many relatively prime positive +integers a, b, and c with a + b = c for which rad(abc) < c. We call such triples (a, b, c) an abc +triple. +The “simplistic abc conjecture” is false, as demonstrated by the triple +� +1, 32k − 1, 32k� +, +which is an abc triple for each positive integer k. This infinite sequence of abc triples is one of the +first documented counterexamples to the simplistic abc conjecture and was communicated to Lang +by Jastrzebowski and Spielman [Lan90]. A theorem of Stewart [Ste84] leads to similar sequences +of abc triples such as +� +1, 87k − 1, 87k� +, where k is a positive integer [MM16]. Jastrzebowski and +Spielman’s counterexample can also be recovered from the following result: for each odd prime p +and each positive integer k, +� +1, p(p−1)k − 1, p(p−1)k� +is an abc triple [Bar23]. Another construction, +due to Granville and Tucker [GT02], shows that for each odd prime p, +� +1, 2p(p−1) − 1, 2p(p−1)� +is an +abc triple. +In this article, we prove that (1, c−1, c) is an abc triple if and only if cosocle(c−1) > rad(c), where +cosocle(m) = +m +rad(m) for m a positive integer (see Proposition 2.2). We note that the term cosocle +is borrowed from module theory, where the cosocle of an R-module M is the maximal semisimple +quotient of M, or equivalently, +M +rad(M). In our setting, the cosocle plays a crucial role in our results, +from which we recover each of the above mentioned sequences of abc triples. To provide context +2020 Mathematics Subject Classification. Primary 11D75, 11J25. +Key words and phrases. abc conjecture, abc triples, number theory. +1 +arXiv:2301.01376v1 [math.NT] 3 Jan 2023 + +2 +ELISE ALVAREZ-SALAZAR, ALEXANDER J. BARRIOS, CALVIN HENAKU, AND SUMMER SOLLER +for our work, we note that the equivalence above requires us to compute cosocle(c − 1) in order to +deduce whether (1, c−1, c) is an abc triple. The computation of cosocle(c−1) requires knowledge of +the prime factorization of c − 1, which becomes computationally difficult as c gets large. Our main +results provide a recipe for constructing infinitely many abc triples of the form (1, c − 1, c) based on +knowledge of a divisor of c − 1 or c. Our first theorem illustrates this: +Theorem 1. Let c and m be positive integers with c > 1. If m divides c−1 and cosocle(m) > rad(c), +then +� +1, ck − 1, ck� +is an abc triple for each positive integer k.. +We prove Theorem 1 in Section 2. While the proof is elementary, the result allows us to recover +each of the previously mentioned sequences of abc triples. It also leads to new sequences of abc +triples, such as +� +1, n(n−1)k − 1, n(n−1)k� +which is an abc triple for each positive integer k whenever n +is a positive integer that is either odd or even and non-squarefree (see Corollary 3.7). A slight +modification of the proof of Theorem 1 leads us to our next result (which is also proven in Section 2): +Theorem 2. Let b and m be positive integers. If m divides b + 1 and cosocle(m) > rad(b), then +� +1, bk, bk + 1 +� +is an abc triple for each positive odd integer k. +A consequence of Theorem 1 is that if (1, c − 1, c) is an abc triple, then (1, ck − 1, ck) is an +abc triple for each positive integer k (see Corollary 2.4). +Similarly, we obtain from Theorem 2 +that if (1, b, b + 1) is an abc triple, then (1, bk, bk + 1) is an abc triple for each odd integer k (see +Corollary 2.5). These results lead to the following question: given an integer c > 1, for what positive +integers k is (1, ck −1, ck) an abc triple? We answer this question with Theorem 2.8, which provides +necessary and sufficient conditions to determine those integers k which yield an abc triple of the +form (1, ck − 1, ck). +In Section 3, we demonstrate various consequences of Theorems 1 and 2. For example, we prove +that if n > 1 is an integer and p is an odd prime such that p > rad(n), then +� +1, np(p−1)k − 1, np(p−1)k� +is an abc triple for each positive integer k (see Corollary 3.5). In particular, taking (n, k) = (2, 1) +allows us to recover Granville and Tucker’s original construction [GT02]. Another consequence is the +following: if n ≥ 3 is an odd integer and b = nj −1 for some positive integer j, then +� +1, bnk, bnk + 1 +� +is an abc triple for each positive odd integer k (see Corollary 3.12). Taking (n, j) = (3, 1) gives us +that +� +1, 8k, 8k + 1 +� +is an abc triple for each odd integer k. +We conclude the article with Section 4, which is an analysis of the abc triples found by the +ABC@Home Project of the form (1, c − 1, c) with c < 1018. The ABC@Home Project was a network +computing project that was started in 2006 by the Mathematics Department of Leiden University, +together with the Dutch Kennislink Science Institute. By 2011, they found that there are exactly +14 482 065 abc triples (a, b, c) with c < 1018. By the time the project came to a close in 2015, the +ABC@Home Project had found a total of 23 827 716 abc triples (a, b, c) with c < 263. We note that +this list is not exhaustive of all abc triples with c < 263. In particular, the ABC@Home project found +that there are exactly 45 604 abc triples of the form (1, c − 1, c) with c < 1018. Further observations +about the abc triples found by the ABC@Home Project can be found in [Pal14, Chapter 7]. +Motivated by the results in Section 3, we study those abc triples found by the ABC@Home +Project that are of the form (1, nl − 1, nl) or (1, nl, nl + 1) for some integer l > 1. We find that +this amounts to 8 413 abc triples. For abc triples (1, c − 1, c) of the aforementioned form, we show +that approximately 48.7% of the abc triples with c ≤ 106 can be obtained from the results proven +in Section 3. We also find that for abc triples of the form (1, nl − 1, nl), there are only four cases +where there does not exists a proper divisor m of nl − 1 for which cosocle(m) > rad(n). + +ON abc TRIPLES OF THE FORM (1, c − 1, c) +3 +2. Main Results +In this section, we establish Theorems 1 and 2. To do so, we recall the following elementary +property about the radical of a positive integer: +Lemma 2.1. Let m and n be relatively prime positive integers. Then rad(mn) = rad(m) rad(n) +and rad(m) ≤ m. Moreover, rad +� +mk� += rad(m) for each positive integer k. +We will assume Lemma 2.1 implicitly throughout this work. Next, we show an important facet +about abc triples of the form (1, c − 1, c), which showcases the importance of the cosocle in our +arguments: +Proposition 2.2. Let c > 1 be an integer. Then the following are equivalent: +(i) cosocle(c − 1) > rad(c); +(ii) cosocle(c) > rad(c − 1); +(iii) (1, c − 1, c) is an abc triple. +Proof. Suppose that rad(c) < cosocle(c − 1). Since rad(c) = +c +cosocle(c) and cosocle(c − 1) = +c−1 +rad(c−1), +we deduce that +rad(c) < cosocle(c − 1) +⇐⇒ +c +cosocle(c) < +c − 1 +rad(c − 1) +⇐⇒ +rad(c − 1) < c − 1 +c +cosocle(c) . +Since c−1 +c +< 1, we have the desired inequality: rad(c − 1) < cosocle(c). +Next, suppose that rad(c−1) +cosocle(c) < 1. Since rad(c) = +c +cosocle(c), we observe that +rad(c(c − 1)) = rad(c) rad(c − 1) = rad(c − 1) +cosocle(c) c < c, +which shows that (1, c − 1, c) is an abc triple. +Lastly, if (1, c − 1, c) is an abc triple, then rad(c(c − 1)) < c. Consequently, +c > rad(c(c − 1)) = rad(c) rad(c − 1) = rad(c)(c − 1) +cosocle(c − 1) +=⇒ +rad(c) < cosocle(c − 1) +c +c − 1. +Since rad(c) is an integer and +c +c−1 > 1, we deduce that rad(c) ≤ +� +cosocle(c − 1) +c +c−1 +� +, where ⌊x⌋ +denotes the floor function. Since cosocle(c−1) +c−1 +< 1, we observe that +� +cosocle(c − 1) +c +c − 1 +� += +� +cosocle(c − 1) + cosocle(c − 1) +c − 1 +� += cosocle(c − 1). +Lastly, c is relatively prime to c − 1, and thus cosocle(c − 1) > rad(c). +□ +An automatic consequence of Proposition 2.2 is that if c or c − 1 is squarefree, then (1, c − 1, c) is +not an abc triple since the cosocle of a squarefree positive integer is 1. Our next result establishes +that the radical of a positive integer n is preserved if n is divided by the cosocle of any of its divisors: +Lemma 2.3. Let m and n be positive integers. If m divides n, then rad(n) = rad +� +n +cosocle(m) +� +. + +4 +ELISE ALVAREZ-SALAZAR, ALEXANDER J. BARRIOS, CALVIN HENAKU, AND SUMMER SOLLER +Proof. If m = 1, there is nothing to show. +So suppose that m > 1 and let m = �r +i=1 pei +i +be +the unique prime factorization of m, with each pi denoting a distinct prime. Since m divides n, +we have that n = q �r +i=1 pfi +i +where ei ≤ fi for 1 ≤ i ≤ r and q is relatively prime to m. Since +cosocle(m) = �r +i=1 pei−1 +i +, we deduce that +n +cosocle(m) = q +r +� +i=1 +pfi−ei+1 +i +For 1 ≤ i ≤ r, observe that fi − ei + 1 ≥ 1 and thus rad +� +n +cosocle(m) +� += rad(n). +□ +With this lemma, we are now ready to prove Theorem 1: +Proof of Theorem 1. Since ck − 1 = (c − 1) �k−1 +j=0 cj, we deduce that m divides ck − 1 for each +positive integer k. By Lemma 2.3, rad +� +ck − 1 +� += rad +� +ck−1 +cosocle(m) +� +. By assumption, +rad(c) +cosocle(m) < 1 +and thus +rad +� +ck � +ck − 1 +�� += rad(c) rad +� +ck − 1 +cosocle(m) +� +≤ +rad(c) +cosocle(m) +� +ck − 1 +� +< ck − 1. +The result now follows since +ck − rad +� +ck � +ck − 1 +�� +> ck − ck + 1 = 1. +□ +An immediate consequence of Theorem 1 and Proposition 2.2, is the following result: +Corollary 2.4. If (1, c − 1, c) is an abc triple, then +� +1, ck − 1, ck� +is an abc triple for each positive +integer k. +In the next section, we will consider further consequences of Theorem 1 that do not require knowl- +edge of an abc triple at the start. The proof of Theorem 1 relies on the factorization of ck − 1. A +similar factorization holds for bk + 1 if k is odd, and our proof of Theorem 2 makes use of this: +Proof of Theorem 2. If k is a positive odd integer, then bk + 1 = (b + 1) �k−1 +j=0 (−1)j bj. It follows +that m divides bk + 1 for each positive integer k. By Lemma 2.3, rad(bk + 1) = rad +� +bk+1 +cosocle(m) +� +. +Since +rad(b) +cosocle(m) < 1, we observe that +rad +� +bk � +bk + 1 +�� += rad(b) rad +� +bk + 1 +cosocle(m) +� +≤ +rad(b) +cosocle(m) rad(bk + 1) < bk + 1. +Consequently, +bk + 1 − rad +� +bk � +bk + 1 +�� +> bk + 1 − bk − 1 = 0. +□ +Similarly to the deduction of Corollary 2.4, we now recover the following result as an immediate +consequence of Theorem 2 and Proposition 2.2: +Corollary 2.5. If (1, b, b + 1) is an abc triple, then +� +1, bk, bk + 1 +� +is an abc triple for each positive +odd integer k. +Since (1, 8, 9) is an abc triple, we deduce from Corollary 2.5 that (1, 8k, 8k + 1) is an abc triple +for each positive odd integer k. We will also recover this sequence of abc triples as a consequence +of Corollary 3.12. +By Corollary 2.4, we have that if (1, c − 1, c) is an abc triple, then +� +1, ck − 1, ck� +is an abc triple +for each positive integer k. This leads us to ask: given a positive integer c > 1, for what positive + +ON abc TRIPLES OF THE FORM (1, c − 1, c) +5 +integers k is +� +1, ck − 1, ck� +an abc triple? To answer this question, we first recall a few number +theory facts. Given a prime number p and a positive integer n, the p-adic valuation of n, denoted +vp(n), is the unique integer that satisfies n = pvp(n)q for some integer q that is relatively prime to +p. Suppose further that p does not divide n. Then the order of n modulo p, denoted ordp(n), is the +least positive integer for which nordp(n) ≡ 1 mod p. By Fermat’s Little Theorem, ordp(n) divides +p − 1. More generally, nk ≡ 1 mod p if and only if ordp(n) divides k. With this terminology, we +determine the exact power of a prime p that divides ck − 1: +Lemma 2.6. Let c and k be positive integers with c > 1. Then p divides ck −1 if and only if ordp(c) +divides k. Moreover, if p divides ck − 1, then vp +� +ck − 1 +� += fp + wp, where fp = vp +� +cordp(c) − 1 +� +and +wp = vp(k). +Proof. The statement that p divides ck−1 if and only if ordp(c) divides k is a standard number theory +result. So suppose that p divides ck − 1. Then ordp(c) divides k and p − 1. In particular, ordp(c) +is not divisible by p. It follows that k = q1pwp ordp(c) for some integer q1 that is not divisible by p. +By assumption, cordp(c) − 1 = q2pfp for some integer q2 that is not divisible by p. Now observe that +by the Binomial Theorem, +ck = +� +cordp(c)�q1pwp += +� +q2pfp + 1 +�q1pwp += 1 + q1q2pfp+wp + +q1pwp−1 +� +j=2 +�q1pwp +j +� +qj +2pfpj. +For 2 ≤ j ≤ q1pwp − 1, we have that pwp+fp+1 divides +�q1pwp +j +� +qj +2pfpj. Hence ck ≡ 1 mod pwp+fp and +ck ≡ 1 + q1q2pfp+wp mod pfp+wp+1 ̸= 0. In particular, vp +� +ck − 1 +� += fp + wp. +□ +As an immediate consequence of Lemma 2.6 and the Fundamental Theorem of Arithmetic, we +obtain the following factorization for ck − 1: +Corollary 2.7. Let c and k be positive integers with c > 1. Then with notation as in Lemma 2.6, +ck − 1 = +� +ordp(c)|k +pfp+wp. +As a demonstration of Corollary 2.7, let c = 21 and k = 12. With notation as above, we see that +w2 = 2, w3 = 1, and wp = 0 for each prime p ̸= 2, 3. Next, we observe that +(2.1) +5540 − 1 = 24 · 5 · 11 · 13 · 17 · 61 · 421 · 463 · 3181. +By Lemma 2.6, the primes appearing in (2.1) are precisely those primes p for which ordp(21) divides +12. With a computer algebra system, such as SageMath [S+23], it is checked that fp = 1 for each +prime p ̸= 2 appearing in (2.1) and f2 = 2. Thus, 2112 − 1 = � +ordp(21)|12 pfp+wp. +Theorem 2.8. Let c and k be positive integers with c > 1. With notation as in Corollary 2.7, write +ck − 1 = +� +ordp(c)|k +pfp+wp. +Then +� +1, ck − 1, ck� +is an abc triple if and only if one of the following conditions hold: +(i) there exists a prime p > rad(c) such that ordp(c) divides k and either fp ≥ 2 or wp ≥ 1; +(ii) there exists a prime p < rad(c) such that ordp(c) divides k and fp + wp − 1 ≥ mp, where mp +denote the least positive integer such that pmp > rad(c); +(iii) for each prime p such that ordp(c) divides k, there exist a non-negative integer ap ≤ fp+wp−1 +such that � +ordp(c)|k pap > rad(c). + +6 +ELISE ALVAREZ-SALAZAR, ALEXANDER J. BARRIOS, CALVIN HENAKU, AND SUMMER SOLLER +Proof. First suppose that +� +1, ck − 1, ck� +is an abc triple. By Proposition 2.2, this is equivalent to +rad(c) < cosocle +� +ck − 1 +� += +� +ordp(c)|k +pfp+wp−1. +In particular, taking ap = fp + wp − 1 yields (iii). +Now suppose there is a prime p > rad(c) such that ordp(c) divides k and either fp ≥ 2 or wp ≥ 1. +Note that fp ≥ 1 for each prime p such that ordp(c) divides k. Consequently, if fp ≥ 2 or wp ≥ 1, +then fp +wp ≥ 2 and thus p2 divides ck −1. Then +� +1, ck − 1, ck� +is an abc triple by Theorem 1 since +cosocle +� +p2� += p > rad(c). +Next, suppose that there is a prime p < rad(c) such that ordp(c) divides k and fp + wp − 1 ≥ mp. +Then pmp+1 divides ck − 1 and +cosocle +� +pmp+1� += pmp > rad(c) . +By Theorem 1, we deduce that +� +1, ck − 1, ck� +is an abc triple. +Lastly, suppose that for each prime p such that ordp(c) divides k, there exist a positive integer +ap ≤ fp + wp − 1 such that � +ordp(c)|k pap > rad(c). Then � +ordp(c)|k pap+1 divides ck − 1 and the +result now follows by Theorem 1 since +cosocle +� +� +� +ordp(c)|k +pap+1 +� +� = +� +ordp(c)|k +pap > rad(c) . +□ +As an illustration, consider c = 21 and k = 12. In the discussion following Corollary 2.7, we +noted that w2 = f2 = 2. Moreover, for each prime p ̸= 2 appearing in (2.1) we have that fp = 1 +and wp = 0. In particular, we see that statements (i) and (ii) of Theorem 2.8 are not satisfied for +each prime p appearing in (2.1). We also have that statement (iii) is not satisfied as the only prime +for which fp + wp − 1 > 0 is p = 2 and 2f2+w2−1 = 16 < rad(21). It follows that +� +1, 2112 − 1, 2112� +is not an abc triple. In the next section, we will see that 21 is the first odd integer n > 1 for which +� +1, nϕ(n) − 1, nϕ(n)� +is not an abc triple, where ϕ(n) denotes the Euler-totient function. We note +that ϕ(21) = 12. +3. Consequences +In this section, we consider various consequences of Theorems 1 and 2. From these consequences, +we deduce the sequences of abc triples that were mentioned in the introduction. We note that +this article began as an investigation of the following question: for what positive odd integers n +is +� +1, nϕ(n) − 1, nϕ(n)� +an abc triple? Here ϕ(n) denotes the Euler-totient function. The question +was motivated by the following observation: if n is an odd integer such that 3 ≤ n ≤ 99, then +� +1, nϕ(n) − 1, nϕ(n)� +is an abc triple for each n except n = 21, 39, 69, and 87. The fact that the four +exceptions are composites is no surprise, as the question is true for odd primes n [Bar23]. Our +investigation of this phenomenon led to our Theorems 1 and 2, and our first consequence provides +necessary conditions for when +� +1, nϕ(n) − 1, nϕ(n)� +is an abc triple for a positive odd integer n. To +prove this result, we first recall the following result from elementary number theory: +Lemma 3.1. Let n be a positive odd integer. Then n2k ≡ 1 mod 2k+2 for each positive integer k. +Proof. Since n is odd, there is an integer m such that n = 2m + 1. By the Binomial Theorem, +n2k = (2m + 1)2k = +2k +� +j=0 +�2k +j +� +(2m)j = 1 + 2k+1m +� +1 + +� +2k − 1 +� +m +� ++ +2k +� +j=3 +�2k +j +� +(2m)j . + +ON abc TRIPLES OF THE FORM (1, c − 1, c) +7 +Now observe that m +� +1 + +� +2k − 1 +� +m +� +is always even and +�2k +j +� +(2m)j is divisible by 2k+2 for 3 ≤ j ≤ +2k. Consequently, n2k ≡ 1 mod 2k+2. +□ +With this result, we obtain our our first application of Theorem 1: +Corollary 3.2. Let n > 1 be an odd integer and let ϕ denote the Euler-totient function. +Set +d = gcd(n−1, ϕ(n)) and m = 2v2(4ϕ(n))−2v2(d)d2. If cosocle(m) > rad(n), then +� +1, nϕ(n)k − 1, nϕ(n)k� +is an abc triple for each positive integer k. +Proof. Let P = �ϕ(n)−1 +j=0 +nj and observe that nϕ(n) − 1 = (n − 1) P. Since d = gcd(n − 1, ϕ(n)) +divides n − 1, n ≡ 1 mod d and thus +P ≡ +ϕ(n)−1 +� +j=0 +1j mod d = ϕ(n) mod d. +In particular, d divides P. Since nϕ(n) − 1 = (n − 1) P, we deduce that d2 divides nϕ(n) − 1. +Next, write ϕ(n) = 2v2(ϕ(n))r for r an odd integer. By Lemma 3.1, +nϕ(n) − 1 = (nr)2v2(ϕ(n)) − 1 ≡ 0 mod 2v2(ϕ(n))+2. +Hence 2v2(ϕ(n))+2 divides nϕ(n) − 1. It follows that +2v2(ϕ(n))+2 +d2 +2v2(d2) = 2v2(4ϕ(n))−2v2(d)d2 = m +divides nϕ(n) − 1. The result now follows from Theorem 1. +□ +As an illustration, let n = 75. Then with notation as in Corollary 3.2, we observe that ϕ(75) = +40, d = 2, and m = 32. Since cosocle(32) = 16 > rad(75) = 15, we have that +� +1, 7540k − 1, 7540k� +is an abc triple for each positive integer k. We note that the converse to Corollary 3.2 does not +hold. In fact, if 3 ≤ n ≤ 99 is an odd integer such that +� +1, nϕ(n) − 1, nϕ(n)� +is an abc triple, then the +corollary fails to show the cases corresponding to n = 33, 35, 55, 57, 63, 65, 77, 93, 95, and 99. The +following result provides an improvement, but comes at the cost of having to compute vp(nϕ(n) − 1) +for each prime p that divides gcd(nϕ(n) − 1, ϕ(n)): +Corollary 3.3. Let n > 1 be an integer and let ϕ denote the Euler-totient function. Set d = +gcd(nϕ(n) − 1, ϕ(n)) and +m = +� +p|d +pvp(nϕ(n)−1). +If cosocle(m) > rad(n), then +� +1, nϕ(n)k − 1, nϕ(n)k� +is an abc triple for each positive integer k. +Proof. By construction, m divides nϕ(n) − 1. The result now follows from Theorem 1. +□ +For odd integers n such that 3 ≤ n ≤ 99 and +� +1, nϕ(n) − 1, nϕ(n)� +is an abc triple, Corollary 3.3 +allows us to conclude that +� +1, nϕ(n) − 1, nϕ(n)� +is an abc triple for each n except n = 55, 57. When +n = 55, we have that ϕ(55) = 40 and gcd +� +5540 − 1, 40 +� += 8. Then m = 2v2(5540−1) = 64, and thus +cosocle(64) = 32 < rad(55) = 55. Consequently, Corollary 3.3 fails to show that +� +1, 5540 − 1, 5540� +is an abc triple. We note the cosocle +� +5540 − 1 +� += 288, and hence +� +1, 5540k − 1, 5540k� +is an abc triple +for each positive integer k by Proposition 2.2. The failure of Corollaries 3.2 and 3.3 in the n = 55 case +stems from the fact that the primes dividing m must divide ϕ(n). Indeed, cosocle +� +5540 − 1 +� += 32·9 +and 3 ∤ ϕ(55). + +8 +ELISE ALVAREZ-SALAZAR, ALEXANDER J. BARRIOS, CALVIN HENAKU, AND SUMMER SOLLER +To state our next result, we recall the Carmichael function λ : N → N, which has the property +that λ(m) is the least positive integer for which aλ(m) ≡ 1 mod m for each integer a that is relatively +prime to m. In particular, λ(m) divides ϕ(m). +Corollary 3.4. Let λ and ϕ denote the Carmichael function and Euler-totient function, respec- +tively. If m and n are relatively prime positive integers such that cosocle(m) > rad(n) > 1, then +� +1, nλ(m)k − 1, nλ(m)k� +and +� +1, nϕ(m)k − 1, nϕ(m)k� +are abc triples for each positive integer k. +Proof. Since nλ(m) ≡ 1 mod m, we have that m divides nλ(m) − 1. By Theorem 1, we have that +� +1, nλ(m)k − 1, nλ(m)k� +is an abc triple for each positive integer k. Since λ(m) | ϕ(m), we also have +that +� +1, nϕ(m)k − 1, nϕ(m)k� +is an abc triple for each positive integer k. +□ +As an example, choose n = 11 and m = 32. Then cosocle(32) = 16 > rad(11), and therefore +the conditions of Corollary 3.4 are satisfied. As a result, we find that +� +1, 11λ(32)k − 1, 11λ(32)k� += +(1, 118k − 1, 118k) is a sequence of abc triples. More generally, we have the following application of +Corollary 3.4: +Corollary 3.5. Let n > 1 be an integer and let p be an odd prime such that p > rad(n). Then for +each positive integer k, +� +1, np(p−1)k − 1, np(p−1)k� +is an abc triple. +Proof. By assumption, cosocle +� +p2� += p > rad(n). Moreover, λ +� +p2� += p (p − 1) since p is prime. It +follows from Corollary 3.4 that +� +1, nλ(p2)k − 1, nλ(p2)k� += +� +1, np(p−1)k − 1, np(p−1)k� +is an abc triple +for each positive integer k. +□ +Taking (n, k) = (2, 1) in Corollary 3.5 yields that +� +1, 2p(p−1) − 1, 2p(p−1)� +is an abc triple for each +odd prime p. This result is originally due to Granville and Tucker [GT02]. Theorem 2.8 gives the +following refinement of Corollary 3.5: +Corollary 3.6. Let n > 1 be an integer and let p be an odd prime such that p > rad(n). Then for +each positive integer k, +� +1, np ordp(n)k − 1, np ordp(n)k� +is an abc triple. In particular, if n ≡ 1 mod p +and p > rad(n), then +� +1, npk − 1, npk� +is an abc triple for each positive integer n. +Proof. In the notation of Theorem 2.8, we have that wp = vp(p ordp(n)) = 1. Since p > rad(n) and +ordp(n) divides p ordp(n), Theorem 2.8 (i) implies that +� +1, np ordp(n) − 1, np ordp(n)� +is an abc triple. +The result now follows by Corollary 2.4. The second statement is automatic since if n ≡ 1 mod p, +then ordp(n) = 1. +□ +As a demonstration, let n = 16 and p = 5. Then Corollary 3.6 asserts that +� +1, 165k − 1, 165k� +is +an abc triple for each positive integer k. +Corollary 3.7. Let n > 1 be an integer that is either odd or even and non-squarefree. +Then +� +1, n(n−1)k − 1, n(n−1)k� +is an abc triple for each positive integer k. +Proof. Let P = �n−2 +j=0 nj and observe that nn−1 − 1 = (n − 1) P. Moreover, +P ≡ +n−2 +� +j=0 +(1)j mod(n − 1) = 0 mod (n − 1) . +In particular, (n − 1)2 divides nn−1 − 1 and thus +rad(nn−1 − 1) = rad +�nn−1 − 1 +n − 1 +� +. + +ON abc TRIPLES OF THE FORM (1, c − 1, c) +9 +Now suppose that n is odd. We claim that 4 divides P. If n ≡ 1 mod 4, then this follows since P +is divisible by n − 1. So suppose that n ≡ 3 mod 4. Then 4 divides n + 1, and hence 4 divides P +since +P ≡ +n−2 +� +j=0 +(−1)j mod(n + 1) = 0 mod (n + 1) . +Consequently, +(3.1) +rad(nn−1 − 1) = rad +�nn−1 − 1 +2 (n − 1) +� +≤ nn−1 − 1 +2 (n − 1). +Now observe that by (3.1), +cosocle(nn−1 − 1) = +nn−1 − 1 +rad(nn−1 − 1) ≥ 2 (n − 1) > rad(n). +The claim now follows by Theorem 1 with m = nn−1 − 1. +Lastly, suppose that n is an even non-squarefree positive integer. Then n = a2b for some positive +integers a and b with a > 1 and b squarefree. +Then rad(n) = rad(ab) ≤ ab < n − 1. +Since +rad(nn−1 − 1) = rad +� +nn−1−1 +n−1 +� +≤ nn−1−1 +n−1 , we deduce that +cosocle(nn−1 − 1) = +nn−1 − 1 +rad(nn−1 − 1) ≥ n − 1 > rad(n) . +The result follows by Theorem 1 with m = nn−1 − 1. +□ +From Corollary 3.7, we recover that +� +1, 9k − 1, 9k� += +� +1, 32k − 1, 32k� +is a sequence of abc triples. +In particular, we obtain the smallest abc triple (1, 8, 9) as a special case. +Taking n = 8 in +Corollary 3.7 gives us the sequence of abc triples +� +1, 87k − 1, 87k� +, which generalizes the sequence +� +1, 87k − 1, 87k� +that appears in [MM16]. +Corollary 3.8. Let n > 1 be an integer. Then +� +1, n(n+1)k − 1, n(n+1)k� +is an abc triple whenever +(n + 1) k is a positive even integer. +Proof. Let l be a positive even integer and let P = �l−1 +j=0 (−1)j+1 nj. Then nl − 1 = (n + 1) P. We +now proceed by cases. +Case 1. Suppose that n is a positive even integer and let l = 2 (n + 1). Since n ≡ −1 mod(n + 1), +we have that P ≡ �l−1 +j=0 (−1)j+1 = 0 mod(n + 1) and thus +rad(nl − 1) = rad +�nl − 1 +n + 1 +� +≤ nl − 1 +n + 1 . +The claim now holds by Theorem 1 with m = nl − 1 since +cosocle(nl − 1) = +nl − 1 +rad(nl − 1) ≥ n + 1 > rad(n). +Case 2. Suppose that n is a positive odd integer. Then l = n+1 is even and P ≡ 0 mod(n + 1). +A similar argument to that of Case 1 with m = nl −1 shows that the result holds by Theorem 1. +□ +As an example, choose n = 21. As a result, (n + 1)k is even for every positive integer k and by +Corollary 3.8, +� +1, 2122k − 1, 2122k� +is a sequence of abc triples. + +10 +ELISE ALVAREZ-SALAZAR, ALEXANDER J. BARRIOS, CALVIN HENAKU, AND SUMMER SOLLER +Corollary 3.9. Let j ≥ 2 be an integer. Then +� +1, +� +2j − 1 +�2k − 1, +� +2j − 1 +�2k� +is an abc triple for +each positive integer k. +Proof. Observe that rad +�� +2j − 1 +�2� += rad +� +2j − 1 +� +≤ 2j − 1. Since +� +2j − 1 +�2 − 1 = 2j+1 � +2j−1 − 1 +� +, +we deduce that +cosocle +�� +2j − 1 +�2 − 1 +� += 2j+1 � +2j−1 − 1 +� +2 rad(2j−1 − 1) = 2j � +2j−1 − 1 +� +rad(2j−1 − 1) ≥ 2j. +The result now follows from Theorem 1, since cosocle((2j − 1)2 − 1) > rad((2j − 1)2). +□ +The j = 2 and j = 3 cases in Corollary 3.9 result in the sequences of abc triples +� +1, 9k − 1, 9k� +and +� +1, 49k − 1, 49k� +, respectively. Of note is that the proof of the corollary is made possible by the +lower bound, cosocle +�� +2j − 1 +�2 − 1 +� +≥ 2j. This leads us to ask, can Corollary 3.9 be generalized +to deduce sequences of abc triples (1, c − 1, c) with cosocle(c − 1) bounded below by nj for some +positive integer of the form nj? The answer is yes, but we have to take c = +� +nj − 1 +�k for some +positive even integer k that is divisible by n to allow a similar argument to that of Corollary 3.9 to +work. This is shown below: +Corollary 3.10. Let n ≥ 3 and j ≥ 1 be integers. If k is a positive integer such that nk is even, +then +� +1, +� +nj − 1 +�nk − 1, +� +nj − 1 +�nk� +is an abc triple. +Proof. Observe that rad +�� +nj − 1 +�nk� +≤ nj − 1 and +� +nj − 1 +�nk − 1 = −1 + +nk +� +l=0 +�nk +l +� +njl (−1)nk−l = −knj+1 + +nk +� +l=2 +�nk +l +� +njl (−1)nk−l . +Note that in the last expression, each term in the sum is divisible by nj+1. From this, we deduce +that cosocle +�� +nj − 1 +�nk − 1 +� +≥ nj. Hence cosocle +�� +nj − 1 +�nk − 1 +� +> rad +�� +nj − 1 +�nk� +, and the +result now follows by Theorem 1. +□ +As an illustration, consider (n, j) = (3, 1) and k = 2l for some positive integer l. This results in +the sequence of abc triples +� +1, 64l − 1, 64l� +. +Corollary 3.11. Let n be a positive even integer. Then +� +1, n(n+1)k, n(n+1)k + 1 +� +is an abc triple +for each positive odd integer k. +Proof. Observe that nn+1 + 1 = (n + 1) �n +j=0 (−1)j nj. Since n ≡ −1 mod(n + 1), it follows that +n +� +j=0 +(−1)j nj ≡ +n +� +j=0 +1 mod(n + 1) = 0 mod(n + 1) +Hence, rad(nn+1 + 1) = rad +� +nn+1+1 +n+1 +� +≤ nn+1+1 +n+1 . Consequently, +cosocle(nn+1 + 1) = +nn+1 + 1 +rad(nn+1 + 1) ≥ n + 1 > rad(n) . +The result now follows from Theorem 2 by taking m = nn+1 + 1. +□ +As a demonstration of the corollary, take n = 22. Then +� +1, 2223k, 2223k + 1 +� +is a sequence of abc +triples for each positive odd integer k. + +ON abc TRIPLES OF THE FORM (1, c − 1, c) +11 +Corollary 3.12. Let n ≥ 3 be an odd integer and let j ≥ 1 be an integer. Then for each odd +integer k, +� +1, +� +nj − 1 +�nk , +� +nj − 1 +�nk + 1 +� +is an abc triple. +Proof. Observe that rad +�� +nj − 1 +�n� +≤ nj − 1 and +� +nj − 1 +�n + 1 = 1 + +n +� +l=0 +�n +l +� +njl (−1)n−l = nj+1 + +n +� +l=2 +�n +l +� +njl (−1)n−l . +Note that in the last expression, each term in the sum is divisible by nj+1. From this, we conclude +that cosocle +�� +nj − 1 +�n + 1 +� +≥ nj. Hence cosocle +�� +nj − 1 +�n + 1 +� +> rad +�� +nj − 1 +�n� +, and the result +now follows by Theorem 2. +□ +As an example, let n = 3 and j = 1. Then we get the sequence of abc triples +� +1, 8k, 8k + 1 +� +for +each odd integer k. In particular, we recover the abc triple (1, 8, 9) as a special case. +4. abc triples of the form (1, c − 1, c) and the ABC@Home Project +The ABC@Home project found that there are exactly 14 482 065 abc triples (a, b, c) with c < 1018. +The information found by the ABC@Home project is available on Bart de Smit’s webpage [dS23]. +Given an abc triple (a, b, c), we define its quality to be +q(a, b, c) = +log c +log rad(abc). +By definition, we see that since rad(abc) < c, an abc triple (a, b, c) satisfies q(a, b, c) > 1. This +gives us the following restatement of the abc conjecture: For each ϵ > 0, there are finitely many abc +triples (a, b, c) with q(a, b, c) > 1 + ϵ. +The abc triple with the largest known quality is +� +2, 310 · 109, 235� +, which has a quality of approx- +imately 1.6299. In fact, Baker’s [Bak04] explicit abc conjecture asserts that there is no abc triple +(a, b, c) with q(a, b, c) ≥ 7 +4. From this statement, Fermat’s Last Theorem for exponent n > 6 easily +follows. We note that the explicit abc conjecture and the abc conjecture are not equivalent. +Figure 1. Histogram of the quality of abc triples (1, c − 1, c) with c < 1018 + +240 +175 +150 +75 +50 +25 +0 + +1D5 +110 +115 +120 +125 +130 +135 +140 +Quality12 +ELISE ALVAREZ-SALAZAR, ALEXANDER J. BARRIOS, CALVIN HENAKU, AND SUMMER SOLLER +Let S denote the set of abc triples of the form (1, c − 1, c) with c < 1018. From the ABC@Home +project, we have that #S = 45 603. The largest quality occurring in S corresponds to the abc +triple (1, 4374, 4375), which has quality approximately equal to 1.5679. Figure 1 summarize the +distribution of the quality of all abc triples in S. The bin size in the histogram is set to 5 000. +We note that all computations done in this section were done on SageMath [S+23], and our code is +available on GitHub [ASBHS23]. +Table 1 lists the first fifteen abc triples of the form (1, c−1, c), their quality, and whether they arise +from one of the results proven in Section 3. The only abc triple in the table that is not of the form +� +1, nl − 1, nl� +or +� +1, nl, nl + 1 +� +for some integer l > 1 is (1, 1215, 1216). However, most abc triples +in S are not of the aforementioned form. More precisely, S contains 7 376 (resp. 1 038) abc triples +of the form +� +1, nl − 1, nl� +(resp. +� +1, nl, nl + 1 +� +) for some integer l > 1. We note that (1, 8, 9) is +the only double-counted element since Mih˘ailescu’s Theorem [Mih04] (formerly known as Catalan’s +conjecture) asserts that 2 and 3 are the only two consecutive perfect powers. Consequently, +T = +� +(1, c − 1, c) ∈ S | c = nl or c = nl + 1 for some l > 1 +� +has 8 413 elements. The highest quality abc triple in T is (1, 2400, 2401), with a quality of approx- +imately 1.4557. Observe that this abc triple is obtained from Corollary 3.9 since (1, 2400, 2401) = +� +1, 74 − 1, 74� +. +Table 1. The first fifteen abc triples of the form (1, c − 1, c) +(1, c − 1, c) +q(1, c − 1, c) +Arises from result in Section 3? +(1, 8, 9) +1.2263 +Yes; Corollary 3.7 with (n, k) = (3, 1) +(1, 48, 49) +1.0412 +Yes; Corollary 3.9 with (j, k) = (3, 1) +(1, 63, 64) +1.1127 +Yes; Corollary 3.10 with (n, j, k) = (3, 1, 1) +(1, 80, 81) +1.2920 +Yes; Corollary 3.8 with (n, k) = (3, 1) +(1, 224, 225) +1.0129 +Yes; Corollary 3.9 with (j, k) = (4, 1) +(1, 242, 243) +1.3111 +No +(1, 288, 289) +1.2252 +No +(1, 512, 513) +1.3176 +Yes; Corollary 3.12 with (n, j, k) = (3, 1, 2) +(1, 624, 625) +1.0790 +Yes; Corollary 3.7 with (n, k) = (5, 1) +(1, 675, 676) +1.0922 +No +(1, 728, 729) +1.0459 +Yes; Corollary 3.7 with (n, k) = (3, 3) +(1, 960, 961) +1.0048 +Yes; Corollary 3.9 with (j, k) = (5, 1) +(1, 1024, 1025) +1.1523 +Yes; Corollary 3.11 with (n, k) = (4, 1) +(1, 1215, 1216) +1.1194 +No +(1, 2303, 2304) +1.0204 +No +Now suppose that +� +1, nl − 1, nl� +is an abc triple for some integer l > 1. By Proposition 2.2, we +know that cosocle +� +nl − 1 +� +> rad(n). However, checking that +� +1, nl − 1, nl� +is an abc triple via this +criteria gets more difficult as nl grows. By Theorem 1, we can deduce that +� +1, nl − 1, nl� +is an abc +triple if there is a divisor m of nl−1 such that cosocle(m) > rad(n). By considering those elements in +T of the form +� +1, nl − 1, nl� +for some integer l > 1, we find that m can be taken to be a proper divisor + +ON abc TRIPLES OF THE FORM (1, c − 1, c) +13 +of nl − 1, except for the abc triples (1, c − 1, c) where c ∈ {9, 676, 11309769, 17380816062160329}. +Indeed, rad(676) = 26 and 675 = 3352. +The only divisor of 675 satisfying cosocle(m) > 26 is +m = 675. +The above leads us to ask: given +� +1, nl − 1, nl� +∈ T with l > 1 an integer, what is the least +divisor m of nl − 1 for which cosocle(m) > rad(n)? Using SageMath [S+23], we answered this +question, and our datafile can be accessed in [ASBHS23, triples for thm1.csv]. Table 2 gives the +first fifteen elements (a, b, c) in T of the form +� +1, nl − 1, nl� +, where n and l are listed, as well as the +least divisor m of nl − 1 for which cosocle(m) > rad(n) holds. The quality of the abc triple is also +given. +Table 2. The first fifteen abc triples (a, b, c) of the form +� +1, nl − 1, nl� +for l > 1, +with m the least divisor of nl − 1 satisfying cosocle(m) > rad(n) +(a, b, c) +n +l +m +q(a, b, c) +(1, 8, 9) +3 +2 +8 +1.2263 +(1, 48, 49) +7 +2 +16 +1.0412 +(1, 63, 64) +2 +6 +9 +1.1127 +(1, 80, 81) +3 +4 +8 +1.2920 +(1, 224, 225) +15 +2 +32 +1.0129 +(1, 242, 243) +3 +5 +121 +1.3111 +(1, 288, 289) +17 +2 +144 +1.2252 +(1, 624, 625) +5 +4 +16 +1.0790 +(1, 675, 676) +26 +2 +675 +1.0922 +(1, 728, 729) +3 +6 +8 +1.0459 +(1, 960, 961) +31 +2 +64 +1.0048 +(1, 2303, 2304) +48 +2 +49 +1.0204 +(1, 2400, 2401) +7 +4 +16 +1.4557 +(1, 3024, 3025) +55 +2 +432 +1.0348 +(1, 3968, 3969) +63 +2 +64 +1.1554 +Similarly, we ask the same question in the setting of Theorem 2. That is, given +� +1, nl, nl + 1 +� +∈ T +with l > 1 an odd integer, what is the least positive divisor m of nl+1 for which cosocle(m) > rad(n)? +We note that T has 596 elements of the form +� +1, nl, nl + 1 +� +for some integer l > 1. We also answer +this question through SageMath, and our datafile is found in [ASBHS23, triples for thm2.csv]. +Table 3 gives the first fifteen elements (a, b, c) in T of the form +� +1, nl, nl + 1 +� +, where n and l are +listed, as well as the least divisor m of nl + 1 for which cosocle(m) > rad(n) holds. In particular, +we find that (1, 8, 9) is the only abc triple of the form (1, nl, nl + 1) in T with l > 1 an odd integer +for which there is no proper divisor m of nl + 1 satisfying cosocle(m) > rad(n). + +14 +ELISE ALVAREZ-SALAZAR, ALEXANDER J. BARRIOS, CALVIN HENAKU, AND SUMMER SOLLER +Table 3. The first fifteen abc triples (a, b, c) of the form +� +1, nl, nl + 1 +� +for l > 1 an +odd integer, with m the least divisor of nl + 1 satisfying cosocle(m) > rad(n) +(a, b, c) +n +l +m +q(a, b, c) +(1, 8, 9) +2 +3 +9 +1.2263 +(1, 512, 513) +2 +9 +9 +1.3176 +(1, 6859, 6860) +19 +3 +343 +1.2281 +(1, 12167, 12168) +23 +3 +676 +1.2555 +(1, 17576, 17577) +26 +3 +81 +1.0039 +(1, 29791, 29792) +31 +3 +784 +1.1424 +(1, 32768, 32769) +2 +15 +9 +1.0406 +(1, 110592, 110593) +48 +3 +49 +1.0135 +(1, 250047, 250048) +63 +3 +64 +1.0351 +(1, 279936, 279937) +6 +7 +49 +1.0124 +(1, 512000, 512001) +80 +3 +81 +1.4433 +(1, 1953125, 1953126) +5 +9 +27 +1.0423 +(1, 2097152, 2097153) +2 +21 +9 +1.0287 +(1, 3176523, 3176524) +147 +3 +676 +1.0145 +(1, 7077888, 7077889) +192 +3 +169 +1.0515 +Next, we investigate how many elements of T arise from the results proven in Section 3. Indeed, +each abc triple produced by the results of that section are of the form +� +1, nl − 1, nl� +or +� +1, nl, nl + 1 +� +for some integer l > 1. Moreover, for each abc triple obtained from one of our corollaries in Section 3, +we apply the following result from [vdH10, Section 2.3]: +Proposition 4.1. Let (1, c − 1, c) be an abc triple. Then the following are abc triples: +� +1, (c − 1)3 , c +� +c2 − 3c + 3 +�� +and +� +1, c (c − 2) , (c − 1)2� +. +As a demonstration, the abc triple (1, 2303, 2304) is obtained from the abc triple (1, 48, 49) since +2304 = 482. In particular, (1, 2303, 2304) can now be viewed as a consequence of Corollary 3.9 +and Proposition 4.1. Proposition 4.1 is part of a more general result in [vdH10, Section 2.3], which +provides a way of mapping an abc triple (a, b, c) to a new abc triple by applying polynomial identities. +The more general result arises by splitting the binomial formula (a + b)n to obtain the following +family of identities: +an−k +� +� +k +� +j=0 +�n +j +� +ak−jbj +� +� + bk+1 +� +� +n−k−1 +� +j=0 +�n +j +� +ajbn−k−1−j +� +� = cn +Taking k = 0 yields Corollary 2.4. Therefore, the two non-trivial polynomials identities with a = 1 +are those occurring in Proposition 4.1. +Corollaries 3.3 through 3.12 provide us with a recipe for constructing abc triples. For each of +these corollaries, we consider the set +Ci = {(1, c − 1, c) ∈ T | (1, c − 1, c) is obtained from Corollary 3.i} , + +ON abc TRIPLES OF THE FORM (1, c − 1, c) +15 +where 3 ≤ i ≤ 12. By Table 1, we see that (1, 224, 225) ∈ C9, but (1, 242, 243) ̸∈ Ci for each i. +Using SageMath, we find that +i +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +#Ci +32 +58 +12 +17 +41 +29 +81 +46 +18 +36 +The low number of abc triples in T occurring in each Ci is expected. Indeed, for Corollary 3.5 to yield +an abc triples in T, we require that n > 1 be an integer, p be an odd prime such that p > rad(n), +and np(p−1)k < 1018 for some integer k. For n an odd integer, the only possible (n, p, k) is (3, 5, 1), +which gives the abc triple (1, 3486784400, 3486784401). We also note that since Corollary 3.5 is a +special case of Corollary 3.4, we have that C5 ⊆ C4. Now let +C = +� +3≤i≤12 +Ci. +We find that #C = 164. +Lastly, let D be the set of abc triples in T with the property that an element of D is in C or can be +obtained from an abc triple in C after successive applications of Proposition 4.1 and Corollaries 2.4 +and 2.5. As an illustration, the abc triple (1, 12214672127, 12214672128) ̸∈ C, but it is in D. To see +this, recall that (1, 2303, 2304) is obtained from the abc triple (1, 48, 49) via Proposition 4.1. Then, +(1, 12214672127, 12214672128) = +� +1, (c − 1)3 , c +� +c2 − 3c + 3 +�� +, +where c = 2304, which shows that the abc triple is in D. In fact, with the exception of the abc triple +(1, 1215, 1216), every abc triple appearing in Table 1 is in D. Using SageMath, we find that D has +311 elements. +We conclude this article by considering the percentage of abc triples (1, c − 1, c) in S and T, that +are also in D. More precisely, for sets X and Y such that X ⊆ Y ⊆ S, we define +δX,Y (x) = # {(1, c − 1, c) ∈ X | c ≤ x} +# {(1, c − 1, c) ∈ Y | c ≤ x} . +In particular, δX,Y (x) gives the percentage of abc triples (1, c − 1, c) of Y with c ≤ x that are in X. +The table below gives some values of δT,S(x) , δD,S(x), and δD,T (x): +x +104 +106 +108 +1010 +1012 +1014 +1016 +1018 +δT,S(x) +80% +57.8% +45.2% +35.1% +30.0% +24.6% +20.9% +18.4% +δD,S(x) +53.3% +28.1% +13.5% +7.03% +3.79% +2.06% +1.14% +0.68% +δD,T (x) +66.7% +48.7% +29.9% +20.0% +12.6% +8.40% +5.47% +3.70% +In particular, we see that D contains nearly half of the abc triples (1, c − 1, c) in T with c ≤ 106. +Acknowledgments. The authors would like to thank the National Science Foundation, Pomona +College, Edray Goins, Renee Bell, Cory Colbert, Bianca Thompson, and the staff and students +of the Pomona Research in Mathematics Experience (PRiME) for their support and camaraderie +as this work was being undertaken. Research at PRiME was supported by the National Science +Foundation award DMS-2113782. +The authors also thank Andrew Granville for his comments and suggestions on an earlier preprint. +In particular, we are grateful for his observations regarding those integers k for which (1, ck − 1, ck) +is an abc triple. This led to the deduction of Theorem 2.8. +We also would like to thank the High Performance Computing Program team at California +State University San Bernardino and the National Research Platform, especially Youngsu Kim, for + +16 +ELISE ALVAREZ-SALAZAR, ALEXANDER J. BARRIOS, CALVIN HENAKU, AND SUMMER SOLLER +providing us access to SageMath in said computing program. In particular, this work was supported +in part by National Science Foundation awards CNS-1730158, ACI-1540112, ACI-1541349, OAC- +1826967, OAC-2112167, CNS-2120019, the University of California Office of the President, and the +University of California San Diego’s California Institute for Telecommunications and Information +Technology/Qualcomm Institute. Thanks to CENIC for the 100Gbps networks. +Any opinions, findings, and conclusions or recommendations expressed in this article are those +of the author(s) and do not necessarily reflect the views of the National Science Foundation. +References +[ASBHS23] Elise Alvarez-Salazar, Alexander J. Barrios, Calvin Henaku, and Summer Soller, Code for abc triples of +the form (1, c − 1, c), https://github.com/alexanderbarrios/abc_triples/, 2023. +[Bak04] +Alan Baker, Experiments on the abc-conjecture, Publ. Math. Debrecen 65 (2004), no. 3-4, 253–260. +MR 2107944 +[Bar23] +Alexander J. Barrios, Good elliptic curves with a specified torsion subgroup, J. Number Theory 242 (2023), +21–43. MR 4474850 +[dS23] +Bart de Smit, Abc-triples, 2023, http://www.math.leidenuniv.nl/~desmit/abc/ . +[Elk91] +Noam D. Elkies, ABC implies Mordell, Internat. Math. Res. Notices (1991), no. 7, 99–109. MR 1141316 +[GT02] +Andrew Granville and Thomas J. Tucker, It’s as easy as abc, Notices Amer. Math. Soc. 49 (2002), no. 10, +1224–1231. MR 1930670 +[Lan90] +Serge Lang, Old and new conjectured Diophantine inequalities, Bull. Amer. Math. Soc. (N.S.) 23 (1990), +no. 1, 37–75. MR 1005184 +[Mas17] +D. W. Masser, Abcological anecdotes, Mathematika 63 (2017), no. 3, 713–714. MR 3731300 +[Mih04] +Preda Mih˘ailescu, Primary cyclotomic units and a proof of Catalan’s conjecture, J. Reine Angew. Math. +572 (2004), 167–195. MR 2076124 +[MM16] +Greg Martin and Winnie Miao, abc triples, Funct. Approx. Comment. Math. 55 (2016), no. 2, 145–176. +MR 3584566 +[Oes88] +Joseph Oesterl´e, Nouvelles approches du “th´eor`eme” de Fermat, Ast´erisque (1988), no. 161-162, Exp. No. +694, 4, 165–186 (1989), S´eminaire Bourbaki, Vol. 1987/88. MR 992208 +[Pal14] +Willem Jan Palenstijn, Finding abc-triples using elliptic curves, 2014, Thesis (Ph.D.)–Universiteit Leiden. +[S+23] +W. A. Stein et al., Sage Mathematics Software (Version 9.7), The Sage Development Team, 2023, +http://www.sagemath.org. +[Ste84] +C. L. Stewart, A note on the product of consecutive integers, Topics in classical number theory, Vol. I, II +(Budapest, 1981), Colloq. Math. Soc. J´anos Bolyai, vol. 34, North-Holland, Amsterdam, 1984, pp. 1523– +1537. MR 781193 +[vdH10] +Johannes Petrus van der Horst, Finding abc-triples using elliptic curves, 2010, Masters Thesis – +Universiteit Leiden. +Department of Mathematics, University of California, Santa Barbara, CA 93106 USA +Email address: ealvarez-salazar@ucsb.edu +Department of Mathematics, University of St. Thomas, St. Paul, MN 55105 USA +Email address: abarrios@stthomas.edu +Department of Mathematics, Washington University in St. Louis, St. Louis, MO 63130 USA +Email address: chenaku@wustl.edu +Department of Mathematics, Colorado State University, Fort Collins, CO 80523 USA +Email address: summer.soller@colostate.edu + diff --git a/g9AzT4oBgHgl3EQfa_wH/content/tmp_files/load_file.txt b/g9AzT4oBgHgl3EQfa_wH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7f4ec208126047a0ab23f7628b804ec63686199f --- /dev/null +++ b/g9AzT4oBgHgl3EQfa_wH/content/tmp_files/load_file.txt @@ -0,0 +1,704 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf,len=703 +page_content='ON abc TRIPLES OF THE FORM (1, c − 1, c) ELISE ALVAREZ-SALAZAR, ALEXANDER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' BARRIOS, CALVIN HENAKU, AND SUMMER SOLLER Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By an abc triple, we mean a triple (a, b, c) of relatively prime positive integers a, b, and c such that a+b = c and rad(abc) < c, where rad(n) denotes the product of the distinct prime factors of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The necessity of the ϵ in the abc conjecture is demonstrated by the existence of infinitely many abc triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' For instance, � 1, 9k − 1, 9k� is an abc triple for each positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In this article, we study abc triples of the form (1, c − 1, c) and deduce two general results that allow us to recover existing sequences in the literature of abc triples with a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Introduction In 1985, Masser and Oesterl´e proposed the abc conjecture [Oes88, Mas17], which states: The abc conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' For every ϵ > 0, there are finitely many relatively prime positive integers a, b, and c with a + b = c such that rad(abc)1+ϵ < c, where rad(n) denotes the product of the distinct prime factors of a positive integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Due to its profound implications, this simple-to-state conjecture is one of the most important open questions in number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' For instance, some consequences of the abc conjecture include an asymptotic version of Fermat’s Last Theorem, Faltings’s Theorem, Roth’s Theorem, and Szpiro’s Conjecture [Elk91, Lan90, Oes88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' For further information on the abc conjecture, see the excellent survey article [MM16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The statement of the abc conjecture naturally leads us to ask if the ϵ is necessary?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' This leads us to the “simplistic abc conjecture,” which asks if there are finitely many relatively prime positive integers a, b, and c with a + b = c for which rad(abc) < c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We call such triples (a, b, c) an abc triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The “simplistic abc conjecture” is false, as demonstrated by the triple � 1, 32k − 1, 32k� , which is an abc triple for each positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' This infinite sequence of abc triples is one of the first documented counterexamples to the simplistic abc conjecture and was communicated to Lang by Jastrzebowski and Spielman [Lan90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' A theorem of Stewart [Ste84] leads to similar sequences of abc triples such as � 1, 87k − 1, 87k� , where k is a positive integer [MM16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Jastrzebowski and Spielman’s counterexample can also be recovered from the following result: for each odd prime p and each positive integer k, � 1, p(p−1)k − 1, p(p−1)k� is an abc triple [Bar23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Another construction, due to Granville and Tucker [GT02], shows that for each odd prime p, � 1, 2p(p−1) − 1, 2p(p−1)� is an abc triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In this article, we prove that (1, c−1, c) is an abc triple if and only if cosocle(c−1) > rad(c), where cosocle(m) = m rad(m) for m a positive integer (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We note that the term cosocle is borrowed from module theory, where the cosocle of an R-module M is the maximal semisimple quotient of M, or equivalently, M rad(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In our setting, the cosocle plays a crucial role in our results, from which we recover each of the above mentioned sequences of abc triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' To provide context 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Primary 11D75, 11J25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' abc conjecture, abc triples, number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='01376v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='NT] 3 Jan 2023 2 ELISE ALVAREZ-SALAZAR, ALEXANDER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' BARRIOS, CALVIN HENAKU, AND SUMMER SOLLER for our work, we note that the equivalence above requires us to compute cosocle(c − 1) in order to deduce whether (1, c−1, c) is an abc triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The computation of cosocle(c−1) requires knowledge of the prime factorization of c − 1, which becomes computationally difficult as c gets large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Our main results provide a recipe for constructing infinitely many abc triples of the form (1, c − 1, c) based on knowledge of a divisor of c − 1 or c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Our first theorem illustrates this: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let c and m be positive integers with c > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' If m divides c−1 and cosocle(m) > rad(c), then � 1, ck − 1, ck� is an abc triple for each positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='. We prove Theorem 1 in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' While the proof is elementary, the result allows us to recover each of the previously mentioned sequences of abc triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' It also leads to new sequences of abc triples, such as � 1, n(n−1)k − 1, n(n−1)k� which is an abc triple for each positive integer k whenever n is a positive integer that is either odd or even and non-squarefree (see Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' A slight modification of the proof of Theorem 1 leads us to our next result (which is also proven in Section 2): Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let b and m be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' If m divides b + 1 and cosocle(m) > rad(b), then � 1, bk, bk + 1 � is an abc triple for each positive odd integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' A consequence of Theorem 1 is that if (1, c − 1, c) is an abc triple, then (1, ck − 1, ck) is an abc triple for each positive integer k (see Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Similarly, we obtain from Theorem 2 that if (1, b, b + 1) is an abc triple, then (1, bk, bk + 1) is an abc triple for each odd integer k (see Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' These results lead to the following question: given an integer c > 1, for what positive integers k is (1, ck −1, ck) an abc triple?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We answer this question with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='8, which provides necessary and sufficient conditions to determine those integers k which yield an abc triple of the form (1, ck − 1, ck).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In Section 3, we demonstrate various consequences of Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' For example, we prove that if n > 1 is an integer and p is an odd prime such that p > rad(n), then � 1, np(p−1)k − 1, np(p−1)k� is an abc triple for each positive integer k (see Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In particular, taking (n, k) = (2, 1) allows us to recover Granville and Tucker’s original construction [GT02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Another consequence is the following: if n ≥ 3 is an odd integer and b = nj −1 for some positive integer j, then � 1, bnk, bnk + 1 � is an abc triple for each positive odd integer k (see Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Taking (n, j) = (3, 1) gives us that � 1, 8k, 8k + 1 � is an abc triple for each odd integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We conclude the article with Section 4, which is an analysis of the abc triples found by the ABC@Home Project of the form (1, c − 1, c) with c < 1018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The ABC@Home Project was a network computing project that was started in 2006 by the Mathematics Department of Leiden University, together with the Dutch Kennislink Science Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By 2011, they found that there are exactly 14 482 065 abc triples (a, b, c) with c < 1018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By the time the project came to a close in 2015, the ABC@Home Project had found a total of 23 827 716 abc triples (a, b, c) with c < 263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We note that this list is not exhaustive of all abc triples with c < 263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In particular, the ABC@Home project found that there are exactly 45 604 abc triples of the form (1, c − 1, c) with c < 1018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Further observations about the abc triples found by the ABC@Home Project can be found in [Pal14, Chapter 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Motivated by the results in Section 3, we study those abc triples found by the ABC@Home Project that are of the form (1, nl − 1, nl) or (1, nl, nl + 1) for some integer l > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We find that this amounts to 8 413 abc triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' For abc triples (1, c − 1, c) of the aforementioned form, we show that approximately 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='7% of the abc triples with c ≤ 106 can be obtained from the results proven in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We also find that for abc triples of the form (1, nl − 1, nl), there are only four cases where there does not exists a proper divisor m of nl − 1 for which cosocle(m) > rad(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' ON abc TRIPLES OF THE FORM (1, c − 1, c) 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Main Results In this section, we establish Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' To do so, we recall the following elementary property about the radical of a positive integer: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let m and n be relatively prime positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then rad(mn) = rad(m) rad(n) and rad(m) ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Moreover, rad � mk� = rad(m) for each positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We will assume Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1 implicitly throughout this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Next, we show an important facet about abc triples of the form (1, c − 1, c), which showcases the importance of the cosocle in our arguments: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let c > 1 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then the following are equivalent: (i) cosocle(c − 1) > rad(c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' (ii) cosocle(c) > rad(c − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' (iii) (1, c − 1, c) is an abc triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Suppose that rad(c) < cosocle(c − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since rad(c) = c cosocle(c) and cosocle(c − 1) = c−1 rad(c−1), we deduce that rad(c) < cosocle(c − 1) ⇐⇒ c cosocle(c) < c − 1 rad(c − 1) ⇐⇒ rad(c − 1) < c − 1 c cosocle(c) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since c−1 c < 1, we have the desired inequality: rad(c − 1) < cosocle(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Next, suppose that rad(c−1) cosocle(c) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since rad(c) = c cosocle(c), we observe that rad(c(c − 1)) = rad(c) rad(c − 1) = rad(c − 1) cosocle(c) c < c, which shows that (1, c − 1, c) is an abc triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Lastly, if (1, c − 1, c) is an abc triple, then rad(c(c − 1)) < c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Consequently, c > rad(c(c − 1)) = rad(c) rad(c − 1) = rad(c)(c − 1) cosocle(c − 1) =⇒ rad(c) < cosocle(c − 1) c c − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since rad(c) is an integer and c c−1 > 1, we deduce that rad(c) ≤ � cosocle(c − 1) c c−1 � , where ⌊x⌋ denotes the floor function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since cosocle(c−1) c−1 < 1, we observe that � cosocle(c − 1) c c − 1 � = � cosocle(c − 1) + cosocle(c − 1) c − 1 � = cosocle(c − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Lastly, c is relatively prime to c − 1, and thus cosocle(c − 1) > rad(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' □ An automatic consequence of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2 is that if c or c − 1 is squarefree, then (1, c − 1, c) is not an abc triple since the cosocle of a squarefree positive integer is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Our next result establishes that the radical of a positive integer n is preserved if n is divided by the cosocle of any of its divisors: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let m and n be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' If m divides n, then rad(n) = rad � n cosocle(m) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' 4 ELISE ALVAREZ-SALAZAR, ALEXANDER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' BARRIOS, CALVIN HENAKU, AND SUMMER SOLLER Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' If m = 1, there is nothing to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' So suppose that m > 1 and let m = �r i=1 pei i be the unique prime factorization of m, with each pi denoting a distinct prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since m divides n, we have that n = q �r i=1 pfi i where ei ≤ fi for 1 ≤ i ≤ r and q is relatively prime to m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since cosocle(m) = �r i=1 pei−1 i , we deduce that n cosocle(m) = q r � i=1 pfi−ei+1 i For 1 ≤ i ≤ r, observe that fi − ei + 1 ≥ 1 and thus rad � n cosocle(m) � = rad(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' □ With this lemma, we are now ready to prove Theorem 1: Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since ck − 1 = (c − 1) �k−1 j=0 cj, we deduce that m divides ck − 1 for each positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='3, rad � ck − 1 � = rad � ck−1 cosocle(m) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By assumption, rad(c) cosocle(m) < 1 and thus rad � ck � ck − 1 �� = rad(c) rad � ck − 1 cosocle(m) � ≤ rad(c) cosocle(m) � ck − 1 � < ck − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The result now follows since ck − rad � ck � ck − 1 �� > ck − ck + 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' □ An immediate consequence of Theorem 1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2, is the following result: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' If (1, c − 1, c) is an abc triple, then � 1, ck − 1, ck� is an abc triple for each positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In the next section, we will consider further consequences of Theorem 1 that do not require knowl- edge of an abc triple at the start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The proof of Theorem 1 relies on the factorization of ck − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' A similar factorization holds for bk + 1 if k is odd, and our proof of Theorem 2 makes use of this: Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' If k is a positive odd integer, then bk + 1 = (b + 1) �k−1 j=0 (−1)j bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' It follows that m divides bk + 1 for each positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='3, rad(bk + 1) = rad � bk+1 cosocle(m) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since rad(b) cosocle(m) < 1, we observe that rad � bk � bk + 1 �� = rad(b) rad � bk + 1 cosocle(m) � ≤ rad(b) cosocle(m) rad(bk + 1) < bk + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Consequently, bk + 1 − rad � bk � bk + 1 �� > bk + 1 − bk − 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' □ Similarly to the deduction of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='4, we now recover the following result as an immediate consequence of Theorem 2 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' If (1, b, b + 1) is an abc triple, then � 1, bk, bk + 1 � is an abc triple for each positive odd integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since (1, 8, 9) is an abc triple, we deduce from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='5 that (1, 8k, 8k + 1) is an abc triple for each positive odd integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We will also recover this sequence of abc triples as a consequence of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='4, we have that if (1, c − 1, c) is an abc triple, then � 1, ck − 1, ck� is an abc triple for each positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' This leads us to ask: given a positive integer c > 1, for what positive ON abc TRIPLES OF THE FORM (1, c − 1, c) 5 integers k is � 1, ck − 1, ck� an abc triple?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' To answer this question, we first recall a few number theory facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Given a prime number p and a positive integer n, the p-adic valuation of n, denoted vp(n), is the unique integer that satisfies n = pvp(n)q for some integer q that is relatively prime to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Suppose further that p does not divide n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then the order of n modulo p, denoted ordp(n), is the least positive integer for which nordp(n) ≡ 1 mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By Fermat’s Little Theorem, ordp(n) divides p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' More generally, nk ≡ 1 mod p if and only if ordp(n) divides k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' With this terminology, we determine the exact power of a prime p that divides ck − 1: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let c and k be positive integers with c > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then p divides ck −1 if and only if ordp(c) divides k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Moreover, if p divides ck − 1, then vp � ck − 1 � = fp + wp, where fp = vp � cordp(c) − 1 � and wp = vp(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The statement that p divides ck−1 if and only if ordp(c) divides k is a standard number theory result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' So suppose that p divides ck − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then ordp(c) divides k and p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In particular, ordp(c) is not divisible by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' It follows that k = q1pwp ordp(c) for some integer q1 that is not divisible by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By assumption, cordp(c) − 1 = q2pfp for some integer q2 that is not divisible by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Now observe that by the Binomial Theorem, ck = � cordp(c)�q1pwp = � q2pfp + 1 �q1pwp = 1 + q1q2pfp+wp + q1pwp−1 � j=2 �q1pwp j � qj 2pfpj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' For 2 ≤ j ≤ q1pwp − 1, we have that pwp+fp+1 divides �q1pwp j � qj 2pfpj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Hence ck ≡ 1 mod pwp+fp and ck ≡ 1 + q1q2pfp+wp mod pfp+wp+1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In particular, vp � ck − 1 � = fp + wp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' □ As an immediate consequence of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='6 and the Fundamental Theorem of Arithmetic, we obtain the following factorization for ck − 1: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let c and k be positive integers with c > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then with notation as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='6, ck − 1 = � ordp(c)|k pfp+wp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' As a demonstration of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='7, let c = 21 and k = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' With notation as above, we see that w2 = 2, w3 = 1, and wp = 0 for each prime p ̸= 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Next, we observe that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1) 5540 − 1 = 24 · 5 · 11 · 13 · 17 · 61 · 421 · 463 · 3181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='6, the primes appearing in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1) are precisely those primes p for which ordp(21) divides 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' With a computer algebra system, such as SageMath [S+23], it is checked that fp = 1 for each prime p ̸= 2 appearing in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1) and f2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Thus, 2112 − 1 = � ordp(21)|12 pfp+wp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let c and k be positive integers with c > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' With notation as in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='7, write ck − 1 = � ordp(c)|k pfp+wp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then � 1, ck − 1, ck� is an abc triple if and only if one of the following conditions hold: (i) there exists a prime p > rad(c) such that ordp(c) divides k and either fp ≥ 2 or wp ≥ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' (ii) there exists a prime p < rad(c) such that ordp(c) divides k and fp + wp − 1 ≥ mp, where mp denote the least positive integer such that pmp > rad(c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' (iii) for each prime p such that ordp(c) divides k, there exist a non-negative integer ap ≤ fp+wp−1 such that � ordp(c)|k pap > rad(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' 6 ELISE ALVAREZ-SALAZAR, ALEXANDER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' BARRIOS, CALVIN HENAKU, AND SUMMER SOLLER Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' First suppose that � 1, ck − 1, ck� is an abc triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2, this is equivalent to rad(c) < cosocle � ck − 1 � = � ordp(c)|k pfp+wp−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In particular, taking ap = fp + wp − 1 yields (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Now suppose there is a prime p > rad(c) such that ordp(c) divides k and either fp ≥ 2 or wp ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Note that fp ≥ 1 for each prime p such that ordp(c) divides k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Consequently, if fp ≥ 2 or wp ≥ 1, then fp +wp ≥ 2 and thus p2 divides ck −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then � 1, ck − 1, ck� is an abc triple by Theorem 1 since cosocle � p2� = p > rad(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Next, suppose that there is a prime p < rad(c) such that ordp(c) divides k and fp + wp − 1 ≥ mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then pmp+1 divides ck − 1 and cosocle � pmp+1� = pmp > rad(c) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By Theorem 1, we deduce that � 1, ck − 1, ck� is an abc triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Lastly, suppose that for each prime p such that ordp(c) divides k, there exist a positive integer ap ≤ fp + wp − 1 such that � ordp(c)|k pap > rad(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then � ordp(c)|k pap+1 divides ck − 1 and the result now follows by Theorem 1 since cosocle � � � ordp(c)|k pap+1 � � = � ordp(c)|k pap > rad(c) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' □ As an illustration, consider c = 21 and k = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In the discussion following Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='7, we noted that w2 = f2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Moreover, for each prime p ̸= 2 appearing in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1) we have that fp = 1 and wp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In particular, we see that statements (i) and (ii) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='8 are not satisfied for each prime p appearing in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We also have that statement (iii) is not satisfied as the only prime for which fp + wp − 1 > 0 is p = 2 and 2f2+w2−1 = 16 < rad(21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' It follows that � 1, 2112 − 1, 2112� is not an abc triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In the next section, we will see that 21 is the first odd integer n > 1 for which � 1, nϕ(n) − 1, nϕ(n)� is not an abc triple, where ϕ(n) denotes the Euler-totient function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We note that ϕ(21) = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Consequences In this section, we consider various consequences of Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' From these consequences, we deduce the sequences of abc triples that were mentioned in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We note that this article began as an investigation of the following question: for what positive odd integers n is � 1, nϕ(n) − 1, nϕ(n)� an abc triple?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Here ϕ(n) denotes the Euler-totient function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The question was motivated by the following observation: if n is an odd integer such that 3 ≤ n ≤ 99, then � 1, nϕ(n) − 1, nϕ(n)� is an abc triple for each n except n = 21, 39, 69, and 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The fact that the four exceptions are composites is no surprise, as the question is true for odd primes n [Bar23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Our investigation of this phenomenon led to our Theorems 1 and 2, and our first consequence provides necessary conditions for when � 1, nϕ(n) − 1, nϕ(n)� is an abc triple for a positive odd integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' To prove this result, we first recall the following result from elementary number theory: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let n be a positive odd integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then n2k ≡ 1 mod 2k+2 for each positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since n is odd, there is an integer m such that n = 2m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By the Binomial Theorem, n2k = (2m + 1)2k = 2k � j=0 �2k j � (2m)j = 1 + 2k+1m � 1 + � 2k − 1 � m � + 2k � j=3 �2k j � (2m)j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' ON abc TRIPLES OF THE FORM (1, c − 1, c) 7 Now observe that m � 1 + � 2k − 1 � m � is always even and �2k j � (2m)j is divisible by 2k+2 for 3 ≤ j ≤ 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Consequently, n2k ≡ 1 mod 2k+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' □ With this result, we obtain our our first application of Theorem 1: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let n > 1 be an odd integer and let ϕ denote the Euler-totient function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Set d = gcd(n−1, ϕ(n)) and m = 2v2(4ϕ(n))−2v2(d)d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' If cosocle(m) > rad(n), then � 1, nϕ(n)k − 1, nϕ(n)k� is an abc triple for each positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let P = �ϕ(n)−1 j=0 nj and observe that nϕ(n) − 1 = (n − 1) P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since d = gcd(n − 1, ϕ(n)) divides n − 1, n ≡ 1 mod d and thus P ≡ ϕ(n)−1 � j=0 1j mod d = ϕ(n) mod d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In particular, d divides P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since nϕ(n) − 1 = (n − 1) P, we deduce that d2 divides nϕ(n) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Next, write ϕ(n) = 2v2(ϕ(n))r for r an odd integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1, nϕ(n) − 1 = (nr)2v2(ϕ(n)) − 1 ≡ 0 mod 2v2(ϕ(n))+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Hence 2v2(ϕ(n))+2 divides nϕ(n) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' It follows that 2v2(ϕ(n))+2 d2 2v2(d2) = 2v2(4ϕ(n))−2v2(d)d2 = m divides nϕ(n) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The result now follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' □ As an illustration, let n = 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then with notation as in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2, we observe that ϕ(75) = 40, d = 2, and m = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since cosocle(32) = 16 > rad(75) = 15, we have that � 1, 7540k − 1, 7540k� is an abc triple for each positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We note that the converse to Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2 does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In fact, if 3 ≤ n ≤ 99 is an odd integer such that � 1, nϕ(n) − 1, nϕ(n)� is an abc triple, then the corollary fails to show the cases corresponding to n = 33, 35, 55, 57, 63, 65, 77, 93, 95, and 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The following result provides an improvement, but comes at the cost of having to compute vp(nϕ(n) − 1) for each prime p that divides gcd(nϕ(n) − 1, ϕ(n)): Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let n > 1 be an integer and let ϕ denote the Euler-totient function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Set d = gcd(nϕ(n) − 1, ϕ(n)) and m = � p|d pvp(nϕ(n)−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' If cosocle(m) > rad(n), then � 1, nϕ(n)k − 1, nϕ(n)k� is an abc triple for each positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By construction, m divides nϕ(n) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The result now follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' □ For odd integers n such that 3 ≤ n ≤ 99 and � 1, nϕ(n) − 1, nϕ(n)� is an abc triple, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='3 allows us to conclude that � 1, nϕ(n) − 1, nϕ(n)� is an abc triple for each n except n = 55, 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' When n = 55, we have that ϕ(55) = 40 and gcd � 5540 − 1, 40 � = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then m = 2v2(5540−1) = 64, and thus cosocle(64) = 32 < rad(55) = 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Consequently, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='3 fails to show that � 1, 5540 − 1, 5540� is an abc triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We note the cosocle � 5540 − 1 � = 288, and hence � 1, 5540k − 1, 5540k� is an abc triple for each positive integer k by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The failure of Corollaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='3 in the n = 55 case stems from the fact that the primes dividing m must divide ϕ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Indeed, cosocle � 5540 − 1 � = 32·9 and 3 ∤ ϕ(55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' 8 ELISE ALVAREZ-SALAZAR, ALEXANDER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' BARRIOS, CALVIN HENAKU, AND SUMMER SOLLER To state our next result, we recall the Carmichael function λ : N → N, which has the property that λ(m) is the least positive integer for which aλ(m) ≡ 1 mod m for each integer a that is relatively prime to m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In particular, λ(m) divides ϕ(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let λ and ϕ denote the Carmichael function and Euler-totient function, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' If m and n are relatively prime positive integers such that cosocle(m) > rad(n) > 1, then � 1, nλ(m)k − 1, nλ(m)k� and � 1, nϕ(m)k − 1, nϕ(m)k� are abc triples for each positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since nλ(m) ≡ 1 mod m, we have that m divides nλ(m) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By Theorem 1, we have that � 1, nλ(m)k − 1, nλ(m)k� is an abc triple for each positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since λ(m) | ϕ(m), we also have that � 1, nϕ(m)k − 1, nϕ(m)k� is an abc triple for each positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' □ As an example, choose n = 11 and m = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then cosocle(32) = 16 > rad(11), and therefore the conditions of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='4 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' As a result, we find that � 1, 11λ(32)k − 1, 11λ(32)k� = (1, 118k − 1, 118k) is a sequence of abc triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' More generally, we have the following application of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='4: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let n > 1 be an integer and let p be an odd prime such that p > rad(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then for each positive integer k, � 1, np(p−1)k − 1, np(p−1)k� is an abc triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By assumption, cosocle � p2� = p > rad(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Moreover, λ � p2� = p (p − 1) since p is prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' It follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='4 that � 1, nλ(p2)k − 1, nλ(p2)k� = � 1, np(p−1)k − 1, np(p−1)k� is an abc triple for each positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' □ Taking (n, k) = (2, 1) in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='5 yields that � 1, 2p(p−1) − 1, 2p(p−1)� is an abc triple for each odd prime p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' This result is originally due to Granville and Tucker [GT02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='8 gives the following refinement of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='5: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let n > 1 be an integer and let p be an odd prime such that p > rad(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then for each positive integer k, � 1, np ordp(n)k − 1, np ordp(n)k� is an abc triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In particular, if n ≡ 1 mod p and p > rad(n), then � 1, npk − 1, npk� is an abc triple for each positive integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In the notation of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='8, we have that wp = vp(p ordp(n)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since p > rad(n) and ordp(n) divides p ordp(n), Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='8 (i) implies that � 1, np ordp(n) − 1, np ordp(n)� is an abc triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The result now follows by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The second statement is automatic since if n ≡ 1 mod p, then ordp(n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' □ As a demonstration, let n = 16 and p = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='6 asserts that � 1, 165k − 1, 165k� is an abc triple for each positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let n > 1 be an integer that is either odd or even and non-squarefree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then � 1, n(n−1)k − 1, n(n−1)k� is an abc triple for each positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let P = �n−2 j=0 nj and observe that nn−1 − 1 = (n − 1) P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Moreover, P ≡ n−2 � j=0 (1)j mod(n − 1) = 0 mod (n − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In particular, (n − 1)2 divides nn−1 − 1 and thus rad(nn−1 − 1) = rad �nn−1 − 1 n − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' ON abc TRIPLES OF THE FORM (1, c − 1, c) 9 Now suppose that n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We claim that 4 divides P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' If n ≡ 1 mod 4, then this follows since P is divisible by n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' So suppose that n ≡ 3 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then 4 divides n + 1, and hence 4 divides P since P ≡ n−2 � j=0 (−1)j mod(n + 1) = 0 mod (n + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Consequently, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1) rad(nn−1 − 1) = rad �nn−1 − 1 2 (n − 1) � ≤ nn−1 − 1 2 (n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Now observe that by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1), cosocle(nn−1 − 1) = nn−1 − 1 rad(nn−1 − 1) ≥ 2 (n − 1) > rad(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The claim now follows by Theorem 1 with m = nn−1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Lastly, suppose that n is an even non-squarefree positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then n = a2b for some positive integers a and b with a > 1 and b squarefree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then rad(n) = rad(ab) ≤ ab < n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since rad(nn−1 − 1) = rad � nn−1−1 n−1 � ≤ nn−1−1 n−1 , we deduce that cosocle(nn−1 − 1) = nn−1 − 1 rad(nn−1 − 1) ≥ n − 1 > rad(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The result follows by Theorem 1 with m = nn−1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' □ From Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='7, we recover that � 1, 9k − 1, 9k� = � 1, 32k − 1, 32k� is a sequence of abc triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In particular, we obtain the smallest abc triple (1, 8, 9) as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Taking n = 8 in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='7 gives us the sequence of abc triples � 1, 87k − 1, 87k� , which generalizes the sequence � 1, 87k − 1, 87k� that appears in [MM16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let n > 1 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then � 1, n(n+1)k − 1, n(n+1)k� is an abc triple whenever (n + 1) k is a positive even integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let l be a positive even integer and let P = �l−1 j=0 (−1)j+1 nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then nl − 1 = (n + 1) P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We now proceed by cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Suppose that n is a positive even integer and let l = 2 (n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since n ≡ −1 mod(n + 1), we have that P ≡ �l−1 j=0 (−1)j+1 = 0 mod(n + 1) and thus rad(nl − 1) = rad �nl − 1 n + 1 � ≤ nl − 1 n + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The claim now holds by Theorem 1 with m = nl − 1 since cosocle(nl − 1) = nl − 1 rad(nl − 1) ≥ n + 1 > rad(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Suppose that n is a positive odd integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then l = n+1 is even and P ≡ 0 mod(n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' A similar argument to that of Case 1 with m = nl −1 shows that the result holds by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' □ As an example, choose n = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' As a result, (n + 1)k is even for every positive integer k and by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='8, � 1, 2122k − 1, 2122k� is a sequence of abc triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' 10 ELISE ALVAREZ-SALAZAR, ALEXANDER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' BARRIOS, CALVIN HENAKU, AND SUMMER SOLLER Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let j ≥ 2 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then � 1, � 2j − 1 �2k − 1, � 2j − 1 �2k� is an abc triple for each positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Observe that rad �� 2j − 1 �2� = rad � 2j − 1 � ≤ 2j − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since � 2j − 1 �2 − 1 = 2j+1 � 2j−1 − 1 � , we deduce that cosocle �� 2j − 1 �2 − 1 � = 2j+1 � 2j−1 − 1 � 2 rad(2j−1 − 1) = 2j � 2j−1 − 1 � rad(2j−1 − 1) ≥ 2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The result now follows from Theorem 1, since cosocle((2j − 1)2 − 1) > rad((2j − 1)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' □ The j = 2 and j = 3 cases in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='9 result in the sequences of abc triples � 1, 9k − 1, 9k� and � 1, 49k − 1, 49k� , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Of note is that the proof of the corollary is made possible by the lower bound, cosocle �� 2j − 1 �2 − 1 � ≥ 2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' This leads us to ask, can Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='9 be generalized to deduce sequences of abc triples (1, c − 1, c) with cosocle(c − 1) bounded below by nj for some positive integer of the form nj?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The answer is yes, but we have to take c = � nj − 1 �k for some positive even integer k that is divisible by n to allow a similar argument to that of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='9 to work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' This is shown below: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let n ≥ 3 and j ≥ 1 be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' If k is a positive integer such that nk is even, then � 1, � nj − 1 �nk − 1, � nj − 1 �nk� is an abc triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Observe that rad �� nj − 1 �nk� ≤ nj − 1 and � nj − 1 �nk − 1 = −1 + nk � l=0 �nk l � njl (−1)nk−l = −knj+1 + nk � l=2 �nk l � njl (−1)nk−l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Note that in the last expression, each term in the sum is divisible by nj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' From this, we deduce that cosocle �� nj − 1 �nk − 1 � ≥ nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Hence cosocle �� nj − 1 �nk − 1 � > rad �� nj − 1 �nk� , and the result now follows by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' □ As an illustration, consider (n, j) = (3, 1) and k = 2l for some positive integer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' This results in the sequence of abc triples � 1, 64l − 1, 64l� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let n be a positive even integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then � 1, n(n+1)k, n(n+1)k + 1 � is an abc triple for each positive odd integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Observe that nn+1 + 1 = (n + 1) �n j=0 (−1)j nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Since n ≡ −1 mod(n + 1), it follows that n � j=0 (−1)j nj ≡ n � j=0 1 mod(n + 1) = 0 mod(n + 1) Hence, rad(nn+1 + 1) = rad � nn+1+1 n+1 � ≤ nn+1+1 n+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Consequently, cosocle(nn+1 + 1) = nn+1 + 1 rad(nn+1 + 1) ≥ n + 1 > rad(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The result now follows from Theorem 2 by taking m = nn+1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' □ As a demonstration of the corollary, take n = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then � 1, 2223k, 2223k + 1 � is a sequence of abc triples for each positive odd integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' ON abc TRIPLES OF THE FORM (1, c − 1, c) 11 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let n ≥ 3 be an odd integer and let j ≥ 1 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then for each odd integer k, � 1, � nj − 1 �nk , � nj − 1 �nk + 1 � is an abc triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Observe that rad �� nj − 1 �n� ≤ nj − 1 and � nj − 1 �n + 1 = 1 + n � l=0 �n l � njl (−1)n−l = nj+1 + n � l=2 �n l � njl (−1)n−l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Note that in the last expression, each term in the sum is divisible by nj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' From this, we conclude that cosocle �� nj − 1 �n + 1 � ≥ nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Hence cosocle �� nj − 1 �n + 1 � > rad �� nj − 1 �n� , and the result now follows by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' □ As an example, let n = 3 and j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then we get the sequence of abc triples � 1, 8k, 8k + 1 � for each odd integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In particular, we recover the abc triple (1, 8, 9) as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' abc triples of the form (1, c − 1, c) and the ABC@Home Project The ABC@Home project found that there are exactly 14 482 065 abc triples (a, b, c) with c < 1018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The information found by the ABC@Home project is available on Bart de Smit’s webpage [dS23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Given an abc triple (a, b, c), we define its quality to be q(a, b, c) = log c log rad(abc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By definition, we see that since rad(abc) < c, an abc triple (a, b, c) satisfies q(a, b, c) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' This gives us the following restatement of the abc conjecture: For each ϵ > 0, there are finitely many abc triples (a, b, c) with q(a, b, c) > 1 + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The abc triple with the largest known quality is � 2, 310 · 109, 235� , which has a quality of approx- imately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='6299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In fact, Baker’s [Bak04] explicit abc conjecture asserts that there is no abc triple (a, b, c) with q(a, b, c) ≥ 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' From this statement, Fermat’s Last Theorem for exponent n > 6 easily follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We note that the explicit abc conjecture and the abc conjecture are not equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Histogram of the quality of abc triples (1, c − 1, c) with c < 1018 240 175 150 75 50 25 0 + 1D5 110 115 120 125 130 135 140 Quality12 ELISE ALVAREZ-SALAZAR, ALEXANDER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' BARRIOS, CALVIN HENAKU, AND SUMMER SOLLER Let S denote the set of abc triples of the form (1, c − 1, c) with c < 1018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' From the ABC@Home project, we have that #S = 45 603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The largest quality occurring in S corresponds to the abc triple (1, 4374, 4375), which has quality approximately equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='5679.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Figure 1 summarize the distribution of the quality of all abc triples in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The bin size in the histogram is set to 5 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We note that all computations done in this section were done on SageMath [S+23], and our code is available on GitHub [ASBHS23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Table 1 lists the first fifteen abc triples of the form (1, c−1, c), their quality, and whether they arise from one of the results proven in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The only abc triple in the table that is not of the form � 1, nl − 1, nl� or � 1, nl, nl + 1 � for some integer l > 1 is (1, 1215, 1216).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' However, most abc triples in S are not of the aforementioned form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' More precisely, S contains 7 376 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' 1 038) abc triples of the form � 1, nl − 1, nl� (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' � 1, nl, nl + 1 � ) for some integer l > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We note that (1, 8, 9) is the only double-counted element since Mih˘ailescu’s Theorem [Mih04] (formerly known as Catalan’s conjecture) asserts that 2 and 3 are the only two consecutive perfect powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Consequently, T = � (1, c − 1, c) ∈ S | c = nl or c = nl + 1 for some l > 1 � has 8 413 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The highest quality abc triple in T is (1, 2400, 2401), with a quality of approx- imately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='4557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Observe that this abc triple is obtained from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='9 since (1, 2400, 2401) = � 1, 74 − 1, 74� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The first fifteen abc triples of the form (1, c − 1, c) (1, c − 1, c) q(1, c − 1, c) Arises from result in Section 3?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' (1, 8, 9) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2263 Yes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='7 with (n, k) = (3, 1) (1, 48, 49) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0412 Yes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='9 with (j, k) = (3, 1) (1, 63, 64) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1127 Yes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='10 with (n, j, k) = (3, 1, 1) (1, 80, 81) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2920 Yes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='8 with (n, k) = (3, 1) (1, 224, 225) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0129 Yes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='9 with (j, k) = (4, 1) (1, 242, 243) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='3111 No (1, 288, 289) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2252 No (1, 512, 513) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='3176 Yes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='12 with (n, j, k) = (3, 1, 2) (1, 624, 625) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0790 Yes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='7 with (n, k) = (5, 1) (1, 675, 676) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0922 No (1, 728, 729) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0459 Yes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='7 with (n, k) = (3, 3) (1, 960, 961) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0048 Yes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='9 with (j, k) = (5, 1) (1, 1024, 1025) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1523 Yes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='11 with (n, k) = (4, 1) (1, 1215, 1216) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1194 No (1, 2303, 2304) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0204 No Now suppose that � 1, nl − 1, nl� is an abc triple for some integer l > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2, we know that cosocle � nl − 1 � > rad(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' However, checking that � 1, nl − 1, nl� is an abc triple via this criteria gets more difficult as nl grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By Theorem 1, we can deduce that � 1, nl − 1, nl� is an abc triple if there is a divisor m of nl−1 such that cosocle(m) > rad(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By considering those elements in T of the form � 1, nl − 1, nl� for some integer l > 1, we find that m can be taken to be a proper divisor ON abc TRIPLES OF THE FORM (1, c − 1, c) 13 of nl − 1, except for the abc triples (1, c − 1, c) where c ∈ {9, 676, 11309769, 17380816062160329}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Indeed, rad(676) = 26 and 675 = 3352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The only divisor of 675 satisfying cosocle(m) > 26 is m = 675.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The above leads us to ask: given � 1, nl − 1, nl� ∈ T with l > 1 an integer, what is the least divisor m of nl − 1 for which cosocle(m) > rad(n)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Using SageMath [S+23], we answered this question, and our datafile can be accessed in [ASBHS23, triples for thm1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='csv].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Table 2 gives the first fifteen elements (a, b, c) in T of the form � 1, nl − 1, nl� , where n and l are listed, as well as the least divisor m of nl − 1 for which cosocle(m) > rad(n) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The quality of the abc triple is also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The first fifteen abc triples (a, b, c) of the form � 1, nl − 1, nl� for l > 1, with m the least divisor of nl − 1 satisfying cosocle(m) > rad(n) (a, b, c) n l m q(a, b, c) (1, 8, 9) 3 2 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2263 (1, 48, 49) 7 2 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0412 (1, 63, 64) 2 6 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1127 (1, 80, 81) 3 4 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2920 (1, 224, 225) 15 2 32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0129 (1, 242, 243) 3 5 121 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='3111 (1, 288, 289) 17 2 144 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2252 (1, 624, 625) 5 4 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0790 (1, 675, 676) 26 2 675 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0922 (1, 728, 729) 3 6 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0459 (1, 960, 961) 31 2 64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0048 (1, 2303, 2304) 48 2 49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0204 (1, 2400, 2401) 7 4 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='4557 (1, 3024, 3025) 55 2 432 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0348 (1, 3968, 3969) 63 2 64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1554 Similarly, we ask the same question in the setting of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' That is, given � 1, nl, nl + 1 � ∈ T with l > 1 an odd integer, what is the least positive divisor m of nl+1 for which cosocle(m) > rad(n)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We note that T has 596 elements of the form � 1, nl, nl + 1 � for some integer l > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We also answer this question through SageMath, and our datafile is found in [ASBHS23, triples for thm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='csv].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Table 3 gives the first fifteen elements (a, b, c) in T of the form � 1, nl, nl + 1 � , where n and l are listed, as well as the least divisor m of nl + 1 for which cosocle(m) > rad(n) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In particular, we find that (1, 8, 9) is the only abc triple of the form (1, nl, nl + 1) in T with l > 1 an odd integer for which there is no proper divisor m of nl + 1 satisfying cosocle(m) > rad(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' 14 ELISE ALVAREZ-SALAZAR, ALEXANDER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' BARRIOS, CALVIN HENAKU, AND SUMMER SOLLER Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The first fifteen abc triples (a, b, c) of the form � 1, nl, nl + 1 � for l > 1 an odd integer, with m the least divisor of nl + 1 satisfying cosocle(m) > rad(n) (a, b, c) n l m q(a, b, c) (1, 8, 9) 2 3 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2263 (1, 512, 513) 2 9 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='3176 (1, 6859, 6860) 19 3 343 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2281 (1, 12167, 12168) 23 3 676 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2555 (1, 17576, 17577) 26 3 81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0039 (1, 29791, 29792) 31 3 784 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1424 (1, 32768, 32769) 2 15 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0406 (1, 110592, 110593) 48 3 49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0135 (1, 250047, 250048) 63 3 64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0351 (1, 279936, 279937) 6 7 49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0124 (1, 512000, 512001) 80 3 81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='4433 (1, 1953125, 1953126) 5 9 27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0423 (1, 2097152, 2097153) 2 21 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0287 (1, 3176523, 3176524) 147 3 676 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0145 (1, 7077888, 7077889) 192 3 169 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0515 Next, we investigate how many elements of T arise from the results proven in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Indeed, each abc triple produced by the results of that section are of the form � 1, nl − 1, nl� or � 1, nl, nl + 1 � for some integer l > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Moreover, for each abc triple obtained from one of our corollaries in Section 3, we apply the following result from [vdH10, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='3]: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Let (1, c − 1, c) be an abc triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then the following are abc triples: � 1, (c − 1)3 , c � c2 − 3c + 3 �� and � 1, c (c − 2) , (c − 1)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' As a demonstration, the abc triple (1, 2303, 2304) is obtained from the abc triple (1, 48, 49) since 2304 = 482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In particular, (1, 2303, 2304) can now be viewed as a consequence of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='9 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1 is part of a more general result in [vdH10, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='3], which provides a way of mapping an abc triple (a, b, c) to a new abc triple by applying polynomial identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The more general result arises by splitting the binomial formula (a + b)n to obtain the following family of identities: an−k � � k � j=0 �n j � ak−jbj � � + bk+1 � � n−k−1 � j=0 �n j � ajbn−k−1−j � � = cn Taking k = 0 yields Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Therefore, the two non-trivial polynomials identities with a = 1 are those occurring in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Corollaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='3 through 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='12 provide us with a recipe for constructing abc triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' For each of these corollaries, we consider the set Ci = {(1, c − 1, c) ∈ T | (1, c − 1, c) is obtained from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='i} , ON abc TRIPLES OF THE FORM (1, c − 1, c) 15 where 3 ≤ i ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' By Table 1, we see that (1, 224, 225) ∈ C9, but (1, 242, 243) ̸∈ Ci for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Using SageMath, we find that i 3 4 5 6 7 8 9 10 11 12 #Ci 32 58 12 17 41 29 81 46 18 36 The low number of abc triples in T occurring in each Ci is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Indeed, for Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='5 to yield an abc triples in T, we require that n > 1 be an integer, p be an odd prime such that p > rad(n), and np(p−1)k < 1018 for some integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' For n an odd integer, the only possible (n, p, k) is (3, 5, 1), which gives the abc triple (1, 3486784400, 3486784401).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We also note that since Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='5 is a special case of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='4, we have that C5 ⊆ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Now let C = � 3≤i≤12 Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We find that #C = 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Lastly, let D be the set of abc triples in T with the property that an element of D is in C or can be obtained from an abc triple in C after successive applications of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1 and Corollaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' As an illustration, the abc triple (1, 12214672127, 12214672128) ̸∈ C, but it is in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' To see this, recall that (1, 2303, 2304) is obtained from the abc triple (1, 48, 49) via Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Then, (1, 12214672127, 12214672128) = � 1, (c − 1)3 , c � c2 − 3c + 3 �� , where c = 2304, which shows that the abc triple is in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In fact, with the exception of the abc triple (1, 1215, 1216), every abc triple appearing in Table 1 is in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Using SageMath, we find that D has 311 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We conclude this article by considering the percentage of abc triples (1, c − 1, c) in S and T, that are also in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' More precisely, for sets X and Y such that X ⊆ Y ⊆ S, we define δX,Y (x) = # {(1, c − 1, c) ∈ X | c ≤ x} # {(1, c − 1, c) ∈ Y | c ≤ x} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In particular, δX,Y (x) gives the percentage of abc triples (1, c − 1, c) of Y with c ≤ x that are in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The table below gives some values of δT,S(x) , δD,S(x), and δD,T (x): x 104 106 108 1010 1012 1014 1016 1018 δT,S(x) 80% 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='8% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='2% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='6% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='9% 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='4% δD,S(x) 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='3% 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='1% 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='5% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='03% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='79% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='06% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='14% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='68% δD,T (x) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='7% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='7% 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='9% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='0% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='6% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='40% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='47% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='70% In particular, we see that D contains nearly half of the abc triples (1, c − 1, c) in T with c ≤ 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The authors would like to thank the National Science Foundation, Pomona College, Edray Goins, Renee Bell, Cory Colbert, Bianca Thompson, and the staff and students of the Pomona Research in Mathematics Experience (PRiME) for their support and camaraderie as this work was being undertaken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Research at PRiME was supported by the National Science Foundation award DMS-2113782.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' The authors also thank Andrew Granville for his comments and suggestions on an earlier preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In particular, we are grateful for his observations regarding those integers k for which (1, ck − 1, ck) is an abc triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' This led to the deduction of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' We also would like to thank the High Performance Computing Program team at California State University San Bernardino and the National Research Platform, especially Youngsu Kim, for 16 ELISE ALVAREZ-SALAZAR, ALEXANDER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' BARRIOS, CALVIN HENAKU, AND SUMMER SOLLER providing us access to SageMath in said computing program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' In particular, this work was supported in part by National Science Foundation awards CNS-1730158, ACI-1540112, ACI-1541349, OAC- 1826967, OAC-2112167, CNS-2120019, the University of California Office of the President, and the University of California San Diego’s California Institute for Telecommunications and Information Technology/Qualcomm Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Thanks to CENIC for the 100Gbps networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Any opinions, findings, and conclusions or recommendations expressed in this article are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} 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Leiden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Department of Mathematics, University of California, Santa Barbara, CA 93106 USA Email address: ealvarez-salazar@ucsb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='edu Department of Mathematics, University of St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Thomas, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Paul, MN 55105 USA Email address: abarrios@stthomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='edu Department of Mathematics, Washington University in St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Louis, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content=' Louis, MO 63130 USA Email address: chenaku@wustl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='edu Department of Mathematics, Colorado State University, Fort Collins, CO 80523 USA Email address: summer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='soller@colostate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AzT4oBgHgl3EQfa_wH/content/2301.01376v1.pdf'} diff --git 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a/kdFST4oBgHgl3EQfITgg/content/tmp_files/2301.13728v1.pdf.txt b/kdFST4oBgHgl3EQfITgg/content/tmp_files/2301.13728v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8a8566226b0eb3338e9ce51192bbc85527d90f97 --- /dev/null +++ b/kdFST4oBgHgl3EQfITgg/content/tmp_files/2301.13728v1.pdf.txt @@ -0,0 +1,316 @@ +arXiv:2301.13728v1 [physics.flu-dyn] 31 Jan 2023 +Convolutional autoencoder for the spatiotemporal +latent representation of turbulence +Nguyen Anh Khoa Doan1, Alberto Racca2, and Luca Magri3,4 +1 Delft University of Technology, Delft 2629HS, Netherlands +n.a.k.doan@tudelft.nl +2 University of Cambridge, Cambridge CB2 1PZ, United Kingdom +3 Imperial College London, London SW7 2AZ, United Kingdom +4 The Alan Turing Institute, London NW1 2DB, United Kingdom +Abstract. Turbulence is characterised by chaotic dynamics and a high- +dimensional state space, which make the phenomenon challenging to pre- +dict. However, turbulent flows are often characterised by coherent spa- +tiotemporal structures, such as vortices or large-scale modes, which can +help obtain a latent description of turbulent flows. However, current ap- +proaches are often limited by either the need to use some form of thresh- +olding on quantities defining the isosurfaces to which the flow structures +are associated or the linearity of traditional modal flow decomposition +approaches, such as those based on proper orthogonal decomposition. +This problem is exacerbated in flows that exhibit extreme events, which +are rare and sudden changes in a turbulent state. The goal of this paper +is to obtain an efficient and accurate reduced-order latent representation +of a turbulent flow that exhibits extreme events. Specifically, we em- +ploy a three-dimensional multiscale convolutional autoencoder (CAE) to +obtain such latent representation. We apply it to a three-dimensional tur- +bulent flow. We show that the Multiscale CAE is efficient, requiring less +than 10% degrees of freedom than proper orthogonal decomposition for +compressing the data and is able to accurately reconstruct flow states re- +lated to extreme events. The proposed deep learning architecture opens +opportunities for nonlinear reduced-order modeling of turbulent flows +from data. +Keywords: Chaotic System · Reduced Order Modelling · Convolutional +Autoencoder. +1 +Introduction +Turbulence is a chaotic phenomenon that arises from the nonlinear interactions +between spatiotemporal structures over a wide range of scales. Turbulent flows +are typically high-dimensional systems, which may exhibit sudden and unpre- +dictable bursts of energy/dissipation [3]. The combination of these dynamical +properties makes the study of turbulent flows particularly challenging. Despite +these complexities, advances have been made in the analysis of turbulent flows + +2 +N. A. K. Doan et al. +through the coherent (spatial) structures, such as vortices [13], and modal de- +composition techniques [1]. These achievements showed that there exist energetic +patterns within turbulence, which allow for the development of reduced-order +models. +To efficiently identify these patterns, recent works have used machine learn- +ing [4]. Specifically, Convolutional Neural Networks (CNNs) have been used to +identify spatial features in flows [9] and perform nonlinear modal decomposi- +tion [10,6]. These works showed the advantages of using CNNs over traditional +methods based on Principal Component Analysis (PCA) (also called Proper Or- +thogonal Decomposition, in the fluid mechanics community), providing lower +reconstruction errors in two-dimensional flows. More recently, the use of such +CNN-based architecture has also been extended to a small 3D turbulent chan- +nel flow at a moderate Reynolds number [11]. The works highlight the poten- +tial of deep learning for the analysis and reduced-order modelling of turbulent +flows, but they were restricted to two-dimensional flows or weakly turbulent +with non-extreme dynamics. Therefore, the applicability of deep-learning-based +techniques to obtain an accurate reduced-order representation of 3D flows with +extreme events remains unknown. Specifically, the presence of extreme events +is particularly challenging because they appear rarely in the datasets. In this +paper, we propose a 3D Multiscale Convolutional Autoencoder (CAE) to obtain +such a reduced representation of the 3D Minimal Flow Unit (MFU), which is +a flow that exhibit such extreme events in the form of a sudden and rare in- +termittent quasi-laminar flow state. We explore whether the Multiscale CAE is +able to represent the flow in a latent space with a reduced number of degrees +of freedom with higher accuracy than the traditional PCA, and whether it can +also accurately reconstruct the flow state during the extreme events. +Section 2 describes the MFU and its extreme events. Section 3 presents in +detail the Multiscale CAE framework used to obtain a reduced-order representa- +tion of the MFU. The accuracy of the Multiscale CAE in reconstructing the MFU +state is discussed in Section 4. A summary of the main results and directions for +future work are provided in Section 5. +2 +Minimal Flow Unit +The flow under consideration is the 3D MFU [8]. The MFU is an example of +prototypical near-wall turbulence, which consists of a turbulent channel flow +whose dimensions are smaller than conventional channel flow simulations. The +system is governed by the incompressible Navier-Stokes equations +∇ · u = 0, +∂tu + u · ∇u = 1 +ρf0 − 1 +ρ∇p + ν∆u, +(1) +where u = (u, v, w) is the 3D velocity field and f0 = (f0, 0, 0) is the constant +forcing in the streamwise direction, x; ρ, p, and ν are the density, pressure, and +kinematic viscosity, respectively. In the wall-normal direction, y, we impose a + +CAE for latent representation of turbulence +3 +no-slip boundary condition, u(x, ±δ, z, t) = 0, where δ is half the channel width. +In the the streamwise, x, and spanwise, z, directions we have periodic boundary +conditions. For this study, a channel with dimension Ω ≡ πδ × 2δ × 0.34πδ is +considered, as in [3] with δ = 1.0. The Reynolds number of the flow, which is +based on the bulk velocity and the half-channel width, is set to Re = 3000, which +corresponds to a friction Reynolds number Reτ ≈ 140. An in-house code similar +to that of [2] is used to simulate the MFU, and generate the dataset on which +the Multiscale CAE is trained and assessed. +The extreme events in the MFU are quasi-relaminarization events, which +take place close to either wall. A typical evolution of the flow during a quasi- +relaminarization event is shown in Fig. 1(a-g). Time is normalized by the eddy +turnover time. During an extreme event, (i) the flow at either wall (the upper +wall in Fig. 1) becomes laminar (Fig. 1(a-c)); (ii) the flow remains laminar for +some time (Fig. 1(c-f)), which results in a larger axial velocity close to the +centerline (and therefore an increase in kinetic energy); (iii) the greater velocity +close to the centerline makes the effective Reynolds number of the flow larger, +which in turn makes the flow prone to a turbulence burst on the quasi-laminar +wall; (iv) the turbulence burst occurs on the quasi-laminar wall, which results +in a large increase in the energy dissipation rate; and (v) the flow close to that +quasi-laminar wall becomes turbulent again, which leads to a decrease in the +kinetic energy (Fig. 1(g)). +(a) +(b) +(c) +(d) +(e) +(f) +(g) +x +y +z +(h) +Fig. 1. (a-g) Snapshots of the Q-criterion isosurface (with value Q = 0.1) during an +extreme event, where Q = 0.5(||ω||2 − ||S||2), ω is the vorticity vector, and S is the +strain-rate tensor. (h) Evolution of kinetic energy, k, of the MFU. The red box indicates +the event whose evolution is shown in (a-g). +These quasi-relaminarisation are accompanied by bursts in the total kinetic +energy, k(t) = +� � � +Ω +1 +2u · udxdydz, where Ω is the computational domain. +This can be seen in Fig. 1h, where the normalized kinetic energy, ˜k = (k − +min(k))/(max(k) − min(k)) is shown. +The dataset of the MFU contains 2000 eddy turnover times (i.e., 20000 snap- +shots) on a grid of 32 × 256 × 16, which contains 50 extreme events. The first + +4 +N. A. K. Doan et al. +200 eddy turnover times of the dataset (2000 snapshots) are employed for the +training of the Multiscale CAE, which contains only 4 extreme events. +3 +Multiscale Convolutional Autoencoder +We implement a 3D convolutional autoencoder. A schematic of the proposed +architecture is shown in Fig. 2. +~ +Fig. 2. Schematic of the 3D multiscale convolutional autoencoder. +The three dimensional convolution autoencoder (CAE) learns an efficient +reduced-order representation of the original data, which consists of the flow +state u ∈ RNx×Ny×Nz×Nu, where Nx, Ny and Nz are the number of grid points, +and Nu = 3 is the number of velocity components. On one hand, the encoder +(blue box in Fig. 2) reduces the dimension of the data down to a latent state, +c, with small dimension Nc ≪ NxNyNzNu. This operation can be symbolically +expressed as c = E(u; φE), where φE represents the weights of the encoder. On +the other hand, the decoder (green box in Fig. 2) reconstructs the data from +the latent state back to the original full flow state. This operation is expressed +as �u = D(c; φD), where φD are the trainable weights of the decoder. We em- +ploy a multiscale autoencoder, which was originally developed for image-based +super-resolution analysis [5]. It relies on the use of convolutional kernels of dif- +ferent sizes to analyse the input and improve reconstruction in fluids [7,12]. In +this work, two kernels, (3 × 5 × 3) and (5 × 7 × 5), are employed (represented +schematically by the two parallel streams of encoder/decoder in the blue and +green boxes in Fig. 2). This choice ensures a trade-off between the size of the +3D multiscale autoencoder and the reconstruction accuracy (see Section 4). To +reduce the dimension of the input, the convolution operation in each layer of +the encoder is applied in a strided manner, which means that the convolutional +neural network (CNN) kernel is progressively applied to the input by moving +the CNN kernel by (sx, sy, sz) = (2, 4, 2) grid points. This results in an output +of a smaller dimension than the input. After each convolution layer, to fulfill +the boundary conditions of the MFU, periodic padding is applied in the x and +z directions, while zero padding is applied in the y direction. Three successive +layers of CNN/padding operations are applied to decrease the dimension of the + +CAE for latent representation of turbulence +5 +original field from (32, 256, 16, 3) to (2, 4, 2, Nf), where Nf is the specified num- +ber of filters in the last encoding layer. As a result, the dimension of the latent +space is Nc = 16 × Nf. The decoder mirrors the architecture of the encoder, +where transpose CNN layers [14] are used, which increase the dimension of the +latent space up to the original flow dimension. The end-to-end autoencoder is +trained by minimizing the mean squared error (MSE) between the reconstructed +velocity field, �u, and the original field, u using the ADAM optimizer. +4 +Reconstruction error +We analyze the ability of the CAE to learn a latent space that encodes the flow +state accurately. To do so, we train three CAEs with latent-space dimensions of +Nc = 384, 768, and 1536. We compute their reconstruction errors on the test set +based on the MSE. A typical comparison between a reconstructed velocity field +obtained from the CAE with Nc = 1536 is shown in Fig. 3. The CAE is able to +reconstruct accurately the features of the velocity field. +(a) +(b) +(c) +Fig. 3. Comparison of (a) the actual velocity magnitude (ground truth), (b) the CAE- +reconstructed velocity magnitude, (c) the root-squared difference between (a) and (b) +in the mid-y plane for a typical snapshot in the test set. +To provide a comparison with the CAE, we also compute the reconstruc- +tion error obtained from PCA, whose principal directions are obtained with the +method of snapshots [1] on the same dataset. Figure 4 shows the reconstruction +error. PCA decomposition requires more than 15000 PCA components to reach +the same level of accuracy as the CAE with a latent space of dimension 1536. +This highlights the advantage of learning a latent representation with nonlinear +operations, as in the autoencoder, compared with relying on a linear combination +of components, as in PCA. +The better performance of the CAE with respect to PCA is evident when we +consider the reconstruction accuracy for flow states that correspond to extreme +events, which we extract from the test set. Here, we define an extreme event +as a flow state with normalized kinetic energy above a user-selected threshold +value of 0.7. Hence, for the selected snapshots in the test set, the mean squared +error (MSE) between the reconstructed velocity and the truth is computed using +the CAE and PCA as a function of the latent space size. The resulting MSE is +shown in Fig. 5, where the CAE exhibits an accuracy for the extreme dynamics +similar to reference case of the entire test set (see Fig. 4). On the other hand, the + +6 +N. A. K. Doan et al. +PCA +PCA +CAE +Fig. 4. Reconstruction error with a PCA-based method (blue) and the autoencoder +(yellow) for different dimensions of the latent space, Nc, or number of retained PCA +components, NP CA. The reconstruction error is computed as the mean squared error +between the reconstructed velocity field and the exact field, averaged over the test set. +accuracy of the PCA is lower than the reference case of the entire dataset. This +lack of accuracy is further analysed in Fig. 6, where typical velocity magnitudes, +i.e. the norm of the velocity field u, are shown for the mid-z plane during a +representative extreme event. The velocity reconstructed with the CAE (Fig. +6b) is almost identical to the true velocity field (Fig. 6a). The CAE captures +the smooth variation at the lower wall indicating a quasi-laminar flow state in +that region. In contrast, the velocity field reconstructed using PCA (Fig. 6c) +completely fails at reproducing those features. This is because extreme states +are rare in the training set used to construct the PCA components (and the +CAE). Because of this, the extreme states only have a small contribution to the +PCA components and are accounted for only in higher order PCA components, +which are neglected in the 1536-dimensional latent space. +CAE +P� � +PCA +Fig. 5. Reconstruction error on the extreme events flow states with PCA-based method +(blue) and autoencoder (yellow) for different dimensions of the latent space, Nc, or +number of retained PCA components, NP CA. The reconstruction error is computed as +in Fig. 4 but only on the flow state corresponding to extreme events. + +CAE for latent representation of turbulence +7 +(a) +(b) +(c) +Fig. 6. Comparison of (a) the velocity magnitude (ground truth), (b) the CAE- +reconstructed velocity magnitude, (c) the PCA-reconstructed velocity magnitude in +the mid-z plane for a typical extreme event snapshot in the test set. +5 +Conclusion +In this work, we develop a nonlinear autoencoder to obtain an accurate latent +representation of a turbulent flow that exhibits extreme events. We propose the +3D Multiscale CAE to learn the spatial features of the MFU, which exhibits ex- +treme events in the form of near-wall quasi-relaminarization events. The model +consists of a convolutional autoencoder with multiple channels, which learn an +efficient reduced latent representation of the flow state. We apply the framework +to a three-dimensional turbulent flow with extreme events (MFU). We show that +the Multiscale CAE is able to compress the flow state to a lower-dimensional +latent space by three orders of magnitude to accurately reconstruct the flow +state from this latent space. This constitutes a key improvement over a principal +component analysis (PCA), which requires at least one order of magnitude more +PCA components to achieve an accuracy similar to the CAE. This improvement +in reconstruction accuracy is crucial for the reconstruction of the flow state dur- +ing the extreme events of the MFU. This is because extreme states are rare and, +thus, require a large number of PCA components to be accurately reconstructed. +The proposed method and results open up possibilities for using deep learn- +ing to obtain an accurate latent reduced representation of three-dimensional +turbulent flows. Future work will be devoted to physically interpreting the la- +tent space discovered by the Multiscale CAE, and learning the dynamics in this +latent space. +Acknowledgements The authors thank Dr. Modesti for providing the flow +solver. N.A.K.D and L.M acknowledge that part of this work was performed +during the 2022 Stanford University CTR Summer Program. L.M. acknowledges +financial support from the ERC Starting Grant PhyCo 949388. +References +1. Berkooz, G., Holmes, P., Lumley, J.L.: The proper orthogonal, decomposition in +the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25, 539–575 (1993) +2. Bernardini, M., Pirozzoli, S., Orlandi, P.: Velocity statistics in turbulent channel +flow up to Reτ =4000. J. Fluid Mech. 742, 171–191 (2014) + +8 +N. A. K. Doan et al. +3. Blonigan, P.J., Farazmand, M., Sapsis, T.P.: Are extreme dissipation events pre- +dictable in turbulent fluid flows? Phys. Rev. Fluids 4, 044606 (2019) +4. Brunton, S.L., Noack, B.R., Koumoutsakos, P.: Machine Learning for Fluid Me- +chanics. Annu. Rev. Fluid Mech. 52(1), 477–508 (2020) +5. Du, X., Qu, X., He, Y., Guo, D.: Single image super-resolution based on multi-scale +competitive convolutional neural network. Sensors 18(3), 1–17 (2018) +6. 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Nakamura, T., Fukami, K., Hasegawa, K., Nabae, Y., Fukagata, K.: Convolutional +neural network and long short-term memory based reduced order surrogate for +minimal turbulent channel flow. Phys. Fluids 33, 025116 (2021) +12. Racca, A., Doan, N.A.K., Magri, L.: Modelling spatiotemporal turbulent dynamics +with the convolutional autoencoder echo state network. arXiv (2022) +13. Yao, J., Hussain, F.: A physical model of turbulence cascade via vortex reconnec- +tion sequence and avalanche. J. Fluid Mech. 883, A51 (2020) +14. Zeiler, M.D., Krishnan, D., Taylor, G.W., Fergus, R.: Deconvolutional networks. +Proc. IEEE Comput. Soc. Conf. Comput. Vis. Pattern Recognit. pp. 2528–2535 +(2010) + diff --git a/kdFST4oBgHgl3EQfITgg/content/tmp_files/load_file.txt b/kdFST4oBgHgl3EQfITgg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..62de60a2f7d89049d2b3f93ddd67319cc70221c2 --- /dev/null +++ b/kdFST4oBgHgl3EQfITgg/content/tmp_files/load_file.txt @@ -0,0 +1,300 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf,len=299 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content='13728v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content='flu-dyn] 31 Jan 2023 Convolutional autoencoder for the spatiotemporal latent representation of turbulence Nguyen Anh Khoa Doan1, Alberto Racca2, and Luca Magri3,4 1 Delft University of Technology, Delft 2629HS, Netherlands n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content='doan@tudelft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content='nl 2 University of Cambridge, Cambridge CB2 1PZ, United Kingdom 3 Imperial College London, London SW7 2AZ, United Kingdom 4 The Alan Turing Institute, London NW1 2DB, United Kingdom Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Turbulence is characterised by chaotic dynamics and a high- dimensional state space, which make the phenomenon challenging to pre- dict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' However, turbulent flows are often characterised by coherent spa- tiotemporal structures, such as vortices or large-scale modes, which can help obtain a latent description of turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' However, current ap- proaches are often limited by either the need to use some form of thresh- olding on quantities defining the isosurfaces to which the flow structures are associated or the linearity of traditional modal flow decomposition approaches, such as those based on proper orthogonal decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' This problem is exacerbated in flows that exhibit extreme events, which are rare and sudden changes in a turbulent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The goal of this paper is to obtain an efficient and accurate reduced-order latent representation of a turbulent flow that exhibits extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Specifically, we em- ploy a three-dimensional multiscale convolutional autoencoder (CAE) to obtain such latent representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' We apply it to a three-dimensional tur- bulent flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' We show that the Multiscale CAE is efficient, requiring less than 10% degrees of freedom than proper orthogonal decomposition for compressing the data and is able to accurately reconstruct flow states re- lated to extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The proposed deep learning architecture opens opportunities for nonlinear reduced-order modeling of turbulent flows from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Keywords: Chaotic System · Reduced Order Modelling · Convolutional Autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 1 Introduction Turbulence is a chaotic phenomenon that arises from the nonlinear interactions between spatiotemporal structures over a wide range of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Turbulent flows are typically high-dimensional systems, which may exhibit sudden and unpre- dictable bursts of energy/dissipation [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The combination of these dynamical properties makes the study of turbulent flows particularly challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Despite these complexities, advances have been made in the analysis of turbulent flows 2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Doan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' through the coherent (spatial) structures, such as vortices [13], and modal de- composition techniques [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' These achievements showed that there exist energetic patterns within turbulence, which allow for the development of reduced-order models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' To efficiently identify these patterns, recent works have used machine learn- ing [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Specifically, Convolutional Neural Networks (CNNs) have been used to identify spatial features in flows [9] and perform nonlinear modal decomposi- tion [10,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' These works showed the advantages of using CNNs over traditional methods based on Principal Component Analysis (PCA) (also called Proper Or- thogonal Decomposition, in the fluid mechanics community), providing lower reconstruction errors in two-dimensional flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' More recently, the use of such CNN-based architecture has also been extended to a small 3D turbulent chan- nel flow at a moderate Reynolds number [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The works highlight the poten- tial of deep learning for the analysis and reduced-order modelling of turbulent flows, but they were restricted to two-dimensional flows or weakly turbulent with non-extreme dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Therefore, the applicability of deep-learning-based techniques to obtain an accurate reduced-order representation of 3D flows with extreme events remains unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Specifically, the presence of extreme events is particularly challenging because they appear rarely in the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' In this paper, we propose a 3D Multiscale Convolutional Autoencoder (CAE) to obtain such a reduced representation of the 3D Minimal Flow Unit (MFU), which is a flow that exhibit such extreme events in the form of a sudden and rare in- termittent quasi-laminar flow state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' We explore whether the Multiscale CAE is able to represent the flow in a latent space with a reduced number of degrees of freedom with higher accuracy than the traditional PCA, and whether it can also accurately reconstruct the flow state during the extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Section 2 describes the MFU and its extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Section 3 presents in detail the Multiscale CAE framework used to obtain a reduced-order representa- tion of the MFU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The accuracy of the Multiscale CAE in reconstructing the MFU state is discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' A summary of the main results and directions for future work are provided in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 2 Minimal Flow Unit The flow under consideration is the 3D MFU [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The MFU is an example of prototypical near-wall turbulence, which consists of a turbulent channel flow whose dimensions are smaller than conventional channel flow simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The system is governed by the incompressible Navier-Stokes equations ∇ · u = 0, ∂tu + u · ∇u = 1 ρf0 − 1 ρ∇p + ν∆u, (1) where u = (u, v, w) is the 3D velocity field and f0 = (f0, 0, 0) is the constant forcing in the streamwise direction, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' ρ, p, and ν are the density, pressure, and kinematic viscosity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' In the wall-normal direction, y, we impose a CAE for latent representation of turbulence 3 no-slip boundary condition, u(x, ±δ, z, t) = 0, where δ is half the channel width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' In the the streamwise, x, and spanwise, z, directions we have periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' For this study, a channel with dimension Ω ≡ πδ × 2δ × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content='34πδ is considered, as in [3] with δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The Reynolds number of the flow, which is based on the bulk velocity and the half-channel width, is set to Re = 3000, which corresponds to a friction Reynolds number Reτ ≈ 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' An in-house code similar to that of [2] is used to simulate the MFU, and generate the dataset on which the Multiscale CAE is trained and assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The extreme events in the MFU are quasi-relaminarization events, which take place close to either wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' A typical evolution of the flow during a quasi- relaminarization event is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 1(a-g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Time is normalized by the eddy turnover time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' During an extreme event, (i) the flow at either wall (the upper wall in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 1) becomes laminar (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 1(a-c));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' (ii) the flow remains laminar for some time (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 1(c-f)), which results in a larger axial velocity close to the centerline (and therefore an increase in kinetic energy);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' (iii) the greater velocity close to the centerline makes the effective Reynolds number of the flow larger, which in turn makes the flow prone to a turbulence burst on the quasi-laminar wall;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' (iv) the turbulence burst occurs on the quasi-laminar wall, which results in a large increase in the energy dissipation rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' and (v) the flow close to that quasi-laminar wall becomes turbulent again, which leads to a decrease in the kinetic energy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 1(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' (a) (b) (c) (d) (e) (f) (g) x y z (h) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' (a-g) Snapshots of the Q-criterion isosurface (with value Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content='1) during an extreme event, where Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content='5(||ω||2 − ||S||2), ω is the vorticity vector, and S is the strain-rate tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' (h) Evolution of kinetic energy, k, of the MFU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The red box indicates the event whose evolution is shown in (a-g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' These quasi-relaminarisation are accompanied by bursts in the total kinetic energy, k(t) = � � � Ω 1 2u · udxdydz, where Ω is the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' This can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 1h, where the normalized kinetic energy, ˜k = (k − min(k))/(max(k) − min(k)) is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The dataset of the MFU contains 2000 eddy turnover times (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=', 20000 snap- shots) on a grid of 32 × 256 × 16, which contains 50 extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The first 4 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Doan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 200 eddy turnover times of the dataset (2000 snapshots) are employed for the training of the Multiscale CAE, which contains only 4 extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 3 Multiscale Convolutional Autoencoder We implement a 3D convolutional autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' A schematic of the proposed architecture is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' ~ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Schematic of the 3D multiscale convolutional autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The three dimensional convolution autoencoder (CAE) learns an efficient reduced-order representation of the original data, which consists of the flow state u ∈ RNx×Ny×Nz×Nu, where Nx, Ny and Nz are the number of grid points, and Nu = 3 is the number of velocity components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' On one hand, the encoder (blue box in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 2) reduces the dimension of the data down to a latent state, c, with small dimension Nc ≪ NxNyNzNu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' This operation can be symbolically expressed as c = E(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' φE), where φE represents the weights of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' On the other hand, the decoder (green box in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 2) reconstructs the data from the latent state back to the original full flow state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' This operation is expressed as �u = D(c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' φD), where φD are the trainable weights of the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' We em- ploy a multiscale autoencoder, which was originally developed for image-based super-resolution analysis [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' It relies on the use of convolutional kernels of dif- ferent sizes to analyse the input and improve reconstruction in fluids [7,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' In this work, two kernels, (3 × 5 × 3) and (5 × 7 × 5), are employed (represented schematically by the two parallel streams of encoder/decoder in the blue and green boxes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' This choice ensures a trade-off between the size of the 3D multiscale autoencoder and the reconstruction accuracy (see Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' To reduce the dimension of the input, the convolution operation in each layer of the encoder is applied in a strided manner, which means that the convolutional neural network (CNN) kernel is progressively applied to the input by moving the CNN kernel by (sx, sy, sz) = (2, 4, 2) grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' This results in an output of a smaller dimension than the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' After each convolution layer, to fulfill the boundary conditions of the MFU, periodic padding is applied in the x and z directions, while zero padding is applied in the y direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Three successive layers of CNN/padding operations are applied to decrease the dimension of the CAE for latent representation of turbulence 5 original field from (32, 256, 16, 3) to (2, 4, 2, Nf), where Nf is the specified num- ber of filters in the last encoding layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' As a result, the dimension of the latent space is Nc = 16 × Nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The decoder mirrors the architecture of the encoder, where transpose CNN layers [14] are used, which increase the dimension of the latent space up to the original flow dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The end-to-end autoencoder is trained by minimizing the mean squared error (MSE) between the reconstructed velocity field, �u, and the original field, u using the ADAM optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 4 Reconstruction error We analyze the ability of the CAE to learn a latent space that encodes the flow state accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' To do so, we train three CAEs with latent-space dimensions of Nc = 384, 768, and 1536.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' We compute their reconstruction errors on the test set based on the MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' A typical comparison between a reconstructed velocity field obtained from the CAE with Nc = 1536 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The CAE is able to reconstruct accurately the features of the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Comparison of (a) the actual velocity magnitude (ground truth), (b) the CAE- reconstructed velocity magnitude, (c) the root-squared difference between (a) and (b) in the mid-y plane for a typical snapshot in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' To provide a comparison with the CAE, we also compute the reconstruc- tion error obtained from PCA, whose principal directions are obtained with the method of snapshots [1] on the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Figure 4 shows the reconstruction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' PCA decomposition requires more than 15000 PCA components to reach the same level of accuracy as the CAE with a latent space of dimension 1536.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' This highlights the advantage of learning a latent representation with nonlinear operations, as in the autoencoder, compared with relying on a linear combination of components, as in PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The better performance of the CAE with respect to PCA is evident when we consider the reconstruction accuracy for flow states that correspond to extreme events, which we extract from the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Here, we define an extreme event as a flow state with normalized kinetic energy above a user-selected threshold value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Hence, for the selected snapshots in the test set, the mean squared error (MSE) between the reconstructed velocity and the truth is computed using the CAE and PCA as a function of the latent space size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The resulting MSE is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 5, where the CAE exhibits an accuracy for the extreme dynamics similar to reference case of the entire test set (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' On the other hand, the 6 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Doan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' PCA PCA CAE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Reconstruction error with a PCA-based method (blue) and the autoencoder (yellow) for different dimensions of the latent space, Nc, or number of retained PCA components, NP CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The reconstruction error is computed as the mean squared error between the reconstructed velocity field and the exact field, averaged over the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' accuracy of the PCA is lower than the reference case of the entire dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' This lack of accuracy is further analysed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 6, where typical velocity magnitudes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' the norm of the velocity field u, are shown for the mid-z plane during a representative extreme event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The velocity reconstructed with the CAE (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 6b) is almost identical to the true velocity field (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The CAE captures the smooth variation at the lower wall indicating a quasi-laminar flow state in that region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' In contrast, the velocity field reconstructed using PCA (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 6c) completely fails at reproducing those features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' This is because extreme states are rare in the training set used to construct the PCA components (and the CAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Because of this, the extreme states only have a small contribution to the PCA components and are accounted for only in higher order PCA components, which are neglected in the 1536-dimensional latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' CAE P� � PCA Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Reconstruction error on the extreme events flow states with PCA-based method (blue) and autoencoder (yellow) for different dimensions of the latent space, Nc, or number of retained PCA components, NP CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The reconstruction error is computed as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 4 but only on the flow state corresponding to extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' CAE for latent representation of turbulence 7 (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Comparison of (a) the velocity magnitude (ground truth), (b) the CAE- reconstructed velocity magnitude, (c) the PCA-reconstructed velocity magnitude in the mid-z plane for a typical extreme event snapshot in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 5 Conclusion In this work, we develop a nonlinear autoencoder to obtain an accurate latent representation of a turbulent flow that exhibits extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' We propose the 3D Multiscale CAE to learn the spatial features of the MFU, which exhibits ex- treme events in the form of near-wall quasi-relaminarization events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The model consists of a convolutional autoencoder with multiple channels, which learn an efficient reduced latent representation of the flow state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' We apply the framework to a three-dimensional turbulent flow with extreme events (MFU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' We show that the Multiscale CAE is able to compress the flow state to a lower-dimensional latent space by three orders of magnitude to accurately reconstruct the flow state from this latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' This constitutes a key improvement over a principal component analysis (PCA), which requires at least one order of magnitude more PCA components to achieve an accuracy similar to the CAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' This improvement in reconstruction accuracy is crucial for the reconstruction of the flow state dur- ing the extreme events of the MFU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' This is because extreme states are rare and, thus, require a large number of PCA components to be accurately reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' The proposed method and results open up possibilities for using deep learn- ing to obtain an accurate latent reduced representation of three-dimensional turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Future work will be devoted to physically interpreting the la- tent space discovered by the Multiscale CAE, and learning the dynamics in this latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Acknowledgements The authors thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Modesti for providing the flow solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content='D and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content='M acknowledge that part of this work was performed during the 2022 Stanford University CTR Summer Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' acknowledges financial support from the ERC Starting Grant PhyCo 949388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Berkooz, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=', Holmes, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=', Lumley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content='L.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=', Fergus, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=': Deconvolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' IEEE Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' Pattern Recognit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} +page_content=' 2528–2535 (2010)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFST4oBgHgl3EQfITgg/content/2301.13728v1.pdf'} diff --git a/l9FIT4oBgHgl3EQfsisx/vector_store/index.pkl b/l9FIT4oBgHgl3EQfsisx/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..30802dd96dc5eaa1c36f7c5945158c886a55b2ce --- /dev/null +++ b/l9FIT4oBgHgl3EQfsisx/vector_store/index.pkl 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b/r9E5T4oBgHgl3EQflw-X/content/tmp_files/2301.05674v1.pdf.txt @@ -0,0 +1,1603 @@ +arXiv:2301.05674v1 [cs.GT] 13 Jan 2023 +Altruism in Coalition Formation Games +Anna Maria Kerkmann, Simon Cramer, and J¨org Rothe +Heinrich-Heine-Universit¨at D¨usseldorf, Germany +January 16, 2023 +Abstract +Nguyen et al. [1] introduced altruistic hedonic games in which agents’ utilities depend not only on their +own preferences but also on those of their friends in the same coalition. We propose to extend their model +to coalition formation games in general, considering also the friends in other coalitions. Comparing our +model to altruistic hedonic games, we argue that excluding some friends from the altruistic behavior of +an agent is a major disadvantage that comes with the restriction to hedonic games. After introducing our +model and showing some desirable properties, we additionally study some common stability notions and +provide a computational analysis of the associated verification and existence problems. +1 +Introduction +We consider coalition formation games where agents have to form coalitions based on their preferences. +Among other compact representations of hedonic coalition formation games, Dimitrov et al. [2] in particular +proposed the friends-and-enemies encoding with friend-oriented preferences which involves a network of +friends: a (simple) undirected graph whose vertices are the players and where two players are connected by +an edge exactly if they are friends of each other. Players not connected by an edge consider each other as +enemies. Under friend-oriented preferences, player i prefers a coalition C to a coalition D if C contains more +of i’s friends than D, or C and D have the same number of i’s friends but C contains fewer enemies of i’s +than D. This is a special case of the additive encoding [3]. For more background on these two compact +representations, see Section 2 and the book chapter by Aziz and Savani [4]. +Based on friend-oriented preferences, Nguyen et al. [1] introduced altruistic hedonic games where agents +gain utility not only from their own satisfaction but also from their friends’ satisfaction. However, Nguyen +et al. [1] specifically considered hedonic games only, which require that an agent’s utility only depends on +her own coalition. In their interpretation of altruism, the utility of an agent is composed of the agent’s own +valuation of her coalition and the valuation of all this agent’s friends in this coalition. While Nguyen et al. [1] +used the average when aggregating some agents’ valuations, Wiechers and Rothe [5] proposed a variant of +altruistic hedonic games where some agents’ valuations are aggregated by taking the minimum. +Inspired by the idea of altruism, we extend the model of altruism in hedonic games to coalition formation +games in general. That is, we propose a model where agents behave altruistically to all their friends, not only +to the friends in the same coalition. Not restricting to hedonic games, we aim to capture a more natural notion +of altruism where none of an agent’s friends is excluded from her altruistic behavior. +Example 1. To become acquainted with this idea of altruism, consider the coalition formation game that +is represented by the network of friends in Figure 1. For the coalition structures Γ = {{1,2,3},{4}} and +∆ = {{1,2,4},{3}}, it is clear that player 1 is indifferent between coalitions {1,2,3} and {1,2,4} under +friend-oriented preferences, as both coalitions contain 1’s only friend (player 2) and one of 1’s enemies +(either 3 or 4). Under altruistic hedonic preferences [1], however, player 1 behaves altruistically to her friend +2 (who is friends with 3 but not with 4) and therefore prefers {1,2,3} to {1,2,4}. Now, consider the slightly +modified coalition structures Γ′ = {{1},{2,3},{4}} and ∆′ = {{1},{2,4},{3}}. Intuitively, one would still +expect 1 to behave altruistically to her friend 2. However, under any hedonic preference (which requires the +1 + +1 +2 +3 +4 +Figure 1: Network of friends for Example 1 +players’ preferences to depend only on their own coalitions), player 1 (being in the same coalition for both +Γ′ and ∆′) must be indifferent between Γ′ and ∆′. +In order to model altruism globally, we release the restriction to hedonic games and introduce altruistic +coalition formation games where agents behave altruistically to all their friends, independently of their current +coalition. +1.1 +Related Work +Coalition formation games, as considered here, are closely related to the subclass of hedonic games which has +been broadly studied in the literature, addressing the issue of compactly representing preferences, conducting +axiomatic analyses, dealing with different notions of stability, and investigating the computational complexity +of the associated problems (see, e.g., the book chapter by Aziz and Savani [4]). +Closest related to our work are the altruistic hedonic games by Nguyen et al. [1] (see also the related +minimization-based variant by Wiechers and Rothe [5]), which we modify to obtain our more general models +of altruism. Based on the model due to Nguyen et al. [1], Schlueter and Goldsmith [6] defined super altruistic +hedonic games where friends have a different impact on an agent based on their distances in the underlying +network of friends. More recently, Bullinger and Kober [7] introduced loyalty in cardinal hedonic games +where agents are loyal to all agents in their so-called loyalty set. In their model, the utilities of the agents in +the loyalty set are aggregated by taking the minimum. They then study the loyal variants of common classes +of cardinal hedonic games such as additively separable and friend-oriented hedonic games.1 +Altruism has also been studied for noncooperative games. Most prominently, Ashlagi et al. [8] introduced +social context games where a social context is applied to a strategic game and the costs in the resulting game +depend on the original costs and a graph of neighborhood. Their so-called MinMax collaborations (where +players seek to minimize the maximal cost of their own and their neighbors) are related to our minimization- +based equal-treatment model. Still, the model of Ashlagi et al. [8] differs from ours in that they consider +noncooperative games. Other work considering noncooperative games with social networks is due to Bil`o et +al. [9] who study social context games for other underlying strategic games than Ashlagi et al. [8], Hoefer +et al. [10] who study considerate equilibria in strategic games, and Anagnostopoulos et al. [11] who study +altruism and spite in strategic games. Further work studying altruism in noncooperative games without +social networks is due to Hoefer and Skopalik [12], Chen et al. [13], Apt and Sch¨afer [14], and Rahn and +Sch¨afer [15]. +1.2 +Our Contribution +Conceptually, we extend the models of altruism proposed by Nguyen et al. [1] and Wiechers and Rothe [5] +from hedonic games to general coalition formation games. We argue how this captures a more global notion +of altruism and show that our models fulfill some desirable properties that are violated by the previous mod- +els. We then study the common stability concepts in this model and analyze the associated verification and +existence problems in terms of their computational complexity. +This work extends a preliminary version that appeared in the proceedings of the 29th International Joint +Conference on Artificial Intelligence (IJCAI’20) [16]. Parts of this work were also presented at the 16th and +17th International Symposium on Artificial Intelligence and Mathematics (ISAIM’20 and ISAIM’22) and +at the 8th International Workshop on Computational Social Choice (COMSOC’21), each with nonarchival +proceedings. +1Note that their loyal variant of symmetric friend-oriented hedonic games is equivalent to the minimization-based altruistic hedonic +games under equal treatment introduced by Wiechers and Rothe [5]. +2 + +2 +The Model +In coalition formation games, players divide into groups based on their preferences. Before introducing +altruism, we now give some foundations of such games. +2.1 +Coalition Formation Games +Let N = {1,...,n} be a set of agents (or players). Each subset of N is called a coalition. A coalition structure +Γ is a partition of N, and we denote the set of all possible coalition structures for N by CN. For a player +i ∈ N and a coalition structure Γ ∈ CN, Γ(i) denotes the unique coalition in Γ containing i. Now, a coalition +formation game (CFG) is a pair (N,⪰), where N = {1,...,n} is a set of agents, ⪰ = (⪰1,...,⪰n) is a profile +of preferences, and every preference ⪰i ∈ CN × CN is a complete weak order over all possible coalition +structures. Given two coalition structures Γ, ∆ ∈ CN, we say that i weakly prefers Γ to ∆ if Γ ⪰i ∆. When +Γ ⪰i ∆ but not ∆ ⪰i Γ, we say that i prefers Γ to ∆ (denoted by Γ ≻i ∆), and we say that i is indifferent between +Γ and ∆ (denoted by Γ ∼i ∆) if Γ ⪰i ∆ and ∆ ⪰i Γ. +Note that hedonic games are a special case of coalition formation games where the agents’ preference +relations only depend on the coalitions containing themselves. In a hedonic game (N,⪰), agent i ∈ N is +indifferent between any two coalition structures Γ and ∆ as long as her coalition is the same, i.e., Γ(i) = +∆(i) =⇒ Γ ∼i ∆. Therefore, the preference order of any agent i ∈ N in a hedonic game (N,⪰) is usually +represented by a complete weak order over the set of coalitions containing i. +2.2 +The “Friends and Enemies” Encoding +Since |CN|, the number of all possible coalition structures, is extremely large in the number of agents,2 it is +not reasonable to ask every agent for her complete preference over CN. Instead, we are looking for a way to +compactly represent the agents’ preferences. In the literature, many such representations have been proposed +for hedonic games, such as the additive encoding [19, 3, 20], the singleton encoding due to Cechl´arov´a +and Romero-Medina [21] and further studied by Cechl´arov´a and Hajdukov´a [22], the friends-and-enemies +encoding due to Dimitrov et al. [2], and FEN-hedonic games due to Kerkmann et al. [23] and also used by +Rothe et al. [24]. Here, we use the friends-and-enemies encoding due to Dimitrov et al. [2]. We focus on +their friend-oriented model and will later adapt it to our altruistic model. +In the friend-oriented model, the preferences of the agents in N are given by a network of friends, i.e., +a (simple) undirected graph G = (N,A) whose vertices are the players and where two players i, j ∈ N are +connected by an edge {i, j} ∈ A exactly if they are each other’s friends. Agents not connected by an edge +consider each other as enemies. For an agent i ∈ N, we denote the set of i’s friends by Fi = { j ∈ N |{i, j} ∈ A} +and the set of i’s enemies by Ei = N \ (Fi ∪ {i}). Under friend-oriented preferences as defined by Dimitrov +et al. [2], between any two coalitions players prefer the coalition with more friends, and if there are equally +many friends in both coalitions, they prefer the coalition with fewer enemies: +C ⪰F +i D ⇐⇒ |C ∩Fi| > |D∩Fi| or (|C ∩Fi| = |D∩Fi| and |C ∩Ei| ≤ |D∩Ei|). +This can also be represented additively. Assigning a value of n to each friend and a value of −1 to +each enemy, agent i ∈ N values coalition C containing herself with vi(C) = n|C ∩ Fi| − |C ∩ Ei|. Note that +−(n − 1) ≤ vi(C) ≤ n(n − 1), and vi(C) > 0 if and only if there is at least one friend of i’s in C. For a given +coalition structure Γ ∈ CN, we also write vi(Γ) for player i’s value of Γ(i). +Furthermore, we denote the sum of the values of i’s friends by sumF +i (Γ) = ∑f∈Fi vf (Γ). Analogously, we +also define sumF+ +i +(Γ) = ∑ f∈Fi∪{i} vf (Γ), minF +i (Γ) = minf∈Fi vf (Γ), and minF+ +i +(Γ) = minf∈Fi∪{i}vf (Γ). +2.3 +Three Degrees of Altruism +When we now define altruistic coalition formation games based on the friend-oriented preference model, we +consider the same three degrees of altruism that Nguyen et al. [1] introduced for altruistic hedonic games. +2The number of possible partitions of a set with n elements equals the n-th Bell number [17, 18], defined as Bn = ∑n−1 +k=0 +�n−1 +k +� +Bk with +B0 = B1 = 1. For example, for n = 10 agents, we have B10 = 115,975 possible coalition structures. +3 + +5 +1 +2 +3 +4 +6 +7 +8 +9 10 +Figure 2: Network of friends for Example 2 +However, we adapt them to our model, extending the agents’ altruism to all their friends, not only to their +friends in the same coalition. +• Selfish First (SF): Agents first rank coalition structures based on their own valuations. Only in the +case of a tie between two coalition structures, their friends’ valuations are considered as well. +• Equal Treatment (EQ): Agents treat themselves and their friends the same. That means that an agent +i ∈ N and all of i’s friends have the same impact on i’s utility for a coalition structure. +• Altruistic Treatment (AL): Agents first rank coalition structures based on their friends’ valuations. +They only consider their own valuations in the case of a tie. +We further distinguish between a sum-based and a min-based aggregation of some agents’ valuations. For- +mally, for an agent i ∈ N and a coalition structure Γ ∈ CN, we denote i’s sum-based utility for Γ under SF by +usumSF +i +(Γ), under EQ by usumEQ +i +(Γ), and under AL by usumAL +i +(Γ), and her min-based utility for Γ under SF by +uminSF +i +(Γ), under EQ by uminEQ +i +(Γ), and under AL by uminAL +i +(Γ). For a constant M ≥ n3, they are defined as +usumSF +i +(Γ) = M ·vi(Γ)+ sumF +i (Γ); +uminSF +i +(Γ) = M ·vi(Γ)+ minF +i (Γ); +usumEQ +i +(Γ) = sumF+ +i +(Γ); +uminEQ +i +(Γ) = minF+ +i +(Γ); +usumAL +i +(Γ) = vi(Γ)+ M ·sumF +i (Γ); +uminAL +i +(Γ) = vi(Γ)+ M ·minF +i (Γ). +In the case of Fi = /0, we define the minimum of the empty set to be zero. +For any coalition structures Γ,∆ ∈ CN, agent i’s sum-based SF preference is then defined by Γ ⪰sumSF +i +∆ ⇐⇒ usumSF +i +(Γ) ≥ usumSF +i +(∆). Her other altruistic preferences (⪰sumEQ +i +; ⪰sumAL +i +; ⪰minSF +i +; ⪰minEQ +i +; and +⪰minAL +i +) are defined analogously, using the respective utility functions. The factor M, which is used for the +SF and AL models, ensures that an agent’s utility is first determined by the agent’s own valuation in the SF +model and first determined by the friends’ valuations in the AL model. Similarly as Nguyen et al. [1] prove the +corresponding properties in hedonic games, we can show that for M ≥ n3, vi(Γ) > vi(∆) implies Γ ≻sumSF +i +∆ +and Γ ≻minSF +i +∆, and for M ≥ n2, sumF +i (Γ) > sumF +i (∆) implies Γ ≻sumAL +i +∆ while minF +i (Γ) > minF +i (∆) implies +Γ ≻minAL +i +∆. An altruistic coalition formation game (ACFG) is a coalition formation game where the agents’ +preferences were obtained by a network of friends via one of these cases of altruism. Hence, we distinguish +between sum-based SF, sum-based EQ, sum-based AL, min-based SF, min-based EQ, and min-based AL +ACFGs. For any ACFG, the players’ utilities can obviously be computed in polynomial time. +3 +Monotonicity and Other Properties in ACFGs +Nguyen et al. [1] focus on altruism in hedonic games where an agent’s utility only depends on her own +coalition. As we have already seen in Example 1, there are some aspects of altruistic behavior that cannot be +realized by hedonic games. The following example shows that our model crucially differs from the models +due to Nguyen et al. [1] and Wiechers and Rothe [5]. +Example 2. Consider an ACFG (N,⪰) with the network of friends in Figure 2 and the coalition structures +Γ = {{1,2},{3},{4},...,{10}} and ∆ = {{1,5,...,10},{2,3,4}}. We will now compare agent 1’s prefer- +ences for these two coalition structures under our altruistic models to 1’s preferences under the altruistic +hedonic models [1, 5]. Table 1 shows all relevant values that are needed to compute the utilities of agent 1. +One can observe that agent 1 and all her friends assign a greater value to ∆ than to Γ. Consequently, +also the aggregations of the friends’ values (sumF +1 , sumF+ +1 , minF +1 , minF+ +1 ) are greater for ∆. Hence, 1 prefers +∆ to Γ under all our sum-based and min-based altruistic preferences. +4 + +Table 1: Values for the game in Example 2 with the network of friends in Figure 2 +v1 +v2 +v5 +v6 +sumF +1 +sumF+ +1 +minF +1 +minF+ +1 +Γ +10 +10 +0 +0 +10 +20 +0 +0 +∆ +16 +20 +5 +5 +30 +46 +5 +5 +The hedonic models due to Nguyen et al. [1] and Wiechers and Rothe [5], however, are blind to the fact +that agent 1 and all her friends are better off in ∆ than in Γ. Under their altruistic hedonic preferences, +player 1 compares the two coalition structures Γ and ∆ only based on her own coalitions Γ(1) = {1,2} and +∆(1) = {1,5,...,10}. She then only considers her friends that are in the same coalition, i.e., player 2 for +Γ and players 5 and 6 for ∆. This leads to 1 preferring Γ(1) to ∆(1) under altruistic hedonic EQ and AL +preferences. In particular, the average (and minimum) valuation of 1’s friends in Γ(1) is 10 while the average +(and minimum) valuation of 1’s friends in ∆(1) is 5. Also considering 1’s own value for EQ, the average (and +minimum) in Γ(1) is 10 while the average (respectively, minimum) value in ∆(1) is 8.6 (respectively, 5). +3.1 +Some Basic Properties +As we have seen in Example 2, altruistic hedonic games [1, 5] allow for players that prefer coalition structures +that make themselves and all their friends worse off. To avoid this kind of unreasonable behavior, we focus on +general coalition formation games. In fact, all our altruistic coalition-formation preferences fulfill unanimity: +For an ACFG (N,⪰) and a player i ∈ N, we say that ⪰i is unanimous if, for any two coalition structures +Γ,∆ ∈ CN, va(Γ) > va(∆) for each a ∈ Fi ∪{i} implies Γ ≻i ∆. +This property crucially distinguishes our preference models from the corresponding altruistic hedonic +preferences, which are not unanimous under EQ or AL preferences, as Example 2 shows. Note that Nguyen et +al. [1] define a restricted version of unanimity in altruistic hedonic games by considering only the agents’ own +coalitions. Other desirable properties that were studied by Nguyen et al. [1] for altruistic hedonic preferences +can be generalized to coalition formation games. We show that these desirable properties also hold for our +models. First, we collect some basic observations: +Observation 3. Consider any ACFG (N,⪰) with an underlying network of friends G. +1. All preferences ⪰i, i ∈ N, are reflexive and transitive. +2. For any player i ∈ N and any two coalition structures Γ,∆ ∈ CN, it can be decided in polynomial time +(in the number of agents) whether Γ ⪰i ∆. +3. The preferences ⪰i, i ∈ N, only depend on the structure of G. +Note that the third statement of Observation 3 implies that the properties that Nguyen et al. [1] call +anonymity and symmetry are both satisfied in ACFGs. Another desirable property they consider is called +sovereignty of players and inspired by the axiom of “citizens’ sovereignty” from social choice theory:3 Given +a set of agents N, a coalition structure Γ ∈ CN, and an agent i ∈ N, we say that sovereignty of players is +satisfied if there is a network of friends G on N such that Γ is i’s most preferred coalition structure in any +ACFG induced by G. +Proposition 4. ACFGs satisfy sovereignty of players under all sum-based and min-based SF, EQ, and AL +altruistic preferences. +Proof. +Sovereignty of players in ACFGs can be shown with an analogous construction as in the proof of +Nguyen et al. [1, Theorem 5]: For a given set of players N, player i ∈ N, and coalition structure Γ ∈ CN, +we construct a network of friends where all players in Γ(i) are friends of each other while there are no other +friendship relations. Then Γ is i’s (nonunique) most preferred coalition structure under all sum-based and +min-based SF, EQ, and AL altruistic preferences. +3Informally stated, a voting rule satisfies citizens’ sovereignty if every candidate can be made a winner of an election for a suitably +chosen preference profile. +5 + +3.2 +Monotonicity +The next property describes the monotonicity of preferences and further distinguishes our models from altru- +istic hedonic games. In fact, Nguyen et al. [1] define two types of monotonicity, which we here adapt to our +setting. +Definition 5. Consider any ACFG (N,⪰), agents i, j ∈ N with j ∈ Ei, and coalition structures Γ,∆ ∈ CN. Let +further ⪰′ +i be the preference relation resulting from ⪰i when j turns from being i’s enemy to being i’s friend +(all else remaining equal). We say that ⪰i is +• type-I-monotonic if (1) Γ ≻i ∆, j ∈ Γ(i) ∩ ∆(i), and vj(Γ) ≥ vj(∆) implies Γ ≻′ +i ∆, and (2) Γ ∼i ∆, +j ∈ Γ(i)∩∆(i), and vj(Γ) ≥ vj(∆) implies Γ ⪰′ +i ∆; +• type-II-monotonic if (1) Γ ≻i ∆, j ∈ Γ(i) \ ∆(i), and vj(Γ) ≥ vj(∆) implies Γ ≻′ +i ∆, and (2) Γ ∼i ∆, +j ∈ Γ(i)\ ∆(i), and vj(Γ) ≥ vj(∆) implies Γ ⪰′ +i ∆. +Theorem 6. Let (N,⪰) be an ACFG. +1. If (N,⪰) is sum-based, its preferences satisfy type-I- and type-II-monotonicity. +2. If (N,⪰) is min-based, its preferences satisfy type-II- but not type-I-monotonicity. +Proof. +Let (N,⪰) be an ACFG with an underlying network of friends G = (N,H). Consider i ∈ N, Γ,∆ ∈ +CN, and j ∈ Ei and denote with G′ = (N,H ∪{{i, j}}) the network of friends resulting from G when j turns +from being i’s enemy to being i’s friend (all else being equal). Let (N,⪰′) be the ACFG induced by G′. +For any agent a ∈ N and coalition structure Γ ∈ CN, denote a’s value for Γ in G′ by v′ +a(Γ), a’s preference +relation in (N,⪰′) by ⪰′ +a, and a’s friends and enemies in (N,⪰′) by F′ +a and E′ +a, respectively. That is, we have +F′ +i = Fi ∪{ j}, E′ +i = Ei \ { j}, F′ +j = Fj ∪{i}, and E′ +j = E j \ {i}. Further, v′ +i, v′ +j, and ⪰′ +i might differ from vi, vj, +and ⪰i, while the friends, enemies, and values of all other players stay the same, i.e., F′ +a = Fa, E′ +a = Ea, and +v′ +a = va for all a ∈ N \ {i, j}. +Type-I-monotonicity under sum-based preferences. +Let j ∈ Γ(i)∩∆(i) and vj(Γ) ≥ vj(∆). It then holds +that +v′ +i(Γ) = n|Γ(i)∩F′ +i |− |Γ(i)∩E′ +i| = n|Γ(i)∩Fi|+ n − |Γ(i)∩Ei|+ 1 = vi(Γ)+ n + 1. +Equivalently, v′ +i(∆) = vi(∆)+ n + 1, v′ +j(Γ) = vj(Γ)+ n + 1, and v′ +j(∆) = vj(∆)+ n + 1. Furthermore, +sumF′ +i (Γ) = ∑ +a∈F′ +i +v′ +a(Γ) = +∑ +a∈Fi∪{ j} +v′ +a(Γ) = ∑ +a∈Fi +va(Γ)+ v′ +j(Γ) += sumF +i (Γ)+ vj(Γ)+ n + 1 and +(1) +sumF′ +i (∆) = sumF +i (∆)+ vj(∆)+ n + 1. +(2) +(1) sumSF: If Γ ≻sumSF +i +∆ then either (i) vi(Γ) = vi(∆) and sumF +i (Γ) > sumF +i (∆), or (ii) vi(Γ) > vi(∆). +In case (i), vi(Γ) = vi(∆) implies v′ +i(Γ) = v′ +i(∆). Applying sumF +i (Γ) > sumF +i (∆) and vj(Γ) ≥ vj(∆) to (1) +and (2), we get sumF′ +i (Γ) > sumF′ +i (∆). This together with v′ +i(Γ) = v′ +i(∆) implies Γ ≻sumSF′ +i +∆. +In case (ii), vi(Γ) > vi(∆) implies v′ +i(Γ) > v′ +i(∆). Hence, Γ ≻sumSF′ +i +∆. +If Γ ∼sumSF +i +∆ then vi(Γ) = vi(∆) and sumF +i (Γ) = sumF +i (∆). vi(Γ) = vi(∆) implies v′ +i(Γ) = v′ +i(∆). Applying +sumF +i (Γ) = sumF +i (∆) and vj(Γ) ≥ vj(∆) to (1) and (2), we get sumF′ +i (Γ) ≥ sumF′ +i (∆). This together with +v′ +i(Γ) = v′ +i(∆) implies Γ ⪰sumSF′ +i +∆. +(2) sumEQ: If Γ ≻sumEQ +i +∆ then sumF +i (Γ)+vi(Γ) > sumF +i (∆)+vi(∆). Using (1), (2), v′ +i(Γ) = vi(Γ)+n+1, +v′ +i(∆) = vi(∆)+n+1, and vj(Γ) ≥ vj(∆), this implies sumF′ +i (Γ)+v′ +i(Γ) > sumF′ +i (∆)+v′ +i(∆). Hence, Γ ≻sumEQ′ +i +∆. +If Γ ∼sumEQ +i +∆, using the same equations, Γ ⪰sumEQ′ +i +∆ is implied. +(3) sumAL: If Γ ≻sumAL +i +∆ then either (i) sumF +i (Γ) = sumF +i (∆) and vi(Γ) > vi(∆), or (ii) sumF +i (Γ) > +sumF +i (∆). +6 + +In case (i), sumF +i (Γ) = sumF +i (∆) together with (1), (2), and vj(Γ) ≥ vj(∆) implies sumF′ +i (Γ) ≥ sumF′ +i (∆). +Further, vi(Γ) > vi(∆) together with v′ +i(Γ) = vi(Γ) + n + 1 and v′ +i(∆) = vi(∆) + n + 1 implies v′ +i(Γ) > v′ +i(∆). +Altogether, this implies Γ ≻sumAL′ +i +∆. +In case (ii), sumF′ +i (Γ) > sumF′ +i (∆) is implied and Γ ≻sumAL′ +i +∆ follows. +If Γ ∼sumAL +i +∆ then sumF +i (Γ) = sumF +i (∆) and vi(Γ) = vi(∆). Using the same equations as before, Γ ⪰sumAL′ +i +∆ is implied. +Type-II-monotonicity under sum-based and min-based preferences. +Let j ∈ Γ(i) \ ∆(i) and vj(Γ) ≥ +vj(∆). It follows that v′ +i(Γ) = vi(Γ) + n + 1, v′ +i(∆) = vi(∆), v′ +j(Γ) = vj(Γ) + n + 1, and v′ +j(∆) = vj(∆). Fur- +thermore, +sumF′ +i (Γ) = sumF +i (Γ)+ vj(Γ)+ n + 1, +(3) +sumF′ +i (∆) = sumF +i (∆)+ vj(∆), +(4) +minF′ +i (Γ) = min +� +minF +i (Γ),vj(Γ)+ n + 1 +� +, +(5) +minF′ +i (∆) = min +� +minF +i (∆),vj(∆) +� +, +(6) +minF+′ +i +(Γ) = min +� +minF +i (Γ),vj(Γ)+ n + 1,vi(Γ)+ n + 1 +� +, and +(7) +minF+′ +i +(∆) = min +� +minF +i (∆),vj(∆),vi(∆) +� +. +(8) +(1) sumSF and minSF: If Γ ⪰SF +i +∆ then vi(Γ) ≥ vi(∆). Hence, v′ +i(Γ) = vi(Γ)+ n + 1 ≥ vi(∆)+ n + 1 > +vi(∆) = v′ +i(∆), which implies Γ ≻SF′ +i +∆. +(2) sumEQ: If Γ ⪰sumEQ +i +∆ then sumF +i (Γ)+vi(Γ) ≥ sumF +i (∆)+vi(∆). Together with (3), (4), and vj(Γ) ≥ +vj(∆) this implies sumF′ +i (Γ)+ v′ +j(Γ) > sumF′ +i (∆)+ v′ +j(∆). Hence, Γ ≻sumEQ′ +i +∆. +(3) sumAL: If Γ ⪰sumAL +i +∆ then sumF +i (Γ) ≥ sumF +i (∆). Together with (3), (4), and vj(Γ) ≥ vj(∆) this +implies sumF′ +i (Γ) > sumF′ +i (∆), so Γ ≻sumAL′ +i +∆. +(4) minEQ: First, assume that Γ ≻minEQ +i +∆. We then have min +� +minF +i (Γ),vi(Γ) +� +> min +� +minF +i (∆),vi(∆) +� +. +It follows that Γ ≻minEQ′ +i +∆ because +minF+′ +i +(Γ) = min +� +minF +i (Γ),vj(Γ)+ n + 1,vi(Γ)+ n + 1 +� +(9) +> min +� +minF +i (∆),vj(Γ),vi(∆) +� +≥ min +� +minF +i (∆),vj(∆),vi(∆) +� += minF+′ +i +(∆). +Second, assume Γ ∼minEQ +i +∆. Then min +� +minF +i (Γ),vi(Γ) +� += min +� +minF +i (∆),vi(∆) +� +. Similarly as in (9), it +follows that minF+′ +i +(Γ) ≥ minF+′ +i +(∆). Hence, Γ ⪰minEQ′ +i +∆. +(5) minAL: First, assume Γ ≻minAL +i +∆. Then either (i) minF +i (Γ) > minF +i (∆), or (ii) minF +i (Γ) = minF +i (∆) +and vi(Γ) > vi(∆). +In case of (i), we get Γ ≻minAL′ +i +∆ because +minF′ +i (Γ) = min +� +minF +i (Γ),vj(Γ)+ n + 1 +� +≥ min +� +minF +i (Γ),vj(∆)+ n + 1 +� +(10) +> min +� +minF +i (∆),vj(∆) +� += minF′ +i (∆). +In case of (ii), similarly as in (10), we get minF′ +i (Γ) ≥ minF′ +i (∆). Furthermore, vi(Γ) > vi(∆) implies +v′ +i(Γ) > v′ +i(∆). Hence, Γ ≻minAL′ +i +∆. +Second, assume that Γ ∼minAL +i +∆. Then minF +i (Γ) = minF +i (∆) and vi(Γ) = vi(∆). Similarly as in (10), we +get minF′ +i (Γ) ≥ minF′ +i (∆). Furthermore, vi(Γ) = vi(∆) implies v′ +i(Γ) > v′ +i(∆). Hence, Γ ≻minAL′ +i +∆. +Type-I-monotonicity under min-based preferences. +To see that ⪰minSF is not type-I-monotonic, consider +the game G1 with the network of friends in Figure 3a. Furthermore, consider the coalition structures Γ = +{{1,2},{3,4,5},{6}} and ∆ = {{1,2},{3,4,5,6}} and players i = 1 and j = 2 with 2 ∈ Γ(1) ∩ ∆(1), and +v2(Γ) = −1 = v2(∆). It holds that v1(Γ) = v1(∆) = −1, minF +1 (Γ) = 2n, and minF +1 (∆) = 2n − 1. Hence, +Γ ≻minSF +1 +∆. +7 + +1 +2 +3 +4 +5 +6 +(a) Network of G1 +1 +2 +3 +4 +5 +6 +(b) Network of G ′ +1 +1 +2 +3 +4 +5 +(c) Network of G2 +1 +2 +3 +4 +5 +(d) Network of G ′ +2 +Figure 3: Networks of friends in the proof of Theorem 6 +Now, making 2 a friend of 1’s leads to the game G ′ +1 with the network of friends in Figure 3b. For this +game, we have v′ +1(Γ) = v′ +1(∆) = n and minF′ +1 (Γ) = minF′ +1 (∆) = n. This implies Γ ∼minSF′ +1 +∆, which contradicts +type-I-monotonicity. +To see that ⪰minEQ and ⪰minAL are not type-I-monotonic,consider the game G2 with the network of friends +in Figure 3c. Consider the coalition structures Γ = {{1,2,3,4},{5}} and ∆ = {{1,2,3,4,5}} and players i = +1 and j = 2 with 2 ∈ Γ(1)∩∆(1), and v2(Γ) = −3 > −4 = v2(∆). It holds that minF+ +1 (Γ) = minF +1 (Γ) = 2n−1, +and minF+ +1 (∆) = minF +1 (∆) = 2n − 2. Hence, Γ ≻minEQ +1 +∆ and Γ ≻minAL +1 +∆. +Now, making 2 a friend of 1’s leads to the game G ′ +2 with the network of friends in Figure 3d. For this +game, we have minF+′ +1 +(Γ) = minF′ +1 (Γ) = n and minF+′ +1 +(∆) = minF′ +1 (∆) = n. This implies Γ ∼minEQ′ +1 +∆ and +Γ ∼minAL′ +1 +∆, contradicting type-I-monotonicity and completing the proof. +Note that the hedonic models of altruism [1, 5] violate both type-I- and type-II-monotonicity for EQ and +AL. Hence, it is quite remarkable that all three degrees of our extended sum-based model of altruism satisfy +both types of monotonicity. +4 +Stability in ACFGs +The main question in coalition formation games is which coalition structures might form. There are several +stability concepts that are well-studied for hedonic games, each indicating whether a given coalition structure +would be accepted by the agents or if there are other coalition structures that are more likely to form. Although +we consider more general coalition formation games, we can easily adapt these definitions to our framework. +Let (N,⪰) be an ACFG with preferences ⪰ = (⪰1,...,⪰n) obtained from a network of friends via one of +the three degrees of altruism and with either sum-based or min-based aggregation of the agents’ valuations. +We use the following notation. For a coalition structure Γ ∈ CN, a player i ∈ N, and a coalition C ∈ Γ∪{/0}, +Γi→C denotes the coalition structure that arises from Γ when moving i to C, i.e., +Γi→C = Γ\ {Γ(i),C} ∪{Γ(i)\ {i},C∪{i}}. +In addition, we use ΓC→/0, with C ⊆ N, to denote the coalition structure that arises from Γ when all players +in C leave their respective coalition and form a new one, i.e., +ΓC→/0 = Γ\ {Γ(j)| j ∈ C} ∪{Γ(j)\C| j ∈ C} ∪{C}. +Finally, for any two coalition structures Γ,∆ ∈ CN, let #Γ≻∆ = |{i ∈ N |Γ ≻i ∆}| be the number of players +that prefer Γ to ∆. Now, we are ready to define the common stability notions. +Definition 7. A coalition structure Γ is said to be +• Nash stable if no player prefers moving to another coalition: +(∀i ∈ N)(∀C ∈ Γ∪{/0})[Γ ⪰i Γi→C]; +• individually rational if no player would prefer being alone: +(∀i ∈ N)[Γ ⪰i Γi→/0]; +• individually stable if no player prefers moving to another coalition and could deviate to it without +harming any player in that coalition: +(∀i ∈ N)(∀C ∈ Γ∪{/0}) +� +Γ ⪰i Γi→C ∨(∃j ∈ C)[Γ ≻ j Γi→C] +� +; +8 + +• contractually individually stable if no player prefers another coalition and could deviate to it without +harming any player in the new or the old coalition: +(∀i ∈ N)(∀C ∈ Γ∪{/0}) +� +Γ ⪰i Γi→C ∨(∃j ∈ C)[Γ ≻ j Γi→C]∨(∃k ∈ Γ(i))[Γ ≻k Γi→C] +� +; +• totally individually stable if no player prefers another coalition and could deviate to it without harming +any other player: +(∀i ∈ N)(∀C ∈ Γ∪{/0}) +� +Γ ⪰i Γi→C ∨(∃l ∈ N \ {i})[Γ ≻l Γi→C] +� +; +• core stable if no nonempty coalition blocks Γ: +(∀C ⊆ N,C ̸= /0)(∃i ∈ C)[Γ ⪰i ΓC→/0]; +• strictly core stable if no coalition weakly blocks Γ: +(∀C ⊆ N)(∃i ∈ C)[Γ ≻i ΓC→/0]∨(∀i ∈ C)[Γ ∼i ΓC→/0]; +• popular if for every other coalition structure ∆, at least as many players prefer Γ to ∆ as there are +players who prefer ∆ to Γ: +(∀∆ ∈ CN,∆ ̸= Γ) +� +#Γ≻∆ ≥ #∆≻Γ +� +; +• strictly popular if for every other coalition structure ∆, more players prefer Γ to ∆ than there are players +who prefer ∆ to Γ: +(∀∆ ∈ CN,∆ ̸= Γ) +� +#Γ≻∆ > #∆≻Γ +� +; +• perfect if no player prefers any coalition structure to Γ: +(∀i ∈ N)(∀∆ ∈ CN)[Γ ⪰i ∆]. +Note that “totally individual stability” is a new notion which we introduce here. It strengthens the notion +of contractually individual stability and makes sense in the context of coalition formation games because +players’ preferences may also be influenced by coalitions they are not part of. +We now study the associated verification and existence problems in terms of their computational complex- +ity. We assume the reader to be familiar with the complexity classes P (deterministic polynomial time), NP +(nondeterministic polynomial time) and coNP (the class of complements of NP sets). For more background +on computational complexity, we refer to, e.g., the textbooks by Garey and Johnson [25] and Rothe [26]. +Given a stability concept α, we define: +• α-VERIFICATION: Given an ACFG (N,⪰) and a coalition structure Γ ∈ CN, does Γ satisfy α? +• α-EXISTENCE: Given an ACFG (N,⪰), does there exist a coalition structure Γ ∈ CN that satisfies α? +Table 2 summarizes the results for these problems under sum-based and min-based SF preferences. We +will also give results for EQ and AL in this section. In Table 2, however, we only mark if the results for EQ +and AL match those for SF. +4.1 +Individual Rationality +Verifying individual rationality is easy: We just need to iterate over all agents and compare two coalition +structures in each iteration. Since players’ utilities can be computed in polynomial time, individual rationality +can be verified in time polynomial in the number of agents. The existence problem is trivial, since Γ = +{{1},...,{n}} is always individually rational. Furthermore, we give the following characterization. +Theorem 8. Given an ACFG (N,⪰), a coalition structure Γ ∈ CN is individually rational +9 + +Table 2: Complexity results in sum-based and min-based SF ACFGs +Stability notion α +α-VERIFICATION +α-EXISTENCE +Individual rationality +in P1 +trivial1 +Nash stability +in P1 +trivial1 +Individual stability +in P1 +trivial1 +Core stability +coNP-complete2 +trivial +Strict core stability +in coNP2 +trivial +Popularity +coNP-complete2 +not trivial1 +Strict popularity +coNP-complete2 +coNP-hard +Perfectness +in P2 +in P3 +1 also holds for sum-based and min-based EQ and AL +ACFGs +2 is in coNP for any ACFG +3 is in coNP for sum-based EQ ACFGs +1. under sum-based SF, sum-based EQ, sum-based AL, min-based SF, or min-based AL preferences if +and only if it holds for all players i ∈ N that Γ(i) contains a friend of i’s or i is alone, formally: +(∀i ∈ N)[Γ(i)∩Fi ̸= /0 ∨Γ(i) = {i}]; +2. under min-based EQ preferences if and only if for all players i ∈ N, Γ(i) contains a friend of i’s or i is +alone or there is a friend of i’s whose valuation of Γ is less than or equal to i’s valuation of Γ, formally: +(∀i ∈ N)[Γ(i)∩Fi ̸= /0 ∨Γ(i) = {i} ∨(∃j ∈ Fi)[vj(Γ) ≤ vi(Γ)]]. +Proof. +1. To show the implication from left to right, if Γ is individually rational, we assume for the sake of +contradiction that Γ(i)∩Fi = /0 and Γ(i) ̸= {i} for some player i ∈ N. First, we observe that for all j ∈ Fi we +have vj(Γ) = vj(Γi→/0), as their respective coalition is not affected by i’s move. It directly follows that, for all +considered models of altruism, player i’s utilities for Γ and Γi→/0 only depend on her own valuation, which is +greater for Γi→/0 than for Γ (since there are enemies in Γ(i) but not in Γi→/0(i)). Hence, i prefers Γi→/0 to Γ, so +Γ is not individually rational. This is a contradiction. +The implication from right to left is obvious for all considered models of altruism. +2. From left to right, we have that Γ is individually rational and, for the sake of contradiction, we assume +that there is a player i ∈ N with Γ(i)∩Fi = /0 and Γ(i) ̸= {i} and for all j ∈ Fi we have vj(Γ) > vi(Γ). Since +i is the least satisfied player in Fi ∪ {i}, we have uminEQ +i +(Γ) = vi(Γ). With vj(Γi→/0) = vj(Γ) > vi(Γ) for all +j ∈ Fi and vi(Γi→/0) = 0 > vi(Γ), we immediately obtain uminEQ +i +(Γi→/0) > uminEQ +i +(Γ) and Γi→/0 ≻minEQ +i +Γ. This +is a contradiction to Γ being individually rational. +From right to left, we have to consider two cases. First, if Γ(i)∩Fi ̸= /0 or Γ(i) = {i} for some i ∈ N, we +obviously have Γ ⪰minEQ +i +Γi→/0. Second, if Γ(i) ∩ Fi = /0 and Γ(i) ̸= {i}, we know that there is at least one +j ∈ Fi with vj(Γ) ≤ vi(Γ) < 0. Let j′ denote a least satisfied friend of i’s in Γ (pick one randomly if there +are more than one). Since Γ(i) ∩ Fi = /0, it holds that Γ(j) = Γi→/0(j) for all j ∈ Fi. Consequently, j′ is i’s +least satisfied friend in both coalition structures and we have uminEQ +i +(Γ) = vj′(Γ) = vj′(Γi→/0) = uminEQ +i +(Γi→/0). +Hence, Γ ∼minEQ +i +Γi→/0, so Γ is individually rational. +4.2 +Nash Stability +Since there are at most |N| coalitions in a coalition structure Γ ∈ CN, we can verify Nash stability in polyno- +mial time: We just iterate over all agents i ∈ N and all the (at most |N|+ 1) coalitions C ∈ Γ∪{/0} and check +whether Γ ⪰i Γi→C. Since we can check a player’s altruistic preferences over any two coalition structures in +polynomial time and since we have at most a quadratic number of iterations (|N| · (|N| + 1)), Nash stability +verification is in P for any ACFG. +10 + +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Figure 4: Networks of friends for Example 10 +Nash stability existence is trivially in P for any ACFG; indeed, the same example that Nguyen et al. [1] +gave for altruistic hedonic games works here as well. Specifically, for C = {i ∈ N |Fi = /0} = {c1,...,ck} the +coalition structure {{c1},...,{ck},N \C} is Nash stable. +4.3 +Individual Stability +For individual stability, contractually individual stability, and totally individual stability, existence is also +trivially in P. Nash stability implies all these three concepts, hence, the Nash stable coalition structure given +above is also (contractually; totally) individually stable. +Verification is also in P for these stability concepts. Similarly to Nash stability, we can iterate over all +players and all coalitions and check the respective conditions in polynomial time. +4.4 +Core Stability and Strict Core Stability +We now turn to core stability and state some results for sum-based and min-based SF ACFGs. We first show +that (strict) core stability existence is trivial for SF ACFGs. +Theorem 9. Let (N,⪰SF) be a (sum-based or min-based) SF ACFG with the underlying network of friends +G. Let further C1,...,Ck be the vertex sets of the connected components of G. Then Γ = {C1,...,Ck} is +strictly core stable (and thus core stable). +Proof. +For the sake of contradiction, assume that Γ were not strictly core stable, i.e., that there is a coalition +D ̸= /0 that weakly blocks Γ. Consider some player i ∈ D. Since i weakly prefers deviating from Γ(i) to D, +there have to be at least as many friends of i’s in D as in Γ(i). Since Γ(i) contains all of i’s friends, D also has +to contain all friends of i’s. Then all these friends of i’s also have all their friends in D for the same reason, +and so on. Consequently, D contains all players from the connected component Γ(i), i.e., Γ(i) ⊆ D. +Since D weakly blocks Γ, D cannot be equal to Γ(i) and thus needs to contain some ℓ /∈ Γ(i). Yet, this is a +contradiction, as ℓ is an enemy of i’s and i would prefer Γ to ΓD→/0 if D contains the same number of friends +as Γ(i) but more enemies than Γ(i). +However, the coalition structure from Theorem 9 is not necessarily core stable under EQ and AL prefer- +ences. +Example 10. Let N = {1,...,10} and consider the network of friends G shown in Figure 4. Consider the +coalition structure consisting of the connected component of G (i.e., of only the grand coalition: Γ = {N}) +and the coalition C = {8,9,10}. C blocks Γ under sum-based and min-based EQ and AL preferences. To see +this, consider how players 7, 8, 9, and 10 value Γ and ΓC→/0: +v7(Γ) = v8(Γ) = 30 − 6 = 24, +v7(ΓC→/0) = 20 − 4 = 16, +v9(Γ) = v10(Γ) = 20 − 7 = 13, +v8(ΓC→/0) = v9(ΓC→/0) = v10(ΓC→/0) = 20. +We then obtain +• sumF+ +8 (Γ) = 74 < 76 = sumF+ +8 (ΓC→/0) and sumF+ +9 (Γ) = sumF+ +10 (Γ) = 50 < 60 = sumF+ +9 (ΓC→/0) = +sumF+ +10 (ΓC→/0), so ΓC→/0 ≻sumEQ +i +Γ for all i ∈ C; +• sumF +8 (Γ) = 50 < 56 = sumF +8 (ΓC→/0) and sumF +9 (Γ) = sumF +10(Γ) = 37 < 40 = sumF +9 (ΓC→/0) = sumF +10(ΓC→/0), +so ΓC→/0 ≻sumAL +i +Γ for all i ∈ C; +• minF+ +8 (Γ) = minF +8 (Γ) = 13 < 16 = minF+ +8 (ΓC→/0) = minF +8 (ΓC→/0) and minF+ +9 (Γ) = minF +9 (Γ) = minF+ +10 (Γ) = +minF +10(Γ) = 13 < 20 = minF+ +9 (ΓC→/0) = minF +9 (ΓC→/0) = minF+ +10 (ΓC→/0) = minF +10(ΓC→/0), which implies +ΓC→/0 ≻minEQ +i +Γ and ΓC→/0 ≻minAL +i +Γ for all i ∈ C. +11 + +β1 +... +βb +... +β3k +ζS1... +ζSj +b ∈ Sj +... +ζS3k +αS1,1 +αS1,2 +αS1,3 +δS1,1 +... +δS1,4k−3 +αS3k,1 +αS3k,2 +αS3k,3 +δS3k,1 +... +δS3k,4k−3 +... +... +Beta +Zeta +QS1 +... +QS3k +Figure 5: Network of friends in the proof of Theorem 11 that is used to show coNP-hardness of core stability +verification in min-based SF ACFGs. A dashed rectangle around a group of players indicates that all these +players are friends of each other. +Thus C blocks Γ under sum-based and min-based EQ and AL preferences. +Turning to (strict) core stability verification, we can show that this problem is hard under SF preferences, +and we suspect that this hardness also extends to EQ and AL. +Theorem 11. Strict core stability verification and core stability verification are in coNP for any ACFG. For +(sum-based and min-based) SF ACFGs, core stability verification is even coNP-complete. +Proof. +To see that strict core stability verification and core stability verification are in coNP, consider any +coalition structure Γ ∈ CN in an ACFG (N,⪰). Γ is not (strictly) core stable if there is a coalition C ⊆ N that +(weakly) blocks Γ. Hence, we nondeterministically guess a coalition C ⊆ N and check whether C (weakly) +blocks Γ. This can be done in polynomial time since the preferences of the agents in C for the coalition +structures Γ and ΓC→/0 can be verified in polynomial time for all our altruistic models. +To show coNP-hardness of core stability verification under min-based SF ACFGs, we use RX3C, which +is a restricted variant of EXACT COVER BY 3-SETS and known to be NP-complete [25, 27]. We provide a +polynomial-time many-one reduction from RX3C to the complement of our verification problem. Let (B,S ) +be an instance of RX3C, consisting of a set B = {1,...,3k} and a collection S = {S1,...,S3k} of 3-element +subsets of B, where each element of B occurs in exactly three sets in S . The question is whether there exists +an exact cover for B in S , i.e., a subset S ′ ⊆ S with |S ′| = k and � +S∈S ′ S = B. We assume that k > 4. +From (B,S ) we now construct the following ACFG. The set of players is N = {βb|b ∈ B}∪{ζS,αS,1,αS,2, +αS,3,δS,1,...,δS,4k−3 |S ∈ S } and we define the sets +Beta += +{βb|b ∈ B}, +Zeta += +{ζS |S ∈ S }, and +QS += +{ζS,αS,1,αS,2,αS,3,δS,1,...,δS,4k−3} for each S ∈ S . +Figure 5 shows the network of friends, where a dashed rectangle around a group of players means that all +these players are friends of each other: +• All players in Beta are friends of each other. +• For every S ∈ S , ζS is friend with every βb with b ∈ S and with αS,1, αS,2, and αS,3. +• For every S ∈ S , αS,1, αS,2, αS,3, and δS,1 are friends of each other. +• For every S ∈ S , all players in {δS,1,...,δS,4k−3} are friends of each other. +Furthermore, consider the coalition structure Γ = {Beta,QS1,...,QS3k}. We will now show that S con- +tains an exact cover for B if and only if Γ is not core stable under the min-based SF model. +Only if: +Assume that there is an exact cover S ′ ⊆ S for B. Then |S ′| = k. Consider coalition C = +Beta∪{ζS |S ∈ S ′}. C blocks Γ, i.e., ΓC→/0 ≻minSF +i +Γ for all i ∈ C, because (a) every βb ∈ Beta has 3k friends +in C but only 3k−1 friends in Beta and (b) every ζS with S ∈ S ′ has 3 friends and 4k−4 enemies in C but 3 +friends and 4k − 3 enemies in QS. +12 + +If: +Assume that Γ is not core stable and let C ⊆ N be a coalition that blocks Γ. Then ΓC→/0 ≻minSF +i +Γ for +all i ∈ C. First, observe that every i ∈ C needs to have at least as many friends in C as in Γ(i). So, if any αS,j +or δS,j is in C, it follows quite directly that QS ⊆ C. However, since QS is a coalition in Γ and since every +other player (from N \ QS) is an enemy of all δ-players, any coalition C with QS ⊆ C cannot be a blocking +coalition for Γ. This contradiction implies that no αS,j or δS,j is in C. +We now have C ⊆ Beta∪Zeta. Since any βb ∈ C has 3k − 1 friends and no enemies in Γ(βb) and prefers +ΓC→/0 to Γ, one of the following holds: (a) βb has at least 3k friends in C or (b) βb has 3k − 1 friends and +no enemies in C and βb’s friends assign a higher value to ΓC→/0 than to Γ. For a contradiction, assume that +(b) holds for some βb ∈ C. First, observe that there are exactly 3k players in C (namely, βb and βb’s 3k − 1 +friends). We now distinguish two cases: +Case 1: All the 3k−1 friends of βb’s are β-players. Then C consists of all β-players, i.e., C = Beta. This +is a contradiction, as Beta is already a coalition in Γ. +Case 2: There are some ζ-players in C that are βb’s friends. Since βb has three ζ-friends in total and no +enemies in C, there are between one and three ζ-players in C. Hence, there are between 3k − 3 and 3k − 1 +β-players in C. Then one of the β-players has no ζ-friend in C. (The at most three ζ-players are friends with +at most nine β-players, but 3k − 3 > 9 for k > 4.) Consequently, this β-player has only the other (at most +3k − 2) β-players as friends in C and does not prefer ΓC→/0 to Γ. This is a contradiction. +Hence, option (a) holds for each βb ∈C. In total, each βb has exactly three ζ-friends and 3k−1 β-friends. +Thus at least 3k −3 of βb’s friends in C are β-players and at least one of βb’s friends in C is a ζ-player. Also +counting βb herself, there are at least 3k − 2 β-players in C. Since all of these 3k − 2 β-players have at least +one ζ-friend in C, there are at least k ζ-players in C. (Note that k−1 ζ-players are friends with at most 3k−3 +β-players.) +Consider some ζS ∈ C. Since ζS has three friends and 4k − 3 enemies in QS, at most three friends in C, +and prefers ΓC→/0 to Γ, ζS has exactly three friends and at most 4k − 3 enemies in C. Hence, C contains at +most 4k − 3 + 3 + 1 = 4k + 1 players. +So far we know that there are at least 3k − 2 β-players in C. If C contains exactly 3k − 2 (or 3k − 1) +β-players then each of this players has only 3k − 3 (or 3k − 2) β-friends in C and additionally needs at +least three (or two) ζ-friends in C. Hence, we have at least (3k − 2) · 3 = 9k − 6 (or 6k − 2) edges between +the β- and ζ-players in C. Then there are at least 3k − 2 (or 2k) ζ-players in C. Thus there are at least +(3k−2)+(3k−2) = 6k−4 (or 5k−1) players in C which is a contradiction because there are at most 4k+1 +players in C. Hence, there are exactly 3k β-players in C. +Summing up, there are exactly 3k β-players, at least k ζ-players, and at most 4k+1 players in C. Hence, +there are k or k + 1 ζ-players in C. For the sake of contradiction, assume that there are k + 1 ζ-players in C. +Then each ζS ∈ C has 4k − 3 enemies in C. Since ζS prefers ΓC→/0 to Γ, this implies that ζS has exactly +three friends and 4k − 3 enemies in C and the minimal value assigned to ΓC→/0 by ζS’s friends is higher than +the minimal value assigned to Γ by ζS’s friends. In both coalition structures, the minimal value is given by +ζS’s α-friends. However, since these α-players lose ζS as a friend when ζS deviates to C, the minimal value +assigned to Γ is higher than for ΓC→/0. This is a contradiction. Hence, there are exactly k ζ-players in C. +Finally, since every of the 3k βb ∈ C has one of the k ζS ∈ C as a friend, it holds that {S|ζS ∈ C} is an exact +cover for B. This completes the coNP-hardness proof for min-based SF ACFGs. +For sum-based SF ACFGs, coNP-hardness of core stability verification can be shown by a similar con- +struction. Again, given an instance (B,S ) of RX3C, with B = {1,...,3k}, S = {S1,...,S3k}, and k > 8, we +construct the following ACFG. The set of players is N = {βb|b ∈ B}∪{ζS,αS,1,αS,2,αS,3,δS,1,...,δS,4k−3 |S ∈ +S }. We define the sets Beta = {βb|b ∈ B} and QS = {αS,1,αS,2,αS,3,δS,1,...,δS,4k−3} for each S ∈ S . The +network of friends is given in Figure 6, where a dashed rectangle around a group of players means that all +these players are friends of each other: +• All players in Beta are friends of each other. +• For every S ∈ S , all players in QS are friends of each other. +• For every S ∈ S , ζS is friend with αS,1, αS,2, and αS,3 and with every βb with b ∈ S. +Similar arguments as above show that the coalition structure Γ = {Beta} ∪ {{ζS} ∪ QS | S ∈ S } is not +core stable under sum-based SF preferences if and only if S contains an exact cover for B. +13 + +β1 +... +βb +... +β3k +ζS1... +ζSj +b ∈ Sj +... +ζS3k +αS1,1 +αS1,2 +αS1,3 +δS1,1 +... +δS1,4k−3 +αS3k,1 +αS3k,2 +αS3k,3 +δS3k,1 +... +δS3k,4k−3 +... +Beta +Zeta +QS1 +... +QS3k +Figure 6: Network of friends in the proof of Theorem 11 that is used to show coNP-hardness of core stability +verification in sum-based SF ACFGs. A dashed rectangle around a group of players indicates that all these +players are friends of each other. +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Figure 7: Network of friends for Example 12 +4.5 +Popularity and Strict Popularity +Now we take a look at popularity and strict popularity. For all considered models of altruism, there are games +for which no (strictly) popular coalition structure exists. +Example 12. Let N = {1,...,10} and consider the network of friends shown in Figure 7. Then there is no +strictly popular and no popular coalition structure for any of the sum-based or min-based degrees of altruism. +Since perfectness implies popularity, there is also no perfect coalition structure for this ACFG. +Recall from Footnote 2 that there are 115,975 possible coalition structures for this game with ten players, +which we all tested for this example by brute force. +We now show that, under sum-based and min-based SF preferences, it is hard to verify if a given coalition +structure is popular or strictly popular, and it is also hard to decide whether there exists a strictly popular +coalition structure for a given SF ACFG. +Theorem 13. Popularity verification and strict popularity verification are in coNP for any ACFG. For +(sum-based and min-based) SF ACFGs, popularity verification and strict popularity verification are coNP- +complete and strict popularity existence is coNP-hard. +Proof. +First, we observe that the verification problems are in coNP: To verify that a given coalition structure +Γ is not (strictly) popular, we can nondeterministically guess a coalition structure ∆, compare both coalition +structures in polynomial time, and accept exactly if ∆ is more popular than (or at least as popular as) Γ. +To show coNP-hardness of strict popularity verification for min-based SF ACFGs, we again employ a +polynomial-time many-one reduction from RX3C. Let (B,S ) be an instance of RX3C, consisting of a set +B = {1,...,3k} and a collection S = {S1,··· ,S3k} of 3-element subsets of B. Recall that every element of B +occurs in exactly three sets in S and the question is whether there is an exact cover S ′ ⊆ S of B. +We now construct a network of friends based on this instance. The set of players is given by N = +{α1,...,α2k} ∪{βb|b ∈ B} ∪{ζS,ηS,1,ηS,2 |S ∈ S }, so in total we have n = 14k players. For convenience, +we define Alpha = {α1,...,α2k}, Beta = {βb | b ∈ B}, and QS = {ζS,ηS,1,ηS,2 | S ∈ S } for S ∈ S . The +network of friends is shown in Figure 8, where a dashed square around a group of players means that all these +players are friends of each other: All players in Alpha∪Beta are friends of each other; for every S ∈ S , all +players in QS are friends of each other; and ζS is a friend of every βb with b ∈ S. +We consider the coalition structure Γ = {Alpha∪Beta}∪{QS|S ∈ S } and will now show that S contains +an exact cover for B if and only if Γ is not strictly popular under min-based SF preferences. +Only if: +Assuming that there is an exact cover S ′ ⊂ S for B, we define the coalition structure ∆ = +{Alpha ∪ Beta ∪ � +S∈S ′ QS} ∪ {QS | S ∈ S \ S ′}. We will now show that ∆ is as popular as Γ under min- +based SF preferences. +14 + +α1 +... +α2k +β1... +βb... +β3k +ζS1 +ζSj +b ∈ Sj +ζS3k +ηS1,1 +ηS1,2 +... +ηSj,1 +ηSj,2 +... +ηS3k,1 ηS3k,2 +QS1 +QSj +QS3k +Alpha∪Beta +Figure 8: Network of friends in the proof of Theorem 13 that is used to show coNP-hardness of strict pop- +ularity verification in min-based SF ACFGs. A dashed rectangle around a group of players indicates that all +these players are friends of each other. +First, all 2k α-players prefer Γ to ∆, since they only add enemies to their coalition in ∆. Second, the 3k +β-players prefer ∆ to Γ, as each β-player gains a ζ-friend and then has 5k friends instead of 5k − 1. Next, +we consider the QS-groups for S ∈ S ′, i.e., the groups that were added to Alpha ∪ Beta in ∆. We observe +that every ζS-player in these QS-groups prefers ∆ to Γ, since ζS gains three additional β-friends. For the +η-players, on the other hand, the new coalition only contains more enemies, so the η-players prefer Γ to ∆. +Since we have |S ′| = k, this means k ζ-players prefer ∆ to Γ, and 2k η-players prefer Γ to ∆. Finally, we +consider the remaining QS-groups with S ∈ S \ S ′. Here, the coalition containing these players is the same +in Γ and ∆. Hence, for each player p ∈ QS, we have vp(Γ) = vp(∆). Thus the players have to ask their friends +for their valuations. For ζS ∈ QS with S ∈ S \ S ′, the minimum value of her friends is in both structures +given by an η-friend, since ηS,1 and ηS,2 value Γ and ∆ both with n·2, while the β-friends of ζS assign values +n·(5k−1) to Γ and n·5k−(3k−1) to ∆. So we have uminSF +ζS +(Γ) = uminSF +ζS +(∆) and, therefore, 2k ζ-players that +are indifferent. The η-players in QS, S ∈ S \ S ′, are also indifferent, as all their friends value Γ and ∆ the +same. In total, #∆≻Γ = |Beta ∪ {ζS | S ∈ S ′}| = 4k = |Alpha ∪ {ηS,1,ηS,2 |S ∈ S ′}| = #Γ≻∆ and, therefore, +∆ is exactly as popular as Γ, so Γ is not strictly popular. +If: +Assuming that Γ is not strictly popular, there is some coalition structure ∆ ∈ CN with ∆ ̸= Γ such +that ∆ is at least as popular as Γ under min-based SF preferences. We will now show that this implies the +existence of an exact cover for B in S . +First of all, we observe that all α-players’ most preferred coalition is Alpha∪Beta, as it contains all their +friends and no enemies. Thus we have Γ ≻minSF +α +∆ if Alpha∪Beta /∈ ∆ and Γ ∼minSF +α +∆ if Alpha∪Beta ∈ ∆. +For the sake of contradiction, we assume that Alpha∪Beta ∈ ∆. As ∆ ̸= Γ, the players in the QS-groups +have to be partitioned differently. However, that would not increase any player’s valuation since every player +in QS can only lose friends and gain enemies. That means that no β-player prefers ∆ to Γ, as they are in the +same coalition as in Γ and their friends are not more satisfied. We also have at least three players of a QS- +group that are no longer in the same coalition, so they prefer Γ to ∆. This is a contradiction, as we assumed +that ∆ is at least as popular as Γ. Thus we have Alpha∪Beta /∈ ∆. +Now consider the η-players. For every S ∈ S , we know that QS is the best valued coalition for ηS,1 +and ηS,2. So again, ηS,1 and ηS,2 prefer Γ to ∆ if and only if QS /∈ ∆, and they are indifferent otherwise. +Define k′ = |{S ∈ S | QS /∈ ∆}|. So 2k′ is the number of η-players that prefer Γ to ∆, and the remaining +6k − 2k′ η-players are indifferent between Γ and ∆. We first collect some observations: +1. All 2k α-players prefer Γ to ∆. +2. 2k′ η-players prefer Γ to ∆, and 6k − 2k′ η-players are indifferent. +3. 3k−k′ ζ-players are in the same coalition in both coalition structures, so their utilities depend on their +friends’ valuations. In Γ, the minimum value of their friends is given by an η-player. Since this η- +player is also in the same coalition in ∆ and thus assigns the same value, it is not possible that the +minimum value of the friends is higher in ∆ than in Γ. So 3k − k′ ζ-players are indifferent or prefer Γ +to ∆. +4. We have 14k players in total, so we can have at most 14k − 2k − 2k′ − (6k − 2k′)− (3k − k′) = 3k + k′ +players that prefer ∆ to Γ. +15 + +α1 +... +α5k +β1... +βb... +β3k +ζS1 +ζSj +b ∈ Sj +ζS3k +ηS1 +... +ηSj +... +ηS3k +QS1 +QSj +QS3k +Alpha∪Beta +Figure 9: Network of friends in the proof of Theorem 13 that is used to show coNP-hardness of strict popu- +larity verification in sum-based SF ACFGs. A dashed rectangle around a group of players indicates that all +these players are friends of each other. +Next, we show that k′ = k. First, assume that k′ > k: We have #Γ≻∆ ≥ 2k +2k′, and since k′ > k, we have +2k + 2k′ > 3k + k′ ≥ #∆≻Γ. This is a contradiction to #Γ≻∆ ≤ #∆≻Γ, so we obtain k′ ≤ k. +Second, let us assume k′ < k: Since every ζ-player has three β-friends and there are k′ ζ-players that are +not in their respective QS coalition in ∆, there are at most 3k′ β-players that gain a ζ-friend in ∆. The 3k−3k′ +other β-players have at most 5k−1 friends in ∆, namely all other α- and β-players. But as Alpha∪Beta /∈ ∆, +they would also gain at least one enemy, so we have 3k − 3k′ β-players that prefer Γ. That means we +have #Γ≻∆ ≥ 2k + 2k′ + 3k − 3k′ = 5k − k′ and #∆≻Γ ≤ 3k + k′ − (3k − 3k′) = 4k′. Since k′ < k, we have +5k − k′ > 5k − k = 4k > 4k′, and therefore, #Γ≻∆ > #∆≻Γ, which again is a contradiction. Thus we conclude +that k′ ≥ k and, in total, k′ = k. +Consequently, we know that 4k players prefer Γ to ∆, namely all α-players and the 2k η-players that are +not in QS anymore. Subtracting all the indifferent players, we observe that all other players have to prefer ∆ +to Γ in order to ensure #Γ≻∆ ≤ #∆≻Γ. These other players are the 3k β-players and the k ζ-players that are +not in QS anymore. Finally, that is only possible if every β-player gains a ζ-friend in ∆. Hence each one of +those k ζ-players has to be friends with three different β-players. Therefore, the set {S ∈ S |QS /∈ ∆} is an +exact cover for B. +To show coNP-hardness of strict popularity verification for sum-based SF ACFGs, we use a similar con- +struction. For an instance (B,S ) of RX3C with B = {1,...,3k} and S = {S1,...,S3k}, where each element +of B occurs in exactly three sets in S , we construct the following ACFG. The set of players is given by +N = {α1,...,α5k} ∪ {βb| b ∈ B} ∪ {ζS,ηS | S ∈ S }. Let Alpha = {α1,...,α5k}, Beta = {βb| b ∈ B}, and +QS = {ζS,ηS} for each S ∈ S . The network of friends is given in Figure 9, where a dashed rectangle around +a group of players means that all these players are friends of each other: All players in Alpha ∪ Beta are +friends of each other and, for every S ∈ S , ζS is friends with ηS and every βb with b ∈ S. +Consider the coalition structure Γ = {Alpha ∪ Beta,QS1,...,QS3k}. We show that S contains an exact +cover for B if and only if Γ is not strictly popular. +Only if: +Assuming that there is an exact cover S ′ ⊆ S for B and considering coalition structure ∆ = +{Alpha ∪ Beta ∪ � +S∈S ′ QS} ∪ {QS | S ∈ S \ S ′}, it can be shown with similar arguments as before that +#∆≻Γ = |{β1,...,β3k,ζS1,...,ζS3k}| = 6k = |{α1,...,α5k} ∪ {ηS | S ∈ S ′}| = #Γ≻∆. Hence, ∆ and Γ are +equally popular. +If: +Assuming that Γ is not strictly popular, i.e., that there is a coalition structure ∆ ∈ CN, ∆ ̸= Γ, with +#Γ≻∆ ≤ #∆≻Γ, it can be shown similarly as before that the set {S ∈ S |QS /∈ ∆} is an exact cover for B. +The results for strict popularity existence and popularity verification can be shown by slightly modifying +the above reductions. +To show that strict popularity existence is coNP-hard for min-based and sum-based SF ACFGs, we con- +sider the same two reductions as before but the coalition structures Γ are not given as a part of the problem +instances. Then, there is an exact cover for B if and only if there is no strictly popular coalition structure. +In particular, if there is an exact cover for B, Γ and ∆ as defined in the proofs above are in a tie and every +other coalition structure is beaten by Γ. And if there is no exact cover for B then Γ beats every other coalition +structure and thus is strictly popular. +Popularity verification for min-based and sum-based SF ACFGs can be shown to be coNP-complete by +using the same constructions as for strict popularity verification (see Figure 8 and 9) but reducing the numbers +16 + +of α-players to 2k − 1 and 5k − 1, respectively. Then there is an exact cover for B if and only if Γ, as defined +above, is not popular. +4.6 +Perfectness +Turning now to perfectness, we start with the SF model. +Theorem 14. For any sum-based or min-based SF ACFG (N,⪰) with an underlying network of friends G, +a coalition structure Γ ∈ CN is perfect if and only if it consists of the connected components of G and all of +them are cliques. +Proof. +From left to right, assume that the coalition structure Γ ∈ CN is perfect. It then holds for all agents +i ∈ N and all coalition structures ∆ ∈ CN, ∆ ̸= Γ, that i weakly prefers Γ to ∆. It follows that vi(Γ) ≥ vi(∆) +for all ∆ ∈ CN, ∆ ̸= Γ, and i ∈ N. Hence, every agent i ∈ N has the maximal valuation vi(Γ) = n ·|Fi| and is +together with all of her friends and none of her enemies. This implies that each coalition in Γ is a connected +component and a clique. +The implication from right to left is obvious. +Since it is easy to check this characterization, perfect coalition structures can be verified in polynomial +time for sum-based and min-based SF ACFGs. It follows directly from Theorem 14 that the corresponding +existence problem is also in P. +Corollary 15. For any sum-based or min-based SF ACFG (N,⪰) with an underlying network of friends G, +there exists a perfect coalition structure if and only if all connected components of G are cliques. +We further get the following upper bounds. +Proposition 16. For any ACFG, perfectness verification is in coNP. +Proof. +Consider any ACFG (N,⪰). A coalition structure Γ ∈ CN is not perfect if and only if there is an +agent i ∈ N and a coalition structure ∆ ∈ CN such that ∆ ≻i Γ. Hence, we can nondeterministically guess an +agent i ∈ N and a coalition structure ∆ ∈ CN and verify in polynomial time whether ∆ ≻i Γ. +Furthermore, we initiate the characterization of perfectness in ACFGs. The diameter of a connected graph +component is the greatest distance between any two of its vertices. For sum-based EQ ACFGs, we get the +following implication. +Proposition 17. For any sum-based EQ ACFG with an underlying network of friends G, it holds that if +a coalition structure Γ is perfect for it, then Γ consists of the connected components of G and all these +components have a diameter of at most two. +Proof. +We first show that, in a perfect coalition structure, all agents have to be together with all their +friends. For the sake of contradiction, assume that Γ is perfect but there are i, j ∈ N with j ∈ Fi and j /∈ Γ(i). +We distinguish two cases. +Case 1: All f ∈ Fi ∩ Γ(i) have a friend in Γ(j). Consider the coalition structure ∆ that results from the +union of Γ(i) and Γ(j), i.e., ∆ = Γ \ {Γ(i),Γ(j)} ∪ {Γ(i) ∪ Γ(j)}. It holds that i and all friends of i’s either +gain an additional friend in ∆ or their coalition stays the same: First, i keeps all friends from Γ(i) and gets j as +an additional friend. Hence, i has at least one friend more in ∆ than in Γ and we have vi(∆) > vi(Γ). Second, +all friends f ∈ Fi ∩ Γ(i) have a friend in Γ(j) and therefore also gain at least one additional friend from the +union of the two coalitions. Hence, vf (∆) > vf (Γ) for all f ∈ Fi ∩Γ(i). Third, all friends f ∈ Fi ∩Γ(j) have i +as friend. Hence, they also gain one friend from the union. Thus vf (∆) > vf (Γ) for all f ∈ Fi ∩Γ(j). Finally, +all f ∈ Fi who are not in Γ(i) or Γ(j) value Γ and ∆ the same because their coalition is the same in both +coalition structures. Hence, vf (∆) = vf (Γ) for all f ∈ Fi with f /∈ Γ(j) and f /∈ Γ(i). Summing up, we have +usumEQ +i +(∆) > usumEQ +i +(Γ), so i prefers ∆ to Γ, which is a contradiction to Γ being perfect. +Case 2: There is an f ∈ Fi ∩ Γ(i) who has no friends in Γ(j). Consider the coalition structure ∆ that +results from j moving to Γ(i), i.e., ∆ = Γj→Γ(i). Let k ∈ Fi ∩Γ(i) be one of the agents who have no friends in +17 + +1 +2 +3 +4 +5 +9 +7 +8 +6 +Figure 10: Network of friends for Example 19 +Γ(j). Then vk(∆) = vk(Γ)−1; vi(∆) = vi(Γ)+n; for all f ∈ Fk ∩Γ(i), f ̸= i, we have vf (∆) ≥ vf (Γ)−1; and +for all f ∈ Fk, f /∈ Γ(i) (and f /∈ Γ(j)), we have vf (∆) = vf (Γ). Hence, +usumEQ +k +(∆) = +∑ +a∈Fk∪{k} +va(∆) = +∑ +a∈Fk∩Γ(i),a̸=i +va(∆)+ +∑ +a∈Fk\Γ(i) +va(∆)+ vk(∆)+ vi(∆) +≥ +∑ +a∈Fk∩Γ(i),a̸=i +va(Γ)− 1 + +∑ +a∈Fk\Γ(i) +va(Γ)+ vk(Γ)− 1 + vi(Γ)+ n += +∑ +a∈Fk∪{k} +va(Γ)− (|Fk ∩Γ(i)|− 1)− 1 + n += usumEQ +k +(Γ)− |Fk ∩Γ(i)| +� +�� +� + usumEQ +k +(Γ). +Therefore, k prefers ∆ to Γ, which again is a contradiction to Γ being perfect. +Next, assume that Γ is perfect but there is a coalition C in Γ that has a diameter greater than two. Then +there are agents i, j ∈ C with a distance greater than two. Thus j is an enemy of i’s and an enemy of all of i’s +friends. It follows that i prefers coalition structure Γj→/0 to Γ, which is a contradiction to Γ being perfect. +Summing up, in a perfect coalition structure Γ for a sum-based EQ ACFG every agent is together with all +her friends and every coalition in Γ has a diameter of at most two. Together this implies that Γ consists of the +connected components of G and all these components have a diameter of at most two. +From Propositions 16 and 17, we get the following corollary. +Corollary 18. For sum-based EQ ACFGs, perfectness existence is in coNP. +However, Proposition 17 is not an equivalence. The converse does not hold, as the following example +shows. +Example 19. Consider the sum-based EQ ACFG (N,⪰sumEQ) with the network of friends G in Figure 10. +The coalition structure Γ = {N} consists of the only connected component of G, which has a diameter of two. +However, agent 1 prefers ∆ = {{1,...,6},{7,8,9}} to Γ because +usumEQ +1 +(Γ) = v1(Γ)+ ···+ v5(Γ)+ v9(Γ) = (9 ·5 − 3)+ 4 ·(9·2−6)+(9·3−5)= 112 +< 113 = (9 ·4 − 1)+ 4 ·(9·2−3)+(9·2−0) = v1(∆)+ ···+ v5(∆)+ v9(∆) += usumEQ +1 +(∆). +Hence, Γ is not perfect. +5 +Conclusions and Open Problems +We have proposed to extend the models of altruistic hedonic games due to Nguyen et al. [1] and Wiechers and +Rothe [5] to coalition formation games in general. We have compared our more general models to altruism +in hedonic games and have motivated our work by removing some crucial disadvantages that come with the +restriction to hedonic games. In particular, we have shown that all degrees of our general altruistic preferences +are unanimous while this is not the case for all altruistic hedonic preferences. Furthermore, all our sum-based +degrees of altruism fulfill two types of monotonicity that are violated by the corresponding hedonic equal- +and altruistic-treatment preferences. +We have furthermore studied the common stability notions and have initiated a computational analysis +of the associated verification and existence problems (see Table 2 for an overview of our results). We also +18 + +gave characterizations for some of the stability notions, using graph-theoretical properties of the underlying +network of friends. For future work, we propose to complete this analysis, close all gaps between complexity- +theoretic upper and lower bounds, and get a full characterization for all stability notions. +Acknowledgments +We thank the anonymous IJCAI’20 reviewers for helpful comments. This work was supported in part by DFG +grants RO 1202/14-2 and RO 1202/21-1. The first and third author have been supported in part by the research +project “Online Participation” within the North Rhine-Westphalian funding scheme “Forschungskollegs.” +References +[1] Nguyen, N., Rey, A., Rey, L., Rothe, J., Schend, L.: Altruistic hedonic games. In: Proceedings of the +15th International Conference on Autonomous Agents and Multiagent Systems, pp. 251–259. IFAA- +MAS, Singapore (2016) +[2] Dimitrov, D., Borm, P., Hendrickx, R., Sung, S.-C.: Simple priorities and core stability in hedonic +games. 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Theoretical Computer Science +38, 293–306 (1985) +20 + diff --git a/r9E5T4oBgHgl3EQflw-X/content/tmp_files/load_file.txt b/r9E5T4oBgHgl3EQflw-X/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..35458295433069b947cc00c1d16913ea7b9ae930 --- /dev/null +++ b/r9E5T4oBgHgl3EQflw-X/content/tmp_files/load_file.txt @@ -0,0 +1,979 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf,len=978 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='05674v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='GT] 13 Jan 2023 Altruism in Coalition Formation Games Anna Maria Kerkmann, Simon Cramer, and J¨org Rothe Heinrich-Heine-Universit¨at D¨usseldorf, Germany January 16, 2023 Abstract Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [1] introduced altruistic hedonic games in which agents’ utilities depend not only on their own preferences but also on those of their friends in the same coalition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We propose to extend their model to coalition formation games in general, considering also the friends in other coalitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Comparing our model to altruistic hedonic games, we argue that excluding some friends from the altruistic behavior of an agent is a major disadvantage that comes with the restriction to hedonic games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' After introducing our model and showing some desirable properties, we additionally study some common stability notions and provide a computational analysis of the associated verification and existence problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 1 Introduction We consider coalition formation games where agents have to form coalitions based on their preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Among other compact representations of hedonic coalition formation games, Dimitrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [2] in particular proposed the friends-and-enemies encoding with friend-oriented preferences which involves a network of friends: a (simple) undirected graph whose vertices are the players and where two players are connected by an edge exactly if they are friends of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Players not connected by an edge consider each other as enemies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Under friend-oriented preferences, player i prefers a coalition C to a coalition D if C contains more of i’s friends than D, or C and D have the same number of i’s friends but C contains fewer enemies of i’s than D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This is a special case of the additive encoding [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For more background on these two compact representations, see Section 2 and the book chapter by Aziz and Savani [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Based on friend-oriented preferences, Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [1] introduced altruistic hedonic games where agents gain utility not only from their own satisfaction but also from their friends’ satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' However, Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [1] specifically considered hedonic games only, which require that an agent’s utility only depends on her own coalition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In their interpretation of altruism, the utility of an agent is composed of the agent’s own valuation of her coalition and the valuation of all this agent’s friends in this coalition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' While Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [1] used the average when aggregating some agents’ valuations, Wiechers and Rothe [5] proposed a variant of altruistic hedonic games where some agents’ valuations are aggregated by taking the minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Inspired by the idea of altruism, we extend the model of altruism in hedonic games to coalition formation games in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' That is, we propose a model where agents behave altruistically to all their friends, not only to the friends in the same coalition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Not restricting to hedonic games, we aim to capture a more natural notion of altruism where none of an agent’s friends is excluded from her altruistic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' To become acquainted with this idea of altruism, consider the coalition formation game that is represented by the network of friends in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For the coalition structures Γ = {{1,2,3},{4}} and ∆ = {{1,2,4},{3}}, it is clear that player 1 is indifferent between coalitions {1,2,3} and {1,2,4} under friend-oriented preferences, as both coalitions contain 1’s only friend (player 2) and one of 1’s enemies (either 3 or 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Under altruistic hedonic preferences [1], however, player 1 behaves altruistically to her friend 2 (who is friends with 3 but not with 4) and therefore prefers {1,2,3} to {1,2,4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Now, consider the slightly modified coalition structures Γ′ = {{1},{2,3},{4}} and ∆′ = {{1},{2,4},{3}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Intuitively, one would still expect 1 to behave altruistically to her friend 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' However, under any hedonic preference (which requires the 1 1 2 3 4 Figure 1: Network of friends for Example 1 players’ preferences to depend only on their own coalitions), player 1 (being in the same coalition for both Γ′ and ∆′) must be indifferent between Γ′ and ∆′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In order to model altruism globally, we release the restriction to hedonic games and introduce altruistic coalition formation games where agents behave altruistically to all their friends, independently of their current coalition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='1 Related Work Coalition formation games, as considered here, are closely related to the subclass of hedonic games which has been broadly studied in the literature, addressing the issue of compactly representing preferences, conducting axiomatic analyses, dealing with different notions of stability, and investigating the computational complexity of the associated problems (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=', the book chapter by Aziz and Savani [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Closest related to our work are the altruistic hedonic games by Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [1] (see also the related minimization-based variant by Wiechers and Rothe [5]), which we modify to obtain our more general models of altruism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Based on the model due to Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [1], Schlueter and Goldsmith [6] defined super altruistic hedonic games where friends have a different impact on an agent based on their distances in the underlying network of friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' More recently, Bullinger and Kober [7] introduced loyalty in cardinal hedonic games where agents are loyal to all agents in their so-called loyalty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In their model, the utilities of the agents in the loyalty set are aggregated by taking the minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' They then study the loyal variants of common classes of cardinal hedonic games such as additively separable and friend-oriented hedonic games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='1 Altruism has also been studied for noncooperative games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Most prominently, Ashlagi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [8] introduced social context games where a social context is applied to a strategic game and the costs in the resulting game depend on the original costs and a graph of neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Their so-called MinMax collaborations (where players seek to minimize the maximal cost of their own and their neighbors) are related to our minimization- based equal-treatment model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Still, the model of Ashlagi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [8] differs from ours in that they consider noncooperative games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Other work considering noncooperative games with social networks is due to Bil`o et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [9] who study social context games for other underlying strategic games than Ashlagi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [8], Hoefer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [10] who study considerate equilibria in strategic games, and Anagnostopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [11] who study altruism and spite in strategic games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Further work studying altruism in noncooperative games without social networks is due to Hoefer and Skopalik [12], Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [13], Apt and Sch¨afer [14], and Rahn and Sch¨afer [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='2 Our Contribution Conceptually, we extend the models of altruism proposed by Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [1] and Wiechers and Rothe [5] from hedonic games to general coalition formation games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We argue how this captures a more global notion of altruism and show that our models fulfill some desirable properties that are violated by the previous mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We then study the common stability concepts in this model and analyze the associated verification and existence problems in terms of their computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This work extends a preliminary version that appeared in the proceedings of the 29th International Joint Conference on Artificial Intelligence (IJCAI’20) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Parts of this work were also presented at the 16th and 17th International Symposium on Artificial Intelligence and Mathematics (ISAIM’20 and ISAIM’22) and at the 8th International Workshop on Computational Social Choice (COMSOC’21), each with nonarchival proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 1Note that their loyal variant of symmetric friend-oriented hedonic games is equivalent to the minimization-based altruistic hedonic games under equal treatment introduced by Wiechers and Rothe [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 2 2 The Model In coalition formation games, players divide into groups based on their preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Before introducing altruism, we now give some foundations of such games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='1 Coalition Formation Games Let N = {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',n} be a set of agents (or players).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Each subset of N is called a coalition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' A coalition structure Γ is a partition of N, and we denote the set of all possible coalition structures for N by CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For a player i ∈ N and a coalition structure Γ ∈ CN, Γ(i) denotes the unique coalition in Γ containing i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Now, a coalition formation game (CFG) is a pair (N,⪰), where N = {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',n} is a set of agents, ⪰ = (⪰1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',⪰n) is a profile of preferences, and every preference ⪰i ∈ CN × CN is a complete weak order over all possible coalition structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Given two coalition structures Γ, ∆ ∈ CN, we say that i weakly prefers Γ to ∆ if Γ ⪰i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' When Γ ⪰i ∆ but not ∆ ⪰i Γ, we say that i prefers Γ to ∆ (denoted by Γ ≻i ∆), and we say that i is indifferent between Γ and ∆ (denoted by Γ ∼i ∆) if Γ ⪰i ∆ and ∆ ⪰i Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Note that hedonic games are a special case of coalition formation games where the agents’ preference relations only depend on the coalitions containing themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In a hedonic game (N,⪰), agent i ∈ N is indifferent between any two coalition structures Γ and ∆ as long as her coalition is the same, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=', Γ(i) = ∆(i) =⇒ Γ ∼i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Therefore, the preference order of any agent i ∈ N in a hedonic game (N,⪰) is usually represented by a complete weak order over the set of coalitions containing i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='2 The “Friends and Enemies” Encoding Since |CN|, the number of all possible coalition structures, is extremely large in the number of agents,2 it is not reasonable to ask every agent for her complete preference over CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Instead, we are looking for a way to compactly represent the agents’ preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In the literature, many such representations have been proposed for hedonic games, such as the additive encoding [19, 3, 20], the singleton encoding due to Cechl´arov´a and Romero-Medina [21] and further studied by Cechl´arov´a and Hajdukov´a [22], the friends-and-enemies encoding due to Dimitrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [2], and FEN-hedonic games due to Kerkmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [23] and also used by Rothe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Here, we use the friends-and-enemies encoding due to Dimitrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We focus on their friend-oriented model and will later adapt it to our altruistic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In the friend-oriented model, the preferences of the agents in N are given by a network of friends, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=', a (simple) undirected graph G = (N,A) whose vertices are the players and where two players i, j ∈ N are connected by an edge {i, j} ∈ A exactly if they are each other’s friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Agents not connected by an edge consider each other as enemies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For an agent i ∈ N, we denote the set of i’s friends by Fi = { j ∈ N |{i, j} ∈ A} and the set of i’s enemies by Ei = N \\ (Fi ∪ {i}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Under friend-oriented preferences as defined by Dimitrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [2], between any two coalitions players prefer the coalition with more friends, and if there are equally many friends in both coalitions, they prefer the coalition with fewer enemies: C ⪰F i D ⇐⇒ |C ∩Fi| > |D∩Fi| or (|C ∩Fi| = |D∩Fi| and |C ∩Ei| ≤ |D∩Ei|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This can also be represented additively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Assigning a value of n to each friend and a value of −1 to each enemy, agent i ∈ N values coalition C containing herself with vi(C) = n|C ∩ Fi| − |C ∩ Ei|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Note that −(n − 1) ≤ vi(C) ≤ n(n − 1), and vi(C) > 0 if and only if there is at least one friend of i’s in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For a given coalition structure Γ ∈ CN, we also write vi(Γ) for player i’s value of Γ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Furthermore, we denote the sum of the values of i’s friends by sumF i (Γ) = ∑f∈Fi vf (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Analogously, we also define sumF+ i (Γ) = ∑ f∈Fi∪{i} vf (Γ), minF i (Γ) = minf∈Fi vf (Γ), and minF+ i (Γ) = minf∈Fi∪{i}vf (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='3 Three Degrees of Altruism When we now define altruistic coalition formation games based on the friend-oriented preference model, we consider the same three degrees of altruism that Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [1] introduced for altruistic hedonic games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 2The number of possible partitions of a set with n elements equals the n-th Bell number [17, 18], defined as Bn = ∑n−1 k=0 �n−1 k � Bk with B0 = B1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For example, for n = 10 agents, we have B10 = 115,975 possible coalition structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 3 5 1 2 3 4 6 7 8 9 10 Figure 2: Network of friends for Example 2 However, we adapt them to our model, extending the agents’ altruism to all their friends, not only to their friends in the same coalition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Selfish First (SF): Agents first rank coalition structures based on their own valuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Only in the case of a tie between two coalition structures, their friends’ valuations are considered as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Equal Treatment (EQ): Agents treat themselves and their friends the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' That means that an agent i ∈ N and all of i’s friends have the same impact on i’s utility for a coalition structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Altruistic Treatment (AL): Agents first rank coalition structures based on their friends’ valuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' They only consider their own valuations in the case of a tie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We further distinguish between a sum-based and a min-based aggregation of some agents’ valuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For- mally, for an agent i ∈ N and a coalition structure Γ ∈ CN, we denote i’s sum-based utility for Γ under SF by usumSF i (Γ), under EQ by usumEQ i (Γ), and under AL by usumAL i (Γ), and her min-based utility for Γ under SF by uminSF i (Γ), under EQ by uminEQ i (Γ), and under AL by uminAL i (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For a constant M ≥ n3, they are defined as usumSF i (Γ) = M ·vi(Γ)+ sumF i (Γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' uminSF i (Γ) = M ·vi(Γ)+ minF i (Γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' usumEQ i (Γ) = sumF+ i (Γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' uminEQ i (Γ) = minF+ i (Γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' usumAL i (Γ) = vi(Γ)+ M ·sumF i (Γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' uminAL i (Γ) = vi(Γ)+ M ·minF i (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In the case of Fi = /0, we define the minimum of the empty set to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For any coalition structures Γ,∆ ∈ CN, agent i’s sum-based SF preference is then defined by Γ ⪰sumSF i ∆ ⇐⇒ usumSF i (Γ) ≥ usumSF i (∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Her other altruistic preferences (⪰sumEQ i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' ⪰sumAL i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' ⪰minSF i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' ⪰minEQ i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' and ⪰minAL i ) are defined analogously, using the respective utility functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The factor M, which is used for the SF and AL models, ensures that an agent’s utility is first determined by the agent’s own valuation in the SF model and first determined by the friends’ valuations in the AL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Similarly as Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [1] prove the corresponding properties in hedonic games, we can show that for M ≥ n3, vi(Γ) > vi(∆) implies Γ ≻sumSF i ∆ and Γ ≻minSF i ∆, and for M ≥ n2, sumF i (Γ) > sumF i (∆) implies Γ ≻sumAL i ∆ while minF i (Γ) > minF i (∆) implies Γ ≻minAL i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' An altruistic coalition formation game (ACFG) is a coalition formation game where the agents’ preferences were obtained by a network of friends via one of these cases of altruism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, we distinguish between sum-based SF, sum-based EQ, sum-based AL, min-based SF, min-based EQ, and min-based AL ACFGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For any ACFG, the players’ utilities can obviously be computed in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 3 Monotonicity and Other Properties in ACFGs Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [1] focus on altruism in hedonic games where an agent’s utility only depends on her own coalition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' As we have already seen in Example 1, there are some aspects of altruistic behavior that cannot be realized by hedonic games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The following example shows that our model crucially differs from the models due to Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [1] and Wiechers and Rothe [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Consider an ACFG (N,⪰) with the network of friends in Figure 2 and the coalition structures Γ = {{1,2},{3},{4},.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',{10}} and ∆ = {{1,5,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',10},{2,3,4}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We will now compare agent 1’s prefer- ences for these two coalition structures under our altruistic models to 1’s preferences under the altruistic hedonic models [1, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Table 1 shows all relevant values that are needed to compute the utilities of agent 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' One can observe that agent 1 and all her friends assign a greater value to ∆ than to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Consequently, also the aggregations of the friends’ values (sumF 1 , sumF+ 1 , minF 1 , minF+ 1 ) are greater for ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, 1 prefers ∆ to Γ under all our sum-based and min-based altruistic preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 4 Table 1: Values for the game in Example 2 with the network of friends in Figure 2 v1 v2 v5 v6 sumF 1 sumF+ 1 minF 1 minF+ 1 Γ 10 10 0 0 10 20 0 0 ∆ 16 20 5 5 30 46 5 5 The hedonic models due to Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [1] and Wiechers and Rothe [5], however, are blind to the fact that agent 1 and all her friends are better off in ∆ than in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Under their altruistic hedonic preferences, player 1 compares the two coalition structures Γ and ∆ only based on her own coalitions Γ(1) = {1,2} and ∆(1) = {1,5,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' She then only considers her friends that are in the same coalition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=', player 2 for Γ and players 5 and 6 for ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This leads to 1 preferring Γ(1) to ∆(1) under altruistic hedonic EQ and AL preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In particular, the average (and minimum) valuation of 1’s friends in Γ(1) is 10 while the average (and minimum) valuation of 1’s friends in ∆(1) is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Also considering 1’s own value for EQ, the average (and minimum) in Γ(1) is 10 while the average (respectively, minimum) value in ∆(1) is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='6 (respectively, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='1 Some Basic Properties As we have seen in Example 2, altruistic hedonic games [1, 5] allow for players that prefer coalition structures that make themselves and all their friends worse off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' To avoid this kind of unreasonable behavior, we focus on general coalition formation games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In fact, all our altruistic coalition-formation preferences fulfill unanimity: For an ACFG (N,⪰) and a player i ∈ N, we say that ⪰i is unanimous if, for any two coalition structures Γ,∆ ∈ CN, va(Γ) > va(∆) for each a ∈ Fi ∪{i} implies Γ ≻i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This property crucially distinguishes our preference models from the corresponding altruistic hedonic preferences, which are not unanimous under EQ or AL preferences, as Example 2 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Note that Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [1] define a restricted version of unanimity in altruistic hedonic games by considering only the agents’ own coalitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Other desirable properties that were studied by Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [1] for altruistic hedonic preferences can be generalized to coalition formation games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We show that these desirable properties also hold for our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' First, we collect some basic observations: Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Consider any ACFG (N,⪰) with an underlying network of friends G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' All preferences ⪰i, i ∈ N, are reflexive and transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For any player i ∈ N and any two coalition structures Γ,∆ ∈ CN, it can be decided in polynomial time (in the number of agents) whether Γ ⪰i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The preferences ⪰i, i ∈ N, only depend on the structure of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Note that the third statement of Observation 3 implies that the properties that Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [1] call anonymity and symmetry are both satisfied in ACFGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Another desirable property they consider is called sovereignty of players and inspired by the axiom of “citizens’ sovereignty” from social choice theory:3 Given a set of agents N, a coalition structure Γ ∈ CN, and an agent i ∈ N, we say that sovereignty of players is satisfied if there is a network of friends G on N such that Γ is i’s most preferred coalition structure in any ACFG induced by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' ACFGs satisfy sovereignty of players under all sum-based and min-based SF, EQ, and AL altruistic preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Sovereignty of players in ACFGs can be shown with an analogous construction as in the proof of Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [1, Theorem 5]: For a given set of players N, player i ∈ N, and coalition structure Γ ∈ CN, we construct a network of friends where all players in Γ(i) are friends of each other while there are no other friendship relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Then Γ is i’s (nonunique) most preferred coalition structure under all sum-based and min-based SF, EQ, and AL altruistic preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 3Informally stated, a voting rule satisfies citizens’ sovereignty if every candidate can be made a winner of an election for a suitably chosen preference profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='2 Monotonicity The next property describes the monotonicity of preferences and further distinguishes our models from altru- istic hedonic games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In fact, Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [1] define two types of monotonicity, which we here adapt to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Consider any ACFG (N,⪰), agents i, j ∈ N with j ∈ Ei, and coalition structures Γ,∆ ∈ CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Let further ⪰′ i be the preference relation resulting from ⪰i when j turns from being i’s enemy to being i’s friend (all else remaining equal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We say that ⪰i is type-I-monotonic if (1) Γ ≻i ∆, j ∈ Γ(i) ∩ ∆(i), and vj(Γ) ≥ vj(∆) implies Γ ≻′ i ∆, and (2) Γ ∼i ∆, j ∈ Γ(i)∩∆(i), and vj(Γ) ≥ vj(∆) implies Γ ⪰′ i ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' type-II-monotonic if (1) Γ ≻i ∆, j ∈ Γ(i) \\ ∆(i), and vj(Γ) ≥ vj(∆) implies Γ ≻′ i ∆, and (2) Γ ∼i ∆, j ∈ Γ(i)\\ ∆(i), and vj(Γ) ≥ vj(∆) implies Γ ⪰′ i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Let (N,⪰) be an ACFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' If (N,⪰) is sum-based, its preferences satisfy type-I- and type-II-monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' If (N,⪰) is min-based, its preferences satisfy type-II- but not type-I-monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Let (N,⪰) be an ACFG with an underlying network of friends G = (N,H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Consider i ∈ N, Γ,∆ ∈ CN, and j ∈ Ei and denote with G′ = (N,H ∪{{i, j}}) the network of friends resulting from G when j turns from being i’s enemy to being i’s friend (all else being equal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Let (N,⪰′) be the ACFG induced by G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For any agent a ∈ N and coalition structure Γ ∈ CN, denote a’s value for Γ in G′ by v′ a(Γ), a’s preference relation in (N,⪰′) by ⪰′ a, and a’s friends and enemies in (N,⪰′) by F′ a and E′ a, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' That is, we have F′ i = Fi ∪{ j}, E′ i = Ei \\ { j}, F′ j = Fj ∪{i}, and E′ j = E j \\ {i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Further, v′ i, v′ j, and ⪰′ i might differ from vi, vj, and ⪰i, while the friends, enemies, and values of all other players stay the same, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=', F′ a = Fa, E′ a = Ea, and v′ a = va for all a ∈ N \\ {i, j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Type-I-monotonicity under sum-based preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Let j ∈ Γ(i)∩∆(i) and vj(Γ) ≥ vj(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' It then holds that v′ i(Γ) = n|Γ(i)∩F′ i |− |Γ(i)∩E′ i| = n|Γ(i)∩Fi|+ n − |Γ(i)∩Ei|+ 1 = vi(Γ)+ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Equivalently, v′ i(∆) = vi(∆)+ n + 1, v′ j(Γ) = vj(Γ)+ n + 1, and v′ j(∆) = vj(∆)+ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Furthermore, sumF′ i (Γ) = ∑ a∈F′ i v′ a(Γ) = ∑ a∈Fi∪{ j} v′ a(Γ) = ∑ a∈Fi va(Γ)+ v′ j(Γ) = sumF i (Γ)+ vj(Γ)+ n + 1 and (1) sumF′ i (∆) = sumF i (∆)+ vj(∆)+ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' (2) (1) sumSF: If Γ ≻sumSF i ∆ then either (i) vi(Γ) = vi(∆) and sumF i (Γ) > sumF i (∆), or (ii) vi(Γ) > vi(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In case (i), vi(Γ) = vi(∆) implies v′ i(Γ) = v′ i(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Applying sumF i (Γ) > sumF i (∆) and vj(Γ) ≥ vj(∆) to (1) and (2), we get sumF′ i (Γ) > sumF′ i (∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This together with v′ i(Γ) = v′ i(∆) implies Γ ≻sumSF′ i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In case (ii), vi(Γ) > vi(∆) implies v′ i(Γ) > v′ i(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, Γ ≻sumSF′ i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' If Γ ∼sumSF i ∆ then vi(Γ) = vi(∆) and sumF i (Γ) = sumF i (∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' vi(Γ) = vi(∆) implies v′ i(Γ) = v′ i(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Applying sumF i (Γ) = sumF i (∆) and vj(Γ) ≥ vj(∆) to (1) and (2), we get sumF′ i (Γ) ≥ sumF′ i (∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This together with v′ i(Γ) = v′ i(∆) implies Γ ⪰sumSF′ i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' (2) sumEQ: If Γ ≻sumEQ i ∆ then sumF i (Γ)+vi(Γ) > sumF i (∆)+vi(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Using (1), (2), v′ i(Γ) = vi(Γ)+n+1, v′ i(∆) = vi(∆)+n+1, and vj(Γ) ≥ vj(∆), this implies sumF′ i (Γ)+v′ i(Γ) > sumF′ i (∆)+v′ i(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, Γ ≻sumEQ′ i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' If Γ ∼sumEQ i ∆, using the same equations, Γ ⪰sumEQ′ i ∆ is implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' (3) sumAL: If Γ ≻sumAL i ∆ then either (i) sumF i (Γ) = sumF i (∆) and vi(Γ) > vi(∆), or (ii) sumF i (Γ) > sumF i (∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 6 In case (i), sumF i (Γ) = sumF i (∆) together with (1), (2), and vj(Γ) ≥ vj(∆) implies sumF′ i (Γ) ≥ sumF′ i (∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Further, vi(Γ) > vi(∆) together with v′ i(Γ) = vi(Γ) + n + 1 and v′ i(∆) = vi(∆) + n + 1 implies v′ i(Γ) > v′ i(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Altogether, this implies Γ ≻sumAL′ i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In case (ii), sumF′ i (Γ) > sumF′ i (∆) is implied and Γ ≻sumAL′ i ∆ follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' If Γ ∼sumAL i ∆ then sumF i (Γ) = sumF i (∆) and vi(Γ) = vi(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Using the same equations as before, Γ ⪰sumAL′ i ∆ is implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Type-II-monotonicity under sum-based and min-based preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Let j ∈ Γ(i) \\ ∆(i) and vj(Γ) ≥ vj(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' It follows that v′ i(Γ) = vi(Γ) + n + 1, v′ i(∆) = vi(∆), v′ j(Γ) = vj(Γ) + n + 1, and v′ j(∆) = vj(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Fur- thermore, sumF′ i (Γ) = sumF i (Γ)+ vj(Γ)+ n + 1, (3) sumF′ i (∆) = sumF i (∆)+ vj(∆), (4) minF′ i (Γ) = min � minF i (Γ),vj(Γ)+ n + 1 � , (5) minF′ i (∆) = min � minF i (∆),vj(∆) � , (6) minF+′ i (Γ) = min � minF i (Γ),vj(Γ)+ n + 1,vi(Γ)+ n + 1 � , and (7) minF+′ i (∆) = min � minF i (∆),vj(∆),vi(∆) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' (8) (1) sumSF and minSF: If Γ ⪰SF i ∆ then vi(Γ) ≥ vi(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, v′ i(Γ) = vi(Γ)+ n + 1 ≥ vi(∆)+ n + 1 > vi(∆) = v′ i(∆), which implies Γ ≻SF′ i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' (2) sumEQ: If Γ ⪰sumEQ i ∆ then sumF i (Γ)+vi(Γ) ≥ sumF i (∆)+vi(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Together with (3), (4), and vj(Γ) ≥ vj(∆) this implies sumF′ i (Γ)+ v′ j(Γ) > sumF′ i (∆)+ v′ j(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, Γ ≻sumEQ′ i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' (3) sumAL: If Γ ⪰sumAL i ∆ then sumF i (Γ) ≥ sumF i (∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Together with (3), (4), and vj(Γ) ≥ vj(∆) this implies sumF′ i (Γ) > sumF′ i (∆), so Γ ≻sumAL′ i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' (4) minEQ: First, assume that Γ ≻minEQ i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We then have min � minF i (Γ),vi(Γ) � > min � minF i (∆),vi(∆) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' It follows that Γ ≻minEQ′ i ∆ because minF+′ i (Γ) = min � minF i (Γ),vj(Γ)+ n + 1,vi(Γ)+ n + 1 � (9) > min � minF i (∆),vj(Γ),vi(∆) � ≥ min � minF i (∆),vj(∆),vi(∆) � = minF+′ i (∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Second, assume Γ ∼minEQ i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Then min � minF i (Γ),vi(Γ) � = min � minF i (∆),vi(∆) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Similarly as in (9), it follows that minF+′ i (Γ) ≥ minF+′ i (∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, Γ ⪰minEQ′ i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' (5) minAL: First, assume Γ ≻minAL i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Then either (i) minF i (Γ) > minF i (∆), or (ii) minF i (Γ) = minF i (∆) and vi(Γ) > vi(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In case of (i), we get Γ ≻minAL′ i ∆ because minF′ i (Γ) = min � minF i (Γ),vj(Γ)+ n + 1 � ≥ min � minF i (Γ),vj(∆)+ n + 1 � (10) > min � minF i (∆),vj(∆) � = minF′ i (∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In case of (ii), similarly as in (10), we get minF′ i (Γ) ≥ minF′ i (∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Furthermore, vi(Γ) > vi(∆) implies v′ i(Γ) > v′ i(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, Γ ≻minAL′ i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Second, assume that Γ ∼minAL i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Then minF i (Γ) = minF i (∆) and vi(Γ) = vi(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Similarly as in (10), we get minF′ i (Γ) ≥ minF′ i (∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Furthermore, vi(Γ) = vi(∆) implies v′ i(Γ) > v′ i(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, Γ ≻minAL′ i ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Type-I-monotonicity under min-based preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' To see that ⪰minSF is not type-I-monotonic, consider the game G1 with the network of friends in Figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Furthermore, consider the coalition structures Γ = {{1,2},{3,4,5},{6}} and ∆ = {{1,2},{3,4,5,6}} and players i = 1 and j = 2 with 2 ∈ Γ(1) ∩ ∆(1), and v2(Γ) = −1 = v2(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' It holds that v1(Γ) = v1(∆) = −1, minF 1 (Γ) = 2n, and minF 1 (∆) = 2n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, Γ ≻minSF 1 ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 7 1 2 3 4 5 6 (a) Network of G1 1 2 3 4 5 6 (b) Network of G ′ 1 1 2 3 4 5 (c) Network of G2 1 2 3 4 5 (d) Network of G ′ 2 Figure 3: Networks of friends in the proof of Theorem 6 Now, making 2 a friend of 1’s leads to the game G ′ 1 with the network of friends in Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For this game, we have v′ 1(Γ) = v′ 1(∆) = n and minF′ 1 (Γ) = minF′ 1 (∆) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This implies Γ ∼minSF′ 1 ∆, which contradicts type-I-monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' To see that ⪰minEQ and ⪰minAL are not type-I-monotonic,consider the game G2 with the network of friends in Figure 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Consider the coalition structures Γ = {{1,2,3,4},{5}} and ∆ = {{1,2,3,4,5}} and players i = 1 and j = 2 with 2 ∈ Γ(1)∩∆(1), and v2(Γ) = −3 > −4 = v2(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' It holds that minF+ 1 (Γ) = minF 1 (Γ) = 2n−1, and minF+ 1 (∆) = minF 1 (∆) = 2n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, Γ ≻minEQ 1 ∆ and Γ ≻minAL 1 ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Now, making 2 a friend of 1’s leads to the game G ′ 2 with the network of friends in Figure 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For this game, we have minF+′ 1 (Γ) = minF′ 1 (Γ) = n and minF+′ 1 (∆) = minF′ 1 (∆) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This implies Γ ∼minEQ′ 1 ∆ and Γ ∼minAL′ 1 ∆, contradicting type-I-monotonicity and completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Note that the hedonic models of altruism [1, 5] violate both type-I- and type-II-monotonicity for EQ and AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, it is quite remarkable that all three degrees of our extended sum-based model of altruism satisfy both types of monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 4 Stability in ACFGs The main question in coalition formation games is which coalition structures might form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' There are several stability concepts that are well-studied for hedonic games, each indicating whether a given coalition structure would be accepted by the agents or if there are other coalition structures that are more likely to form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Although we consider more general coalition formation games, we can easily adapt these definitions to our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Let (N,⪰) be an ACFG with preferences ⪰ = (⪰1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',⪰n) obtained from a network of friends via one of the three degrees of altruism and with either sum-based or min-based aggregation of the agents’ valuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We use the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For a coalition structure Γ ∈ CN, a player i ∈ N, and a coalition C ∈ Γ∪{/0}, Γi→C denotes the coalition structure that arises from Γ when moving i to C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=', Γi→C = Γ\\ {Γ(i),C} ∪{Γ(i)\\ {i},C∪{i}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In addition, we use ΓC→/0, with C ⊆ N, to denote the coalition structure that arises from Γ when all players in C leave their respective coalition and form a new one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=', ΓC→/0 = Γ\\ {Γ(j)| j ∈ C} ∪{Γ(j)\\C| j ∈ C} ∪{C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Finally, for any two coalition structures Γ,∆ ∈ CN, let #Γ≻∆ = |{i ∈ N |Γ ≻i ∆}| be the number of players that prefer Γ to ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Now, we are ready to define the common stability notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' A coalition structure Γ is said to be Nash stable if no player prefers moving to another coalition: (∀i ∈ N)(∀C ∈ Γ∪{/0})[Γ ⪰i Γi→C];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' individually rational if no player would prefer being alone: (∀i ∈ N)[Γ ⪰i Γi→/0];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' individually stable if no player prefers moving to another coalition and could deviate to it without harming any player in that coalition: (∀i ∈ N)(∀C ∈ Γ∪{/0}) � Γ ⪰i Γi→C ∨(∃j ∈ C)[Γ ≻ j Γi→C] � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 8 contractually individually stable if no player prefers another coalition and could deviate to it without harming any player in the new or the old coalition: (∀i ∈ N)(∀C ∈ Γ∪{/0}) � Γ ⪰i Γi→C ∨(∃j ∈ C)[Γ ≻ j Γi→C]∨(∃k ∈ Γ(i))[Γ ≻k Γi→C] � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' totally individually stable if no player prefers another coalition and could deviate to it without harming any other player: (∀i ∈ N)(∀C ∈ Γ∪{/0}) � Γ ⪰i Γi→C ∨(∃l ∈ N \\ {i})[Γ ≻l Γi→C] � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' core stable if no nonempty coalition blocks Γ: (∀C ⊆ N,C ̸= /0)(∃i ∈ C)[Γ ⪰i ΓC→/0];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' strictly core stable if no coalition weakly blocks Γ: (∀C ⊆ N)(∃i ∈ C)[Γ ≻i ΓC→/0]∨(∀i ∈ C)[Γ ∼i ΓC→/0];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' popular if for every other coalition structure ∆, at least as many players prefer Γ to ∆ as there are players who prefer ∆ to Γ: (∀∆ ∈ CN,∆ ̸= Γ) � #Γ≻∆ ≥ #∆≻Γ � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' strictly popular if for every other coalition structure ∆, more players prefer Γ to ∆ than there are players who prefer ∆ to Γ: (∀∆ ∈ CN,∆ ̸= Γ) � #Γ≻∆ > #∆≻Γ � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' perfect if no player prefers any coalition structure to Γ: (∀i ∈ N)(∀∆ ∈ CN)[Γ ⪰i ∆].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Note that “totally individual stability” is a new notion which we introduce here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' It strengthens the notion of contractually individual stability and makes sense in the context of coalition formation games because players’ preferences may also be influenced by coalitions they are not part of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We now study the associated verification and existence problems in terms of their computational complex- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We assume the reader to be familiar with the complexity classes P (deterministic polynomial time), NP (nondeterministic polynomial time) and coNP (the class of complements of NP sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For more background on computational complexity, we refer to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=', the textbooks by Garey and Johnson [25] and Rothe [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Given a stability concept α, we define: α-VERIFICATION: Given an ACFG (N,⪰) and a coalition structure Γ ∈ CN, does Γ satisfy α?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' α-EXISTENCE: Given an ACFG (N,⪰), does there exist a coalition structure Γ ∈ CN that satisfies α?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Table 2 summarizes the results for these problems under sum-based and min-based SF preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We will also give results for EQ and AL in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In Table 2, however, we only mark if the results for EQ and AL match those for SF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='1 Individual Rationality Verifying individual rationality is easy: We just need to iterate over all agents and compare two coalition structures in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Since players’ utilities can be computed in polynomial time, individual rationality can be verified in time polynomial in the number of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The existence problem is trivial, since Γ = {{1},.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',{n}} is always individually rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Furthermore, we give the following characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Given an ACFG (N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='⪰),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' a coalition structure Γ ∈ CN is individually rational ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='Table 2: Complexity results in sum-based and min-based SF ACFGs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='Stability notion α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='α-VERIFICATION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='α-EXISTENCE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='Individual rationality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='in P1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='trivial1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='Nash stability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='in P1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='trivial1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='Individual stability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='in P1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='trivial1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='Core stability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='coNP-complete2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='trivial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='Strict core stability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='in coNP2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='trivial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='Popularity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='coNP-complete2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='not trivial1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='Strict popularity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='coNP-complete2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='coNP-hard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='Perfectness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='in P2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='in P3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='1 also holds for sum-based and min-based EQ and AL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='ACFGs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='2 is in coNP for any ACFG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='3 is in coNP for sum-based EQ ACFGs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' under sum-based SF, sum-based EQ, sum-based AL, min-based SF, or min-based AL preferences if and only if it holds for all players i ∈ N that Γ(i) contains a friend of i’s or i is alone, formally: (∀i ∈ N)[Γ(i)∩Fi ̸= /0 ∨Γ(i) = {i}];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' under min-based EQ preferences if and only if for all players i ∈ N, Γ(i) contains a friend of i’s or i is alone or there is a friend of i’s whose valuation of Γ is less than or equal to i’s valuation of Γ, formally: (∀i ∈ N)[Γ(i)∩Fi ̸= /0 ∨Γ(i) = {i} ∨(∃j ∈ Fi)[vj(Γ) ≤ vi(Γ)]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' To show the implication from left to right, if Γ is individually rational, we assume for the sake of contradiction that Γ(i)∩Fi = /0 and Γ(i) ̸= {i} for some player i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' First, we observe that for all j ∈ Fi we have vj(Γ) = vj(Γi→/0), as their respective coalition is not affected by i’s move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' It directly follows that, for all considered models of altruism, player i’s utilities for Γ and Γi→/0 only depend on her own valuation, which is greater for Γi→/0 than for Γ (since there are enemies in Γ(i) but not in Γi→/0(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, i prefers Γi→/0 to Γ, so Γ is not individually rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The implication from right to left is obvious for all considered models of altruism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' From left to right, we have that Γ is individually rational and, for the sake of contradiction, we assume that there is a player i ∈ N with Γ(i)∩Fi = /0 and Γ(i) ̸= {i} and for all j ∈ Fi we have vj(Γ) > vi(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Since i is the least satisfied player in Fi ∪ {i}, we have uminEQ i (Γ) = vi(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' With vj(Γi→/0) = vj(Γ) > vi(Γ) for all j ∈ Fi and vi(Γi→/0) = 0 > vi(Γ), we immediately obtain uminEQ i (Γi→/0) > uminEQ i (Γ) and Γi→/0 ≻minEQ i Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This is a contradiction to Γ being individually rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' From right to left, we have to consider two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' First, if Γ(i)∩Fi ̸= /0 or Γ(i) = {i} for some i ∈ N, we obviously have Γ ⪰minEQ i Γi→/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Second, if Γ(i) ∩ Fi = /0 and Γ(i) ̸= {i}, we know that there is at least one j ∈ Fi with vj(Γ) ≤ vi(Γ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Let j′ denote a least satisfied friend of i’s in Γ (pick one randomly if there are more than one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Since Γ(i) ∩ Fi = /0, it holds that Γ(j) = Γi→/0(j) for all j ∈ Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Consequently, j′ is i’s least satisfied friend in both coalition structures and we have uminEQ i (Γ) = vj′(Γ) = vj′(Γi→/0) = uminEQ i (Γi→/0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, Γ ∼minEQ i Γi→/0, so Γ is individually rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='2 Nash Stability Since there are at most |N| coalitions in a coalition structure Γ ∈ CN, we can verify Nash stability in polyno- mial time: We just iterate over all agents i ∈ N and all the (at most |N|+ 1) coalitions C ∈ Γ∪{/0} and check whether Γ ⪰i Γi→C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Since we can check a player’s altruistic preferences over any two coalition structures in polynomial time and since we have at most a quadratic number of iterations (|N| · (|N| + 1)), Nash stability verification is in P for any ACFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 10 1 2 3 4 5 6 7 8 9 10 Figure 4: Networks of friends for Example 10 Nash stability existence is trivially in P for any ACFG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' indeed, the same example that Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [1] gave for altruistic hedonic games works here as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Specifically, for C = {i ∈ N |Fi = /0} = {c1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',ck} the coalition structure {{c1},.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',{ck},N \\C} is Nash stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='3 Individual Stability For individual stability, contractually individual stability, and totally individual stability, existence is also trivially in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Nash stability implies all these three concepts, hence, the Nash stable coalition structure given above is also (contractually;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' totally) individually stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Verification is also in P for these stability concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Similarly to Nash stability, we can iterate over all players and all coalitions and check the respective conditions in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='4 Core Stability and Strict Core Stability We now turn to core stability and state some results for sum-based and min-based SF ACFGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We first show that (strict) core stability existence is trivial for SF ACFGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Let (N,⪰SF) be a (sum-based or min-based) SF ACFG with the underlying network of friends G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Let further C1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',Ck be the vertex sets of the connected components of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Then Γ = {C1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',Ck} is strictly core stable (and thus core stable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For the sake of contradiction, assume that Γ were not strictly core stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=', that there is a coalition D ̸= /0 that weakly blocks Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Consider some player i ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Since i weakly prefers deviating from Γ(i) to D, there have to be at least as many friends of i’s in D as in Γ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Since Γ(i) contains all of i’s friends, D also has to contain all friends of i’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Then all these friends of i’s also have all their friends in D for the same reason, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Consequently, D contains all players from the connected component Γ(i), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=', Γ(i) ⊆ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Since D weakly blocks Γ, D cannot be equal to Γ(i) and thus needs to contain some ℓ /∈ Γ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Yet, this is a contradiction, as ℓ is an enemy of i’s and i would prefer Γ to ΓD→/0 if D contains the same number of friends as Γ(i) but more enemies than Γ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' However, the coalition structure from Theorem 9 is not necessarily core stable under EQ and AL prefer- ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Let N = {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',10} and consider the network of friends G shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Consider the coalition structure consisting of the connected component of G (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=', of only the grand coalition: Γ = {N}) and the coalition C = {8,9,10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' C blocks Γ under sum-based and min-based EQ and AL preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' To see this, consider how players 7, 8, 9, and 10 value Γ and ΓC→/0: v7(Γ) = v8(Γ) = 30 − 6 = 24, v7(ΓC→/0) = 20 − 4 = 16, v9(Γ) = v10(Γ) = 20 − 7 = 13, v8(ΓC→/0) = v9(ΓC→/0) = v10(ΓC→/0) = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We then obtain sumF+ 8 (Γ) = 74 < 76 = sumF+ 8 (ΓC→/0) and sumF+ 9 (Γ) = sumF+ 10 (Γ) = 50 < 60 = sumF+ 9 (ΓC→/0) = sumF+ 10 (ΓC→/0), so ΓC→/0 ≻sumEQ i Γ for all i ∈ C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' sumF 8 (Γ) = 50 < 56 = sumF 8 (ΓC→/0) and sumF 9 (Γ) = sumF 10(Γ) = 37 < 40 = sumF 9 (ΓC→/0) = sumF 10(ΓC→/0), so ΓC→/0 ≻sumAL i Γ for all i ∈ C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' minF+ 8 (Γ) = minF 8 (Γ) = 13 < 16 = minF+ 8 (ΓC→/0) = minF 8 (ΓC→/0) and minF+ 9 (Γ) = minF 9 (Γ) = minF+ 10 (Γ) = minF 10(Γ) = 13 < 20 = minF+ 9 (ΓC→/0) = minF 9 (ΓC→/0) = minF+ 10 (ΓC→/0) = minF 10(ΓC→/0), which implies ΓC→/0 ≻minEQ i Γ and ΓC→/0 ≻minAL i Γ for all i ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 11 β1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' βb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' β3k ζS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' ζSj b ∈ Sj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' ζS3k αS1,1 αS1,2 αS1,3 δS1,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' δS1,4k−3 αS3k,1 αS3k,2 αS3k,3 δS3k,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' δS3k,4k−3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Beta Zeta QS1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' QS3k Figure 5: Network of friends in the proof of Theorem 11 that is used to show coNP-hardness of core stability verification in min-based SF ACFGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' A dashed rectangle around a group of players indicates that all these players are friends of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Thus C blocks Γ under sum-based and min-based EQ and AL preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Turning to (strict) core stability verification, we can show that this problem is hard under SF preferences, and we suspect that this hardness also extends to EQ and AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Strict core stability verification and core stability verification are in coNP for any ACFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For (sum-based and min-based) SF ACFGs, core stability verification is even coNP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' To see that strict core stability verification and core stability verification are in coNP, consider any coalition structure Γ ∈ CN in an ACFG (N,⪰).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Γ is not (strictly) core stable if there is a coalition C ⊆ N that (weakly) blocks Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, we nondeterministically guess a coalition C ⊆ N and check whether C (weakly) blocks Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This can be done in polynomial time since the preferences of the agents in C for the coalition structures Γ and ΓC→/0 can be verified in polynomial time for all our altruistic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' To show coNP-hardness of core stability verification under min-based SF ACFGs, we use RX3C, which is a restricted variant of EXACT COVER BY 3-SETS and known to be NP-complete [25, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We provide a polynomial-time many-one reduction from RX3C to the complement of our verification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Let (B,S ) be an instance of RX3C, consisting of a set B = {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',3k} and a collection S = {S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',S3k} of 3-element subsets of B, where each element of B occurs in exactly three sets in S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The question is whether there exists an exact cover for B in S , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=', a subset S ′ ⊆ S with |S ′| = k and � S∈S ′ S = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We assume that k > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' From (B,S ) we now construct the following ACFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The set of players is N = {βb|b ∈ B}∪{ζS,αS,1,αS,2, αS,3,δS,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',δS,4k−3 |S ∈ S } and we define the sets Beta = {βb|b ∈ B}, Zeta = {ζS |S ∈ S }, and QS = {ζS,αS,1,αS,2,αS,3,δS,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',δS,4k−3} for each S ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Figure 5 shows the network of friends, where a dashed rectangle around a group of players means that all these players are friends of each other: All players in Beta are friends of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For every S ∈ S , ζS is friend with every βb with b ∈ S and with αS,1, αS,2, and αS,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For every S ∈ S , αS,1, αS,2, αS,3, and δS,1 are friends of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For every S ∈ S , all players in {δS,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',δS,4k−3} are friends of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Furthermore, consider the coalition structure Γ = {Beta,QS1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',QS3k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We will now show that S con- tains an exact cover for B if and only if Γ is not core stable under the min-based SF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Only if: Assume that there is an exact cover S ′ ⊆ S for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Then |S ′| = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Consider coalition C = Beta∪{ζS |S ∈ S ′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' C blocks Γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=', ΓC→/0 ≻minSF i Γ for all i ∈ C, because (a) every βb ∈ Beta has 3k friends in C but only 3k−1 friends in Beta and (b) every ζS with S ∈ S ′ has 3 friends and 4k−4 enemies in C but 3 friends and 4k − 3 enemies in QS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 12 If: Assume that Γ is not core stable and let C ⊆ N be a coalition that blocks Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Then ΓC→/0 ≻minSF i Γ for all i ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' First, observe that every i ∈ C needs to have at least as many friends in C as in Γ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' So, if any αS,j or δS,j is in C, it follows quite directly that QS ⊆ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' However, since QS is a coalition in Γ and since every other player (from N \\ QS) is an enemy of all δ-players, any coalition C with QS ⊆ C cannot be a blocking coalition for Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This contradiction implies that no αS,j or δS,j is in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We now have C ⊆ Beta∪Zeta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Since any βb ∈ C has 3k − 1 friends and no enemies in Γ(βb) and prefers ΓC→/0 to Γ, one of the following holds: (a) βb has at least 3k friends in C or (b) βb has 3k − 1 friends and no enemies in C and βb’s friends assign a higher value to ΓC→/0 than to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For a contradiction, assume that (b) holds for some βb ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' First, observe that there are exactly 3k players in C (namely, βb and βb’s 3k − 1 friends).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We now distinguish two cases: Case 1: All the 3k−1 friends of βb’s are β-players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Then C consists of all β-players, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=', C = Beta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This is a contradiction, as Beta is already a coalition in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Case 2: There are some ζ-players in C that are βb’s friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Since βb has three ζ-friends in total and no enemies in C, there are between one and three ζ-players in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, there are between 3k − 3 and 3k − 1 β-players in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Then one of the β-players has no ζ-friend in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' (The at most three ζ-players are friends with at most nine β-players, but 3k − 3 > 9 for k > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=') Consequently, this β-player has only the other (at most 3k − 2) β-players as friends in C and does not prefer ΓC→/0 to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, option (a) holds for each βb ∈C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In total, each βb has exactly three ζ-friends and 3k−1 β-friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Thus at least 3k −3 of βb’s friends in C are β-players and at least one of βb’s friends in C is a ζ-player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Also counting βb herself, there are at least 3k − 2 β-players in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Since all of these 3k − 2 β-players have at least one ζ-friend in C, there are at least k ζ-players in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' (Note that k−1 ζ-players are friends with at most 3k−3 β-players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=') Consider some ζS ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Since ζS has three friends and 4k − 3 enemies in QS, at most three friends in C, and prefers ΓC→/0 to Γ, ζS has exactly three friends and at most 4k − 3 enemies in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, C contains at most 4k − 3 + 3 + 1 = 4k + 1 players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' So far we know that there are at least 3k − 2 β-players in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' If C contains exactly 3k − 2 (or 3k − 1) β-players then each of this players has only 3k − 3 (or 3k − 2) β-friends in C and additionally needs at least three (or two) ζ-friends in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, we have at least (3k − 2) · 3 = 9k − 6 (or 6k − 2) edges between the β- and ζ-players in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Then there are at least 3k − 2 (or 2k) ζ-players in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Thus there are at least (3k−2)+(3k−2) = 6k−4 (or 5k−1) players in C which is a contradiction because there are at most 4k+1 players in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, there are exactly 3k β-players in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Summing up, there are exactly 3k β-players, at least k ζ-players, and at most 4k+1 players in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, there are k or k + 1 ζ-players in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For the sake of contradiction, assume that there are k + 1 ζ-players in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Then each ζS ∈ C has 4k − 3 enemies in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Since ζS prefers ΓC→/0 to Γ, this implies that ζS has exactly three friends and 4k − 3 enemies in C and the minimal value assigned to ΓC→/0 by ζS’s friends is higher than the minimal value assigned to Γ by ζS’s friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In both coalition structures, the minimal value is given by ζS’s α-friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' However, since these α-players lose ζS as a friend when ζS deviates to C, the minimal value assigned to Γ is higher than for ΓC→/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, there are exactly k ζ-players in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Finally, since every of the 3k βb ∈ C has one of the k ζS ∈ C as a friend, it holds that {S|ζS ∈ C} is an exact cover for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This completes the coNP-hardness proof for min-based SF ACFGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For sum-based SF ACFGs, coNP-hardness of core stability verification can be shown by a similar con- struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Again, given an instance (B,S ) of RX3C, with B = {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',3k}, S = {S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',S3k}, and k > 8, we construct the following ACFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The set of players is N = {βb|b ∈ B}∪{ζS,αS,1,αS,2,αS,3,δS,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',δS,4k−3 |S ∈ S }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We define the sets Beta = {βb|b ∈ B} and QS = {αS,1,αS,2,αS,3,δS,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',δS,4k−3} for each S ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The network of friends is given in Figure 6, where a dashed rectangle around a group of players means that all these players are friends of each other: All players in Beta are friends of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For every S ∈ S , all players in QS are friends of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For every S ∈ S , ζS is friend with αS,1, αS,2, and αS,3 and with every βb with b ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Similar arguments as above show that the coalition structure Γ = {Beta} ∪ {{ζS} ∪ QS | S ∈ S } is not core stable under sum-based SF preferences if and only if S contains an exact cover for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 13 β1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' βb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' β3k ζS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' ζSj b ∈ Sj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' ζS3k αS1,1 αS1,2 αS1,3 δS1,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' δS1,4k−3 αS3k,1 αS3k,2 αS3k,3 δS3k,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' δS3k,4k−3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Beta Zeta QS1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' QS3k Figure 6: Network of friends in the proof of Theorem 11 that is used to show coNP-hardness of core stability verification in sum-based SF ACFGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' A dashed rectangle around a group of players indicates that all these players are friends of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Figure 7: Network of friends for Example 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='5 Popularity and Strict Popularity Now we take a look at popularity and strict popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For all considered models of altruism, there are games for which no (strictly) popular coalition structure exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Example 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Let N = {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',10} and consider the network of friends shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Then there is no strictly popular and no popular coalition structure for any of the sum-based or min-based degrees of altruism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Since perfectness implies popularity, there is also no perfect coalition structure for this ACFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Recall from Footnote 2 that there are 115,975 possible coalition structures for this game with ten players, which we all tested for this example by brute force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We now show that, under sum-based and min-based SF preferences, it is hard to verify if a given coalition structure is popular or strictly popular, and it is also hard to decide whether there exists a strictly popular coalition structure for a given SF ACFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Popularity verification and strict popularity verification are in coNP for any ACFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For (sum-based and min-based) SF ACFGs, popularity verification and strict popularity verification are coNP- complete and strict popularity existence is coNP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' First, we observe that the verification problems are in coNP: To verify that a given coalition structure Γ is not (strictly) popular, we can nondeterministically guess a coalition structure ∆, compare both coalition structures in polynomial time, and accept exactly if ∆ is more popular than (or at least as popular as) Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' To show coNP-hardness of strict popularity verification for min-based SF ACFGs, we again employ a polynomial-time many-one reduction from RX3C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Let (B,S ) be an instance of RX3C, consisting of a set B = {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',3k} and a collection S = {S1,··· ,S3k} of 3-element subsets of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Recall that every element of B occurs in exactly three sets in S and the question is whether there is an exact cover S ′ ⊆ S of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We now construct a network of friends based on this instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The set of players is given by N = {α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',α2k} ∪{βb|b ∈ B} ∪{ζS,ηS,1,ηS,2 |S ∈ S }, so in total we have n = 14k players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For convenience, we define Alpha = {α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',α2k}, Beta = {βb | b ∈ B}, and QS = {ζS,ηS,1,ηS,2 | S ∈ S } for S ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The network of friends is shown in Figure 8, where a dashed square around a group of players means that all these players are friends of each other: All players in Alpha∪Beta are friends of each other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' for every S ∈ S , all players in QS are friends of each other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' and ζS is a friend of every βb with b ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We consider the coalition structure Γ = {Alpha∪Beta}∪{QS|S ∈ S } and will now show that S contains an exact cover for B if and only if Γ is not strictly popular under min-based SF preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Only if: Assuming that there is an exact cover S ′ ⊂ S for B, we define the coalition structure ∆ = {Alpha ∪ Beta ∪ � S∈S ′ QS} ∪ {QS | S ∈ S \\ S ′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We will now show that ∆ is as popular as Γ under min- based SF preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 14 α1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' α2k β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' βb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' β3k ζS1 ζSj b ∈ Sj ζS3k ηS1,1 ηS1,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' ηSj,1 ηSj,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' ηS3k,1 ηS3k,2 QS1 QSj QS3k Alpha∪Beta Figure 8: Network of friends in the proof of Theorem 13 that is used to show coNP-hardness of strict pop- ularity verification in min-based SF ACFGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' A dashed rectangle around a group of players indicates that all these players are friends of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' First, all 2k α-players prefer Γ to ∆, since they only add enemies to their coalition in ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Second, the 3k β-players prefer ∆ to Γ, as each β-player gains a ζ-friend and then has 5k friends instead of 5k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Next, we consider the QS-groups for S ∈ S ′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=', the groups that were added to Alpha ∪ Beta in ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We observe that every ζS-player in these QS-groups prefers ∆ to Γ, since ζS gains three additional β-friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For the η-players, on the other hand, the new coalition only contains more enemies, so the η-players prefer Γ to ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Since we have |S ′| = k, this means k ζ-players prefer ∆ to Γ, and 2k η-players prefer Γ to ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Finally, we consider the remaining QS-groups with S ∈ S \\ S ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Here, the coalition containing these players is the same in Γ and ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, for each player p ∈ QS, we have vp(Γ) = vp(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Thus the players have to ask their friends for their valuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For ζS ∈ QS with S ∈ S \\ S ′, the minimum value of her friends is in both structures given by an η-friend, since ηS,1 and ηS,2 value Γ and ∆ both with n·2, while the β-friends of ζS assign values n·(5k−1) to Γ and n·5k−(3k−1) to ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' So we have uminSF ζS (Γ) = uminSF ζS (∆) and, therefore, 2k ζ-players that are indifferent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The η-players in QS, S ∈ S \\ S ′, are also indifferent, as all their friends value Γ and ∆ the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In total, #∆≻Γ = |Beta ∪ {ζS | S ∈ S ′}| = 4k = |Alpha ∪ {ηS,1,ηS,2 |S ∈ S ′}| = #Γ≻∆ and, therefore, ∆ is exactly as popular as Γ, so Γ is not strictly popular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' If: Assuming that Γ is not strictly popular, there is some coalition structure ∆ ∈ CN with ∆ ̸= Γ such that ∆ is at least as popular as Γ under min-based SF preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We will now show that this implies the existence of an exact cover for B in S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' First of all, we observe that all α-players’ most preferred coalition is Alpha∪Beta, as it contains all their friends and no enemies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Thus we have Γ ≻minSF α ∆ if Alpha∪Beta /∈ ∆ and Γ ∼minSF α ∆ if Alpha∪Beta ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For the sake of contradiction, we assume that Alpha∪Beta ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' As ∆ ̸= Γ, the players in the QS-groups have to be partitioned differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' However, that would not increase any player’s valuation since every player in QS can only lose friends and gain enemies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' That means that no β-player prefers ∆ to Γ, as they are in the same coalition as in Γ and their friends are not more satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We also have at least three players of a QS- group that are no longer in the same coalition, so they prefer Γ to ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This is a contradiction, as we assumed that ∆ is at least as popular as Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Thus we have Alpha∪Beta /∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Now consider the η-players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For every S ∈ S , we know that QS is the best valued coalition for ηS,1 and ηS,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' So again, ηS,1 and ηS,2 prefer Γ to ∆ if and only if QS /∈ ∆, and they are indifferent otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Define k′ = |{S ∈ S | QS /∈ ∆}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' So 2k′ is the number of η-players that prefer Γ to ∆, and the remaining 6k − 2k′ η-players are indifferent between Γ and ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We first collect some observations: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' All 2k α-players prefer Γ to ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 2k′ η-players prefer Γ to ∆, and 6k − 2k′ η-players are indifferent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 3k−k′ ζ-players are in the same coalition in both coalition structures, so their utilities depend on their friends’ valuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In Γ, the minimum value of their friends is given by an η-player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Since this η- player is also in the same coalition in ∆ and thus assigns the same value, it is not possible that the minimum value of the friends is higher in ∆ than in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' So 3k − k′ ζ-players are indifferent or prefer Γ to ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We have 14k players in total, so we can have at most 14k − 2k − 2k′ − (6k − 2k′)− (3k − k′) = 3k + k′ players that prefer ∆ to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 15 α1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' α5k β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' βb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' β3k ζS1 ζSj b ∈ Sj ζS3k ηS1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' ηSj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' ηS3k QS1 QSj QS3k Alpha∪Beta Figure 9: Network of friends in the proof of Theorem 13 that is used to show coNP-hardness of strict popu- larity verification in sum-based SF ACFGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' A dashed rectangle around a group of players indicates that all these players are friends of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Next, we show that k′ = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' First, assume that k′ > k: We have #Γ≻∆ ≥ 2k +2k′, and since k′ > k, we have 2k + 2k′ > 3k + k′ ≥ #∆≻Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This is a contradiction to #Γ≻∆ ≤ #∆≻Γ, so we obtain k′ ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Second, let us assume k′ < k: Since every ζ-player has three β-friends and there are k′ ζ-players that are not in their respective QS coalition in ∆, there are at most 3k′ β-players that gain a ζ-friend in ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The 3k−3k′ other β-players have at most 5k−1 friends in ∆, namely all other α- and β-players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' But as Alpha∪Beta /∈ ∆, they would also gain at least one enemy, so we have 3k − 3k′ β-players that prefer Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' That means we have #Γ≻∆ ≥ 2k + 2k′ + 3k − 3k′ = 5k − k′ and #∆≻Γ ≤ 3k + k′ − (3k − 3k′) = 4k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Since k′ < k, we have 5k − k′ > 5k − k = 4k > 4k′, and therefore, #Γ≻∆ > #∆≻Γ, which again is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Thus we conclude that k′ ≥ k and, in total, k′ = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Consequently, we know that 4k players prefer Γ to ∆, namely all α-players and the 2k η-players that are not in QS anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Subtracting all the indifferent players, we observe that all other players have to prefer ∆ to Γ in order to ensure #Γ≻∆ ≤ #∆≻Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' These other players are the 3k β-players and the k ζ-players that are not in QS anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Finally, that is only possible if every β-player gains a ζ-friend in ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence each one of those k ζ-players has to be friends with three different β-players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Therefore, the set {S ∈ S |QS /∈ ∆} is an exact cover for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' To show coNP-hardness of strict popularity verification for sum-based SF ACFGs, we use a similar con- struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For an instance (B,S ) of RX3C with B = {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',3k} and S = {S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',S3k}, where each element of B occurs in exactly three sets in S , we construct the following ACFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The set of players is given by N = {α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',α5k} ∪ {βb| b ∈ B} ∪ {ζS,ηS | S ∈ S }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Let Alpha = {α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',α5k}, Beta = {βb| b ∈ B}, and QS = {ζS,ηS} for each S ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The network of friends is given in Figure 9, where a dashed rectangle around a group of players means that all these players are friends of each other: All players in Alpha ∪ Beta are friends of each other and, for every S ∈ S , ζS is friends with ηS and every βb with b ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Consider the coalition structure Γ = {Alpha ∪ Beta,QS1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',QS3k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We show that S contains an exact cover for B if and only if Γ is not strictly popular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Only if: Assuming that there is an exact cover S ′ ⊆ S for B and considering coalition structure ∆ = {Alpha ∪ Beta ∪ � S∈S ′ QS} ∪ {QS | S ∈ S \\ S ′}, it can be shown with similar arguments as before that #∆≻Γ = |{β1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',β3k,ζS1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',ζS3k}| = 6k = |{α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',α5k} ∪ {ηS | S ∈ S ′}| = #Γ≻∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, ∆ and Γ are equally popular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' If: Assuming that Γ is not strictly popular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=', that there is a coalition structure ∆ ∈ CN, ∆ ̸= Γ, with #Γ≻∆ ≤ #∆≻Γ, it can be shown similarly as before that the set {S ∈ S |QS /∈ ∆} is an exact cover for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The results for strict popularity existence and popularity verification can be shown by slightly modifying the above reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' To show that strict popularity existence is coNP-hard for min-based and sum-based SF ACFGs, we con- sider the same two reductions as before but the coalition structures Γ are not given as a part of the problem instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Then, there is an exact cover for B if and only if there is no strictly popular coalition structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In particular, if there is an exact cover for B, Γ and ∆ as defined in the proofs above are in a tie and every other coalition structure is beaten by Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' And if there is no exact cover for B then Γ beats every other coalition structure and thus is strictly popular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Popularity verification for min-based and sum-based SF ACFGs can be shown to be coNP-complete by using the same constructions as for strict popularity verification (see Figure 8 and 9) but reducing the numbers 16 of α-players to 2k − 1 and 5k − 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Then there is an exact cover for B if and only if Γ, as defined above, is not popular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='6 Perfectness Turning now to perfectness, we start with the SF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For any sum-based or min-based SF ACFG (N,⪰) with an underlying network of friends G, a coalition structure Γ ∈ CN is perfect if and only if it consists of the connected components of G and all of them are cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' From left to right, assume that the coalition structure Γ ∈ CN is perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' It then holds for all agents i ∈ N and all coalition structures ∆ ∈ CN, ∆ ̸= Γ, that i weakly prefers Γ to ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' It follows that vi(Γ) ≥ vi(∆) for all ∆ ∈ CN, ∆ ̸= Γ, and i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, every agent i ∈ N has the maximal valuation vi(Γ) = n ·|Fi| and is together with all of her friends and none of her enemies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This implies that each coalition in Γ is a connected component and a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The implication from right to left is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Since it is easy to check this characterization, perfect coalition structures can be verified in polynomial time for sum-based and min-based SF ACFGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' It follows directly from Theorem 14 that the corresponding existence problem is also in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Corollary 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For any sum-based or min-based SF ACFG (N,⪰) with an underlying network of friends G, there exists a perfect coalition structure if and only if all connected components of G are cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We further get the following upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For any ACFG, perfectness verification is in coNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Consider any ACFG (N,⪰).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' A coalition structure Γ ∈ CN is not perfect if and only if there is an agent i ∈ N and a coalition structure ∆ ∈ CN such that ∆ ≻i Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, we can nondeterministically guess an agent i ∈ N and a coalition structure ∆ ∈ CN and verify in polynomial time whether ∆ ≻i Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Furthermore, we initiate the characterization of perfectness in ACFGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The diameter of a connected graph component is the greatest distance between any two of its vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For sum-based EQ ACFGs, we get the following implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Proposition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For any sum-based EQ ACFG with an underlying network of friends G, it holds that if a coalition structure Γ is perfect for it, then Γ consists of the connected components of G and all these components have a diameter of at most two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We first show that, in a perfect coalition structure, all agents have to be together with all their friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For the sake of contradiction, assume that Γ is perfect but there are i, j ∈ N with j ∈ Fi and j /∈ Γ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We distinguish two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Case 1: All f ∈ Fi ∩ Γ(i) have a friend in Γ(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Consider the coalition structure ∆ that results from the union of Γ(i) and Γ(j), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=', ∆ = Γ \\ {Γ(i),Γ(j)} ∪ {Γ(i) ∪ Γ(j)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' It holds that i and all friends of i’s either gain an additional friend in ∆ or their coalition stays the same: First, i keeps all friends from Γ(i) and gets j as an additional friend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, i has at least one friend more in ∆ than in Γ and we have vi(∆) > vi(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Second, all friends f ∈ Fi ∩ Γ(i) have a friend in Γ(j) and therefore also gain at least one additional friend from the union of the two coalitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, vf (∆) > vf (Γ) for all f ∈ Fi ∩Γ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Third, all friends f ∈ Fi ∩Γ(j) have i as friend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, they also gain one friend from the union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Thus vf (∆) > vf (Γ) for all f ∈ Fi ∩Γ(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Finally, all f ∈ Fi who are not in Γ(i) or Γ(j) value Γ and ∆ the same because their coalition is the same in both coalition structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, vf (∆) = vf (Γ) for all f ∈ Fi with f /∈ Γ(j) and f /∈ Γ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Summing up, we have usumEQ i (∆) > usumEQ i (Γ), so i prefers ∆ to Γ, which is a contradiction to Γ being perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Case 2: There is an f ∈ Fi ∩ Γ(i) who has no friends in Γ(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Consider the coalition structure ∆ that results from j moving to Γ(i), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=', ∆ = Γj→Γ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Let k ∈ Fi ∩Γ(i) be one of the agents who have no friends in 17 1 2 3 4 5 9 7 8 6 Figure 10: Network of friends for Example 19 Γ(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Then vk(∆) = vk(Γ)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' vi(∆) = vi(Γ)+n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' for all f ∈ Fk ∩Γ(i), f ̸= i, we have vf (∆) ≥ vf (Γ)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' and for all f ∈ Fk, f /∈ Γ(i) (and f /∈ Γ(j)), we have vf (∆) = vf (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, usumEQ k (∆) = ∑ a∈Fk∪{k} va(∆) = ∑ a∈Fk∩Γ(i),a̸=i va(∆)+ ∑ a∈Fk\\Γ(i) va(∆)+ vk(∆)+ vi(∆) ≥ ∑ a∈Fk∩Γ(i),a̸=i va(Γ)− 1 + ∑ a∈Fk\\Γ(i) va(Γ)+ vk(Γ)− 1 + vi(Γ)+ n = ∑ a∈Fk∪{k} va(Γ)− (|Fk ∩Γ(i)|− 1)− 1 + n = usumEQ k (Γ)− |Fk ∩Γ(i)| � �� � usumEQ k (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Therefore, k prefers ∆ to Γ, which again is a contradiction to Γ being perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Next, assume that Γ is perfect but there is a coalition C in Γ that has a diameter greater than two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Then there are agents i, j ∈ C with a distance greater than two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Thus j is an enemy of i’s and an enemy of all of i’s friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' It follows that i prefers coalition structure Γj→/0 to Γ, which is a contradiction to Γ being perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Summing up, in a perfect coalition structure Γ for a sum-based EQ ACFG every agent is together with all her friends and every coalition in Γ has a diameter of at most two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Together this implies that Γ consists of the connected components of G and all these components have a diameter of at most two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' From Propositions 16 and 17, we get the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Corollary 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For sum-based EQ ACFGs, perfectness existence is in coNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' However, Proposition 17 is not an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The converse does not hold, as the following example shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Example 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Consider the sum-based EQ ACFG (N,⪰sumEQ) with the network of friends G in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The coalition structure Γ = {N} consists of the only connected component of G, which has a diameter of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' However, agent 1 prefers ∆ = {{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=',6},{7,8,9}} to Γ because usumEQ 1 (Γ) = v1(Γ)+ ···+ v5(Γ)+ v9(Γ) = (9 ·5 − 3)+ 4 ·(9·2−6)+(9·3−5)= 112 < 113 = (9 ·4 − 1)+ 4 ·(9·2−3)+(9·2−0) = v1(∆)+ ···+ v5(∆)+ v9(∆) = usumEQ 1 (∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Hence, Γ is not perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' 5 Conclusions and Open Problems We have proposed to extend the models of altruistic hedonic games due to Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' [1] and Wiechers and Rothe [5] to coalition formation games in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We have compared our more general models to altruism in hedonic games and have motivated our work by removing some crucial disadvantages that come with the restriction to hedonic games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' In particular, we have shown that all degrees of our general altruistic preferences are unanimous while this is not the case for all altruistic hedonic preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Furthermore, all our sum-based degrees of altruism fulfill two types of monotonicity that are violated by the corresponding hedonic equal- and altruistic-treatment preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We have furthermore studied the common stability notions and have initiated a computational analysis of the associated verification and existence problems (see Table 2 for an overview of our results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' We also 18 gave characterizations for some of the stability notions, using graph-theoretical properties of the underlying network of friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' For future work, we propose to complete this analysis, close all gaps between complexity- theoretic upper and lower bounds, and get a full characterization for all stability notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' Acknowledgments We thank the anonymous IJCAI’20 reviewers for helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' This work was supported in part by DFG grants RO 1202/14-2 and RO 1202/21-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=' The first and third author have been supported in part by the research project “Online Participation” within the North Rhine-Westphalian funding scheme “Forschungskollegs.” References [1] Nguyen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} +page_content=', Rey, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E5T4oBgHgl3EQflw-X/content/2301.05674v1.pdf'} 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b/rtFJT4oBgHgl3EQfbiyy/content/tmp_files/2301.11540v1.pdf.txt @@ -0,0 +1,4249 @@ +arXiv:2301.11540v1 [math.PR] 27 Jan 2023 +Occupation time fluctuations of an age-dependent critical +binary branching particle system +J.A. L´opez-Mimbela∗ +A. Murillo-Salas† +J.H. Ram´ırez-Gonz´alez‡ +Abstract +We study the limit fluctuations of the rescaled occupation time process of a branching particle +system in Rd, where the particles are subject to symmetric α-stable migration (0 < α ≤ 2), critical +binary branching, and general non-lattice lifetime distribution. +We focus on two different regimes: +lifetime distributions having finite expectation, and Pareto-type lifetime distributions, i.e. distributions +belonging to the normal domain of attraction of a γ-stable law with γ ∈ (0, 1). In the latter case we +show that, for dimensions αγ < d < α(1 + γ), the rescaled occupation time fluctuations converge weakly +to a centered Gaussian process whose covariance function is explicitly calculated, and we call it weighted +sub-fractional Brownian motion. Moreover, in the case of lifetimes with finite mean, we show that for +α < d < 2α the fluctuation limit turns out to be the same as in the case of exponentially distributed +lifetimes studied by Bojdecki et al. [7, 8, 9]. We also investigate the maximal parameter range allowing +existence of the weighted sub-fractional Brownian motion and provide some of its fundamental properties, +such as path continuity, long-range dependence, self-similarity and the lack of Markov property. +Key words and phrases: branching particle systems, critical binary branching, Pareto-type tail +lifetimes, occupation time fluctuations, sub-fractional motion, renewal theorem, long-range dependence. +MSC 2000 subject classifications: 60J80, 60E10. +1 +Introduction and main results +Our aim in this paper is to investigate the occupation time fluctuations of a population in Rd which evolves +as follows. +During its lifetime S, any given individual independently develops a spherically symmetric +α-stable process with infinitesimal generator the fractional power ∆α := −(−∆)α/2 of the Laplacian, +0 < α ≤ 2, and at the end of its life it either disappears, or is replaced at the site where it died by two +newborns, each event occurring with probability 1/2. The population starts off from a Poisson random +∗Centro de Investigaci´on en Matem´aticas, Guanajuato, Mexico. jalfredo@cimat.mx +†Departamento de Matem´aticas, Universidad de Guanajuato, Guanajuato, Mexico. amurillos@ugto.mx +‡Centro de Investigaci´on en Matem´aticas, Guanajuato, Mexico. hermenegildo.ramirez@cimat.mx +1 + +field having the Lebesgue measure Λ as its intensity. +Along with the usual independence assumptions +in branching systems, we also assume that the particle lifetimes have a general non-lattice distribution, +and that any individual in the initial population has age 0. We focus on two different regimes for the +distribution of S: either S has finite mean µ > 0 or S has a distribution function F such that +F(0) = 0, +F(x) < 1 for all x ≥ 0, +and +1 − F(t) ∼ +1 +tγΓ(1 − γ) +as +t → ∞, +(1) +where 0 < γ < 1 and Γ denotes the usual Gamma function. +Let Z(t) be the counting measure in Rd whose atoms are the positions of particles alive at time t, +and let Z ≡ {Z(t), t ≥ 0}. Recall that the occupation time of the measure-valued process Z is again a +measure-valued process J ≡ {J(t), t ≥ 0} which is given by +⟨ϕ, J(t)⟩ := +� t +0 +⟨ϕ, Z(s)⟩ ds, t ≥ 0, +for all bounded measurable functions ϕ : Rd → R+, where the notation ⟨ϕ, ν⟩ means +� +ϕ dν. Following [12] +and [7], for each T > 0 we introduce the rescaled occupation time process JT (t) := J(Tt) defined by +⟨ϕ, JT (t)⟩ = +� Tt +0 +⟨ϕ, Z(s)⟩ds = T +� t +0 +⟨ϕ, Z(Ts)⟩ ds, +t ≥ 0, +and the rescaled occupation time fluctuation process {JT (t), t ≥ 0} given by +⟨ϕ, JT (t)⟩ := +1 +HT +� +⟨ϕ, JT (t)⟩ − E⟨ϕ, JT (t)⟩ +� +, +t ≥ 0, +where HT is a normalization factor such that HT → ∞ as T → ∞. It was shown in [17] that, due to +criticality of the branching and invariance of Λ for the α-stable semigroup, E⟨ϕ, JT (t)⟩ = Tt⟨ϕ, Λ⟩. Hence, +the rescaled occupation time fluctuation process takes the form +⟨ϕ, JT (t)⟩ := +1 +HT +� +⟨ϕ, JT (t)⟩ − Tt⟨ϕ, Λ⟩ +� +, +t ≥ 0. +(2) +The Markovian case, i.e. +the case of exponentially distributed particle lifetimes, has been thoroughly +investigated by T. Bojdecki, L.G. Gorostiza and A. Talarczyk in a series of seminal works, see [5, 7, 8, 9, 10]. +Among other results, they showed that when S possesses an exponential distribution and α < d < 2α, the +occupation time fluctuation process, properly rescaled, converges weakly toward a Gaussian process in the +space C([0, η], S′(Rd)) of continuous paths w : [0, η] → S′(Rd) for any η > 0, where S′(Rd) denotes the space +of tempered distributions, i.e. the strong dual of the space S(Rd) of rapidly decreasing smooth functions. +The limit process has a simple spatial structure whereas the temporal structure is characterized by that of +sub-fractional Brownian motion (sub-fBm), i.e. a continuous centered Gaussian process {ζt, t ≥ 0} with +covariance function +C(s, t) := sh + th − 1 +2 +� +(s + t)h + |s − t|h� +, +s, t ≥ 0, +(3) +with h = 3 − d/α (h ∈ (1, 2)); see [8]. According to [7], sub-fBm exists for all h ∈ (0, 2). For h ̸= 1 this +process does not have stationary increments, but possesses the so-called long-range dependence property, +and for h = 1 it reduces to Brownian motion. +2 + +It is known [2] that the process Z fails to be Markovian if S does not have an exponential distribution. +There are relatively few publications on models related to non-Markovian spatial branching systems. Laws +of large numbers for the occupation times of Z have been investigated in [19] and [17]. Diffusion limit-type +approximations for branching systems with non-exponential particle lifetimes were developed in [14] and +[15]. Existence of a non-trivial equilibrium distribution for such kind of models was studied in [20]. +Assume that F is a general absolutely continuous function obeying (1). In this paper we prove that +for dimensions satisfying αγ < d < α(1 + γ), the occupation time fluctuation limit exists and is a centered +Gaussian process whose covariance function has a simple spatial structure, but its temporal structure is +dictated, for the case d ̸= α, by a fractional noise with covariance function +Q(s, t) := +� d +α − 1 +�−1 � s∧t +0 +rγ−1 � +(s − r)2−d/α + (t − r)2−d/α − (t + s − 2r)2−d/α� +dr, +s, t ≥ 0, +(4) +whereas for the case d = α, the limit is a centered Gaussian process whose covariance function has a +temporal structure determined by +K(s, t) := +� s∧t +0 +rγ−1 � +(s + t − 2r) ln(s + t − 2r) − (s − r) ln(s − r) − (t − r) ln(t − r) +� +dr; +see Theorem 3.1 below. The special but important case of particle lifetimes with finite mean is dealt with +in Theorem 3.2, where we show that for dimensions satisfying α < d < 2α the limit process is centered +Gaussian, with covariance function of the form (3). Hence, Theorem 3.2 extends Theorem 2.2 in [8] to the +case of non-exponential particle lifetimes with finite mean. Moreover, in this case the effect of the lifetime +distribution becomes apparent only through its mean. +To obtain these results we follow the method of proof used in [8], i.e. the space-time random field weak +convergence approach developed in [6], combined with the Feynman-Kac formula. However the adaptation +to our case of such method is far from being straightforward. Besides the lack of Markov property of Z, +in our more general scenario the use of a Feynman-Kac formula is much more involved than in [8] due to +the fact that the renewal function of F is in general nonlinear, in contrast to the linear renewal function +of exponential lifetimes. +Notice that the function (4) is a special case of the function Qa,b given by +Qa,b(s, t) := +1 +1 − b +� s∧t +0 +ra � +(s − r)b + (t − r)b − (t + s − 2r)b� +dr, +s, t ≥ 0, +a, b ∈ R. +(5) +Several other interesting cases arise as special instances of (5); see Remark 4.2 bellow. This motivated our +second goal in this paper, which is to determine suitable values of the parameters a, b ∈ R for which Qa,b +is a covariance function. It turns out that, if the parameters a, b are restricted to the domains a > −1 and +b ∈ [0, 2] with b ̸= 1, the function Qa,b is positive definite; see Theorem 4.1 below. A centered real-valued +Gaussian process with covariance function (5) will be called weighted sub-fractional Brownian motion, in +analogy to the weighted fractional Brownian motion introduced in [5]. We recall that a weighted fractional +3 + +Brownian motion is a centered Gaussian process η := {η(t), t ≥ 0} with covariance function of the form +� s∧t +0 +ra � +(s − r)b + (t − r)b� +dr, +s, t ≥ 0, +for a > −1, |b| ≤ 1 and |b| < 1 + a. In Theorem 4.5 we show that any weighted sub-fractional Brownian +motion {ς(t), t ≥ 0} possesses long memory (also called long-range dependence), in the sense that +E +� +(ς(t + T) − ς(s + T))(ς(v) − ς(r)) +� +∼ T b−2 +b +(a + 1)(a + 2)(t − s)(va+2 − ra+2) +as +T → ∞. +It is worth to mention that the weighted fractional Brownian motion η also exhibits the long-range depen- +dence property. In this case, +E +� +(η(t + T) − η(s + T))(η(v) − η(r)) +� +∼ T b−1 +b +a + 1(t − s)(va+1 − ra+1) +as +T → ∞; +see [5]. +The rest of the paper is organized as follows. In Section 2 we prove a recursive relation for the Laplace +functional of the branching particle system which we will need in the sequel. Section 3 is devoted to the +proof of the main theorems 3.1 and 3.2. Finally, in Section 4 we investigate the maximal range of parameters +a and b for which (5) becomes a covariance function, and provide several fundamental properties of weighted +sub-fractional Brownian motion, such as path continuity, long-range dependence, self-similarity and the +lack of Markov property. In particular, the limit process obtained in Theorem 3.1 (i) enjoys such properties. +2 +Laplace functional +In this section we will compute the Laplace functional of the occupation time process of Z in a general +setting, i.e., we only assume that the branching law is characterized by its probability generating function +h(s) = �∞ +k=0 pksk, |s| ≤ 1, and the particle lifetimes by a general distribution function F with support in +[0, ∞). The symmetric α-stable motion in Rd will be denoted by ξ = {ξt, t ≥ 0} and by T = {Tt, t ≥ 0} +its semigroup. +By definition Zt(A) is the number of individuals living in A ∈ B(Rd) at time t ≥ 0, where B(Rd) denotes +the system of Borel set in Rd. Let {Sk, k ≥ 1} be a sequence of i.i.d. random variables with common +distribution function F, and let +Nt = +∞ +� +k=1 +1{Wk≤t} +and +U(t) = +∞ +� +n=1 +F ∗n(t), +t ≥ 0, +be the respective renewal process and renewal function, where the random sequence {Wk, k ≥ 0} is defined +recursively by +W0 = 0, +Wk+1 = Wk + Sk, +k ≥ 0. +4 + +Define g(s) := h(1 − s) − (1 − s), |s| ≤ 1. Notice that in the case of critical binary branching h(s) = +s + 1 +2(1 − s)2 and g(s) = 1 +2s2. +Now, for any nonnegative Ψ ∈ S(Rd+1), we define the function +vΨ(x, r, t) := Ex +� +1 − e− � t +0 ⟨Ψ(·,s+r),Zs⟩ ds� +, +x ∈ Rd, +r, t ≥ 0, +(6) +where Ex denotes the expectation operator in a population starting with one particle of age 0, located at +the position x ∈ Rd. +Proposition 2.1 The function vΨ(x, r, t) satisfies the integral equation +vΨ(x, r, t) = Ex +� +1 − e− +� t +0 Ψ(ξs,r+s) ds� +− +� t +0 +Ex +� +e− +� u +0 Ψ(ξs,r+s) dsg +� +vΨ(ξu, r + u, t − u) +�� +dU(u). +(7) +Proof: Formula (7) obviously holds for t = 0. Let t > 0. By conditioning on the first branching time we +get, +1−vΨ(x, r, t) = Ex +� +e− +� t +0 Ψ(ξs,r+s)ds1{S1>t} +� ++Ex +� +e− +� S1 +0 +Ψ(ξs,r+s)dsh +� +1 − vΨ +� +ξS1, r + S1, t − S1 +�� +1{S1≤t} +� +, +or equivalently, +vΨ(x, r, t) =Ex +�� +1 − e− � t +0 Ψ(ξs,r+s)ds +� +1{S1>t} + +� +1 − e− � S1 +0 +Ψ(ξs,r+s)ds +� +1{S1≤t} +− e− � S1 +0 +Ψ(ξs,r+s)dsg(vΨ(ξS1, r + S1, t − S1))1{S1≤t} +� ++ Ex +� +e− � S1 +0 +Ψ(ξs,r+s)dsvΨ(ξS1, r + S1, t − S1)1{S1≤t} +� +. +(8) +Next, we consider the event [S1 ≤ t] and write ξx = {ξx +s , s ≥ 0} for a symmetric α-stable motion starting in +x ∈ Rd. Proceeding as above with r, t and x replaced respectively by r+S1, t−S1 and ξS1, and designating +EξS1(·) the expected value starting with a particle at position ξS1, given the σ−algebra σ((ξs)0≤s≤S1 ∪ S1), +we obtain +vΨ(ξx +S1, r + S1, t − S1)1{S1≤t} += +EξS1 + + + +1 − e +− � t−S1 +0 +Ψ +� +ξ +ξx +S1 +u +,r+S1+u +� +du + + 1{W1≤tt} +� ++ Ex +� +− e− +� S1+S2 +0 +Ψ(ξx +u,r+u)dug(vΨ(ξx +S1+S2, r + S1 + S2, t − S1 − S2))1{W2≤t} +� ++ Ex +�� +1 − e− +� t +0 Ψ(ξx +u,r+u)du +� +1{W1≤tt} + 1{W1≤t 0, we get +vΨ(x, r, t) += +Ex +�� +1 − e− � t +0 Ψ(ξs,r+s) ds� ∞ +� +i=1 +1{Wi−1≤t 0, i.e., +g(s) = 1 +2s2 and dU(u) = V du, equation (7) reduces to +vΨ(x, r, t) = Ex +� +1 − e− +� t +0 Ψ(ξs,r+s) ds� +− V +� t +0 +Ex +� +e− +� t−u +0 +Ψ(ξs,r+s) ds +�1 +2(vΨ(ξu, r + t − u, u))2 +�� +du, +hence, from the Feynman-Kac formula we get +∂ +∂tvΨ(x, r, t) += +� +∆α + ∂ +∂r +� +vΨ(x, r, t) + Ψ(x, r)(1 − vΨ(x, r, t)) − V +2 (vΨ(x, r, t))2 +vΨ(x, r, 0) += +0, +which is equation (3.20) in [8]. +For any r ∈ R we set +f(x, r, t) := Ex +� +e− +� t +0 Ψ(ξx +u,r+u) du� +, +x ∈ Rd, +t ≥ 0. +(11) +It follows from the Feynman-Kac formula that f solves in mild sense the partial differential equation +∂f +∂t (x, r, t) = +� +∆α + ∂ +∂r +� +f(x, r, t) − Ψ(x, r)f(x, r, t) +with initial value f(x, r, 0) = 1, i.e. +f(x, r, t) += +1 − +� t +0 +Tu [Ψ(·, r + u)f(·, r + u, t − u)] (x) du. +(12) +The following result will be useful to prove convergence of the finite-dimensional distributions in Theorem +3.1 and Theorem 3.2. +8 + +Lemma 2.3 Assume that U is absolutely continuous with density function U. The function vΨ(x, r, t) in +(7) can be written as +vΨ(x, r, t) += +� t +0 +Tu [Ψ(·, r + u)f(·, r + u, t − u)] (x)du − +� t +0 +Tug(vΨ(·, r + u, t − u))(x) dU(u) ++ +� t +0 +� t−z +0 +Tz +� +Ψ(·, r + z)E· +� +e− +� u +0 Ψ(ξ· +s,r+z+s)dsg(vΨ(ξ· +u, r + u + z, t − z − u)) +�� +(x) U(u + z) du dz. +Proof: Let us define, for some fixed s ∈ R+ , +k(x, r, σ) := Ex +� +e− � σ +0 Ψ(ξx +u,r+u) dug (vΨ(ξx +σ, r + σ, s)) +� +. +(13) +Notice that k(x, r, σ) also depends on the fixed parameter s but we omit such dependency. Using again +the Feynman-Kac formula we have +∂k +∂σ (x, r, σ) = +� +∆α + ∂ +∂r +� +k(x, r, σ) − Ψ(x, r)k(x, r, σ), +or +k(x, r, σ) = Tσg(vΨ(·, r + σ, s))(x) − +� σ +0 +Tσ−w [Ψ(·, r + σ − w)k(·, r + σ − w, w)] (x) dw. +(14) +Due to (11) and (13), equation (7) can be expressed as +vΨ(x, r, t) = 1 − f(x, r, t) − +� t +0 +k(x, r, t − v) U(t − v) dv. +(15) +From (12) and (14) we obtain +vΨ(x, r, t) += +� t +0 +Tu [Ψ(·, r + u)f(·, r + u, t − u)] (x) du − +� t +0 +Tt−vg(vΨ(·, r + t − v, v))(x) U(t − v) dv ++ +� t +0 +� t−v +0 +Tt−v−w [Ψ(·, r + t − v − w)k(·, r + t − v − w, w)] (x) dw U(t − v) dv +(16) +with +k(x, r + t − v − w, w) = Ex +� +e− +� w +0 Ψ(ξx +u,r+t−v−w+u) dug(vΨ(ξx +w, r + t − v − w + w, v) +� +, +x ∈ Rd. +(17) +Using (17) and making the change of variables z = t − v − w, the double integral in (16) transforms into +� t +0 +� t−v +0 +Tz +� +Ψ(·, r + z)E· +� +e− � t−v−z +0 +Ψ(ξ· +u,r+z+u) dug(vΨ(ξ· +t−v−z, r + t − v, v) +�� +(x) dz U(t − v) dv. +Then, firstly applying Tonelli’s Theorem and then making the change of variables u = t − z − v, in the +double integral in (16) we get +� t +0 +� t−v +0 +Tz +� +Ψ(·, r + z)E· +� +e− +� t−v−z +0 +Ψ(ξ· +u,r+z+u)dug(vΨ(ξ· +t−v−z, r + t − v, v)) +�� +(x) dz U(t − v) dv += +� t +0 +� t−z +0 +Tz +� +Ψ(·, r + z)E· +� +e− +� t−v−z +0 +Ψ(ξ· +u,r+z+u) dug(vΨ(ξ· +t−v−z, r + t − v, v)) +�� +(x) U(t − v) dv dz += +� t +0 +� t−z +0 +Tz +� +Ψ(·, r + z)E· +� +e− +� u +0 Ψ(ξ· +s,r+z+s)dsg(vΨ(ξ· +u, r + u + z, t − z − u)) +�� +(x) U(u + z) du dz. +Finally, plugging the last identity into (16) we conclude the proof. +□ +9 + +3 +Main results +We start this section by stating the main results of this work. +Theorem 3.1 Let F be an absolutely continuous lifetime distribution function satisfying (1). Let αγ < +d < α(1 + γ) and HT = T (2+γ−d/α)/2. Then JT ⇒ J in C([0, η], S′(Rd)) as T → ∞ for any η > 0, where +{Jt, t ≥ 0} is a centered Gaussian process whose covariance function is given in the following way: +(i) For d ̸= α, +Cov(⟨ϕ, J (s)⟩, ⟨ψ, J (t)⟩) = +� +γ⟨ϕ, λ⟩⟨ψ, λ⟩ +Γ(γ + 1)(2π)d(2 − d +α) +� +Rd e−|y|αdy +� +Q(s, t), +s, t ≥ 0, +where ϕ, ψ ∈ S(Rd) and +Q(s, t) = +� d +α − 1 +�−1 � s∧t +0 +rγ−1 � +(s − r)2−d/α + (t − r)2−d/α − (t + s − 2r)2−d/α� +dr. +(18) +(ii) For d = α, +Cov(⟨ϕ, J (s)⟩, ⟨ψ, J (t)⟩) = +� γ⟨ϕ, λ⟩⟨ψ, λ⟩ +Γ(γ + 1)(2π)d +� +Rd e−|y|αdy +� +K(s, t), +s, t ≥ 0, +where ϕ, ψ ∈ S(Rd) and +K(s, t) := +� s∧t +0 +rγ−1 � +(s + t − 2r) ln(s + t − 2r) − (s − r) ln(s − r) − (t − r) ln(t − r) +� +dr. +Theorem 3.2 Let F be an absolutely continuous lifetime distribution function with finite mean µ > 0. +Let α < d < 2α and HT = T (3−d/α)/2. Then JT ⇒ J in C([0, η], S′(Rd)) as T → ∞ for any η > 0, where +{Jt, t ≥ 0} i a centered Gaussian process with covariance function +Cov(⟨ϕ, J (s)⟩, ⟨ψ, J (t)⟩) = +⟨ϕ, Λ⟩⟨ψ, Λ⟩Γ(2 − h) +2d−1πd/2µαΓ(d/2)h(h − 1)C(s, t), +s, t ≥ 0, +where h = 3 − d/α, ϕ, ψ ∈ S(Rd) and C(s, t) is given by (3). +As we mentioned before, our proof of Theorem 3.1 will relay on the space-time random field method +developed in [6] and applied in [8] to treat the Markovian case. Briefly described, the space-time random +field method consists in the following. Let η > 0. For every stochastic process X ≡ {X(t), t ≥ 0} with +paths in the Skorokhod space D([0, η], S′(Rd)) of c`adl`ag functions ω : [0, η] → S′(Rd) let ˜X be the random +element of S′(Rd+1) defined by +⟨˜Φ, ˜X⟩ = +� η +0 +⟨˜Φ(·, s), X(s)⟩ ds, +˜Φ ∈ S(Rd+1). +If X is a.s. continuous at η, then the law of ˜X determines that of X. Moreover, if a family {XT , T ≥ 1} +of S′(Rd)-valued processes with paths in C([0, η], S′(Rd)) is tight, and ˜XT converges in distribution in +S′(Rd+1) as T → ∞, then XT ⇒ X in C([0, η], S′(Rd)) as T → ∞ for some S′(Rd)-valued process X. +Without loss of generality, in the sequel we will assume η = 1. +10 + +3.1 +Proof of Theorem 3.1 +3.1.1 +Tightness +We start by proving that the sequence {JT , T ≥ 1} is tight. Recall that for 0 ≤ s ≤ t and ψ, ϕ ∈ S(Rd), +Cov(⟨ϕ, Z(s)⟩, ⟨ψ, Z(t)⟩) = ⟨ϕTt−sψ, λ⟩ + +� s +0 +� +Rd(Ts−rϕ)(x)(Tt−rψ)(x) dx dU(r); +(19) +see [17]. Let ˆϕ be the Fourier transform ˆϕ(x) = +� +Rd eix·yϕ(y) dy, x ∈ Rd, where x · y denotes the inner +product in Rd. Using (19), Plancherel’s formula and the identity � +Ttϕ(x) = e−t|x|α ˆϕ(x), we deduce that +Cov (⟨ϕ, Z(s)⟩⟨ψ, Z(t)⟩) = +1 +(2π)d +� +Rd ˆϕ(y) ˆψ(y) +� +e−(t−s)|y|α + +� s +0 +e−(t+s−2r)|y|αdU(r) +� +dy. +(20) +Due to (20), for any ψ ∈ S(Rd), +E [⟨ψ, JT (t)⟩ − ⟨ψ, JT (s)⟩]2 = T 2 +H2 +T +� t +s +� t +s +Cov(⟨ψ, Z(Tu)⟩, ⟨ψ, Z(Tv)⟩) du dv = I + II +(21) +where +I = 2T d/α−γ +(2π)d +� t +s +� v +s +� +Rd | ˆψ(y)|2e−T(v−u)|y|α dy du dv +and +II += +2T d/α−γ +(2π)d +� t +s +� v +s +� +Rd | ˆψ(y)|2 +� u +0 +e−(Tv+Tu−2r)|y|αdU(r) dy du dv += +2T d/α−γ +(2π)d +� t +s +� v +s +� +Rd | ˆψ(y)|2 +� u +0 +e−T(v+u−2r)|y|αdU(Tr) dy du dv. +We first deal with the term I. For any s, t ∈ [0, 1] with s ≤ t, +� t +s +� v +s +e−T(v−u)|y|α du dv = +1 +T|y|α +� t +s +(1 − e−T|y|α(v−s)) dv = +1 +T|y|α +� t−s +0 +(1 − e−Tv|y|α) dv +≤ +1 +T|y|α +� t−s +0 +(T|y|αv)δ dv = T δ−1 +δ + 1 +1 +|y|α(1−δ) (t − s)δ+1, +where the inequality above follows from the relation 1 − e−x ≤ xδ, valid for x > 0 and 0 < δ ≤ 1. Since by +assumption αγ < d < α(1 + γ), choosing δ = 1 + γ − d/α we get δ ∈ (0, 1] and +I ≤ +2 +(2π)dh +� +Rd +| ˆψ(y)|2 +|y|d−αγ dy × (t − s)h, +with h = 2 + γ − d/α, +(22) +and the last integral is finite because d > αγ and ψ ∈ S(Rd). +To bound from above in a useful way the second term II we proceed as follows. Given s, t ∈ [0, 1] with +s ≤ t, since ψ is bounded we have +II +≤ +Cψ +2T d/α−γ +(2π)d +� t +s +� v +s +� +Rd +� u +0 +e−T(v+u−2r)|y|αdU(Tr) dy du dv += +Cψ +2T d/α−γ +(2π)d +� +Rd e−|y|α dy +� t +s +� v +s +� u +0 +(v + u − 2r)−d/α +T d/α +dU(Tr) du dv += +Cψ +2 +(2π)d +� +Rd e−|y|αdy +� t +s +� v +s +� u +0 +(v + u − 2r)−d/αd +�U(Tr) +T γ +� +du dv. +11 + +Now, following [4, Section 8.6.2] or [1, Thm. +2.2.2] it can be shown that the measure ˆUT defined on +([0, 1], B([0, 1])) by +ˆUT ([0, u]) = +� u +0 +U(Ts) +T γ−1 ds → +� u +0 +γ +Γ(1 + γ)sγ−1 ds +(23) +as T → ∞ for all u ∈ [0, 1]. Therefore, there exists Mγ > 0 such that for all T ≥ Mγ, +II ≤ C(ψ, α, γ)2 +1 +(2π)d +� t +s +� v +s +� u +0 +(v + u − 2r)−d/αrγ−1 dr du dv. +(24) +For u < v, we have +� u +0 +(v + u − 2r)−d/αrγ−1 dr += +2−γ +� 2u +0 +(u + v − r)− d +α rγ−1dr += +2−γ(u + v)γ− d +α +� +2u +u+v +0 +(1 − r)− d +α rγ−1dr. +(25) +To deal with the last integral we work separately the two cases αγ < d < α and α ≤ d < α(1 + γ). +Case αγ < d < α. For the integral that appears in (25), +� +2u +u+v +0 +(1 − r)− d +α rγ−1 dr ≤ +� 1 +0 +(1 − r)− d +α rγ−1 dr = B(1 − d/α, γ) ≡ C < ∞, +where B(p, q) denotes the beta function. It follows that +� t +s +� v +s +� u +0 +(v + u − 2r)−d/αrγ−1 dr du dv +< +2−γC +� t +s +� v +s +(v + u)γ−d/αdu dv += +2−γC +� t +s +� +(2v)1+γ−d/α − (v + s)1+γ−d/α� +dv. +Since condition αγ < d < α implies 0 < 1 + γ − d/α < 1, using H¨older continuity we get +� t +s +� +(2v)1+γ−d/α − (v + s)1+γ−d/α� +dv ≤ C1 +� t +s +(v − s)1+γ−d/α dv = +C1 +2 + γ − d/α(t − s)2+γ−d/α. +We conclude that for sufficiently large T, +II < C(ψ, d, α, γ)(t − s)h +with h = 2 + γ − d/α. +(26) +Case α ≤ d < α(1 + γ). Notice that +� +2u +u+v +0 +(1 − r)− d +α rγ−1 dr + + + + + + + + + + + + + + + +≤ +� +1 +2 +0 +(1 − r)− d +α rγ−1 dr, +if +2u +v+u < 1 +2, += +� +1 +2 +0 +(1 − r)− d +α rγ−1 dr + +� +2u +u+v +1 +2 +(1 − r)− d +α rγ−1 dr, +if +2u +v+u ≥ 1 +2. +12 + +Now, since γ ∈ (0, 1) we have that +� 1 +2 +0 (1 − r)− d +α rγ−1 dr < ∞. +For the case +2u +v+u ≥ +1 +2 we notice that +rγ−1 ≤ 2−(γ−1) for all r ∈ +� +1 +2, +2u +v+u +� +. Thus, if α < d we obtain +� +2u +u+v +1 +2 +(1 − r)− d +α rγ−1 dr +≤ +2−(γ−1) +� +2u +u+v +1 +2 +(1 − r)− d +α dr ≤ 2−(γ−1) +d/α − 1 +�v − u +v + u +�1−d/α +. +In fact, in the case we are dealing with, we have that γ − d/α < 1 − d/α < 0. Therefore, +� +2u +u+v +1 +2 +(1 − r)− d +α rγ−1 dr ≤ 2−(γ−1) +d/α − 1 +�v − u +v + u +�γ−d/α +. +If d = α, +� +2u +u+v +1 +2 +(1 − r)− d +α rγ−1 dr +≤ +�1 +2 +�γ−1 � +2u +u+v +1 +2 +(1 − r)−1 dr ≤ − +�1 +2 +�γ−1 +ln +�v − u +v + u +� += +C +�1 +2 +�γ−1 �v − u +v + u +�γ−d/α +, +for some constant C > 0, where the last equality follows from the boundedness of the function x �→ +−(ln x)/xγ−1 on the interval [1 +2, 1]. Therefore, from (24), (25) and the last estimates we get that for T +large enough, and for some positive constant C, +II ≤ C +� t +s +� v +s +(v − u)γ−d/α du dv ≤ +C +h(h − 1)(t − s)h, +(27) +where h = 2 + γ − d/α. +We are now ready to state and prove the following +Proposition 3.3 Let αγ < d < α(1 + γ) and H2 +T = T 2+γ−d/α. There exists a constant Mγ > 0 such that +the sequence of processes {JT , T ≥ Mγ} is tight, where JT is defined in (2). +Proof: From (21), (22), (26) and (27) it follows that, for T large enough, +E +� +⟨ψ, JT (t)⟩ − ⟨ψ, JT (s)⟩ +�2 +≤ C|t − s|h, +s, t ≥ 0, +(28) +where h = 2+γ −d/α > 1 because 1+γ −d/α > 0 due to the assumption d < α(1+γ). From [3, Theorem +13.5] we get that for each ψ ∈ S(Rd) the sequence of processes {⟨ψ, JT (t)⟩, T ≥ Mγ} is tight for some Mγ +sufficiently large. Using Mitoma’s theorem [18, Theorem 3.1] we get the tightness of {JT , T ≥ Mγ}. +□ +Space-time method: convergence to a Gaussian process +From (2) we deduce that the space-time random field associated to {JT , T ≥ 1} is given by +⟨˜Φ, ˜ +JT ⟩ := T +HT +�� 1 +0 +⟨Ψ(·, s), Z(Ts)⟩ ds − +�� 1 +0 +Ψ(·, s) ds, Λ +�� +, +˜Φ ∈ S(Rd+1), +13 + +with Ψ(x, s) = +� 1 +s ˜Φ(x, t) dt. Since the initial population is a Poisson random field with intensity the +Lebesgue measure Λ, +E +� +e−⟨˜Φ, ˜ +JT ⟩� += +exp +�� +Rd +� T +0 +ΨT(x, s) ds dx + +� +Rd Ex +� +e− � T +0 ⟨ΨT (·,s),Z(s)⟩ ds − 1 +� +dx +� += +exp +�� +Rd +� T +0 +ΨT(x, s) ds dx − +� +Rd vΨT (x, 0, T) dx +� +, +(29) +where vΨT (x, 0, T) is given in (6) and ΨT (x, s) = +1 +HT Ψ(x, s +T ) = +1 +HT +� 1 +s +T +˜Φ(x, t) dt with HT = T (2+γ−d/α)/2. +Proposition 3.4 Let ˜Φ be of the form ˜Φ(x, t) = φ1(x)φ2(t), where φ1 ∈ S(Rd)+ and φ2 ∈ S(R)+. If +αγ < d < α(1 + γ), then +lim +T→∞ E +� +e−⟨˜Φ, ˜ +JT ⟩� += exp +� +γ⟨φ1, λ⟩2 +Γ(1 + γ)(2π)d +�� +Rd e−|z|α dz +� � 1 +0 +� v +0 +� u +0 +(u + v − 2r)−d/αrγ−1 dr χ(u) χ(v) du dv +� +, (30) +where χ(·) = +� 1 +· φ2(s) ds. +Proof: The proof will be divided into four steps. Using Lemma 2.3 we have that +� +Rd +� T +0 +ΨT (x, s) ds dx − +� +Rd vΨT (x, 0, T) dx += +� +Rd +� T +0 +ΨT(x, s) ds dx − +� +Rd +� T +0 +Tu (ΨT(·, u)h(·, u, T − u)) (x) du dx ++ +� +Rd +� T +0 +Tu (g(vΨT (·, u, T − u)) (x) dU(u) dx +− +� +Rd +� T +0 +� T−z +0 +Tz +� +ΨT (·, z)E· +� +e− +� u +0 ΨT (ξ· +s,z+s) dsg(vΨT (ξ· +u, u + z, T − z − u)) +�� +(x)U(u + z) du dz dx += +� +Rd +� T +0 +ΨT(x, s) ds dx − +� T +0 +� +Rd ΨT(x, u)h(x, u, T − u) dx du + +� T +0 +� +Rd g(vΨT (x, u, T − u) dx dU(u) +− +� +Rd +� T +0 +� T−z +0 +Tz +� +ΨT (·, z)E· +� +e− +� u +0 ΨT (ξ· +s,z+s) dsg(vΨT (ξ· +u, u + z, T − z − u)) +�� +(x)U(u + z) du dz dx, +where to get the second equality we have used that the Lebesgue measure is invariant for the α-stable +semigroup. Thus, we write +� +Rd +� T +0 +ΨT(x, s) ds dx − +� +Rd vΨT (x, 0, T) dx ≡ I1(T) + I2(T) + I3(T) + I4(T), +where +I1(T) := +� +Rd +� T +0 +ΨT (x, s) ds dx − +� T +0 +� +Rd ΨT (x, u)h(x, u, T − u) dx du, +(31) +I2(T) := +� T +0 +� +Rd +� +g(vΨT (x, T − s, s)) − 1 +2 +�� s +0 +TuΨT (·, T + u − s)(x) du +�2� +U(T − s) dx ds, +(32) +14 + +I3(T) := 1 +2 +� T +0 +� +Rd +�� s +0 +TuΨT (·, T + u − s)(x) du +�2 +U(T − s) dx ds, +(33) +I4(T) := +(34) +− +� +Rd +� T +0 +� T−r +0 +Tr +� +ΨT (·, r)E· +� +e− +� u +0 ΨT (ξ· +s,r+s)dsg(vΨT (ξ· +u, u + r, T − r − u)) +�� +(x)U(u + r) du dr dx. +We are going to show that +lim +T→∞ Ii(T) = 0, i ∈ {1, 2, 4}, +(35) +and +lim +T→∞ I3(T) = +γ⟨λ, φ1⟩2 +Γ(1 + γ)(2π)d +�� +Rd e−|z|α dz +� � 1 +0 +� v +0 +� u +0 +(u + v − 2r)−d/αrγ−1 dr χ(u) χ(v) du dv; +(36) +here and below we set χ(t) := +� 1 +t φ2(s) ds and χT (t) := χ( t +T ). +Step 1. Proof of (36). Performing suitable changes of variables, (33) can be re-written as +2I3(T) += +� T +0 +� +Rd +�� T +s +Tu−sΨT (·, u)(x) du +�2 +U(s) dx ds += +� T +0 +� +Rd +� T +s +� T +s +Tu−sΨT (·, u)(x)Tv−sΨT (·, v)(x) du dv U(s) dx ds += +1 +H2 +T +� T +0 +� T +s +� T +s +� +Rd Tu−sφ1(·)(x)Tv−sφ1(·)(x) dx χT (u)χT (v) du dv U(s) ds. +(37) +From Plancherel’s formula we have +� +Rd Tu−sφ1(·)(x)Tv−sφ1(·)(x)dx = +1 +(2π)d +� +Rd +� +Tu−sφ1(x) � +Tv−sφ1(x)dx. +Moreover, � +Ttφ1(x) = e−t|x|α ˆφ1(x). Thus, from (37) we get +2I3(T) += +1 +(2π)dH2 +T +� T +0 +� T +s +� T +s +� +Rd e−(u−s)|x|α ˆφ1(x)e−(v−s)|x|α ˆφ1(x)χT (u)χT (v) dx du dv U(s) ds += +1 +(2π)dH2 +T +� T +0 +�� T +s +� T +s +� +Rd e−(u+v−2s)|x|α ��� ˆφ1(x) +��� +2 +χT (u)χT (v) dx du dv +� +U(s) ds. +After the change of variables s = Tr we obtain +2I3(T) += +T +(2π)dH2 +T +� 1 +0 +� T +Tr +� T +Tr +� +Rd e−(u+v−2Tr)|x|α ��� ˆφ1(x) +��� +2 +χT (u)χT (v) dx du dv U(Tr) dr += +T 3 +(2π)dH2 +T +� 1 +0 +� 1 +r +� 1 +r +� +Rd e−(u+v−2r)T|x|α ��� ˆφ1(x) +��� +2 +χ(u)χ(v) dx du dv U(Tr) dr, +where to get the last identity we again changed variables. +Performing further the change of variables +z = ((u + v − 2r)T)1/αx, the last expression is equivalent to +I3(T) += +T 3−d/α +(2π)dH2 +T +� 1 +0 +� 1 +r +� 1 +r +� +Rd e−|z|α ��� ˆφ1 +� +((u + v − 2r)T)−1/αz +���� +2 +×(u + v − 2r)−d/αχ(u)χ(v) dz du dv U(Tr) dr. +15 + +Changing the order of integration and using that H2 +T = T 2+γ−d/α gives +I3(T) += +1 +(2π)d +� 1 +0 +� v +0 +� u +0 +� +Rd e−|z|α ��� ˆφ1 +� +((u + v − 2r)T)−1/αz +���� +2 +× (u + v − 2r)−d/αχ(u)χ(v) dz U(Tr) +T γ−1 dr du dv, +where +� u +0 +U(Tr) +T γ−1 dr → +� u +0 +γ +Γ(1+γ)rγ−1 dr as T → ∞ for all u ∈ [0, 1]. This proves that +lim +T→∞ I3(T) = | ˆφ1(0)|2 +(2π)d +γ +Γ(1 + γ) +�� +Rd e−|z|αdz +� � 1 +0 +� v +0 +� u +0 +(u + v − 2r)−d/αrγ−1 dr χ(u) χ(v) du dv. +Step 2. Proof of (35) for i = 1. Using equation (12) we have that +I1(T) = +� +Rd +� T +0 +ΨT (x, u) +� T−u +0 +Ts (ΨT (·, u + s)f(·, u + s, T − u − s)) (x) ds du dx. +(38) +Notice that f ≤ 1 because by assumption Ψ is nonnegative (see (11)). Therefore, letting c > 0 be an upper +bound for φ2, +I1(T) +≤ +c +H2 +T +� +Rd +� T +0 +φ1(x) +� T−u +0 +(Tsφ1) (x) ds du dx = +c +H2 +T +� T +0 +� s +0 +� +Rd φ1(x) (Tuφ1) (x) dx du ds += +c +(2π)dH2 +T +� T +0 +� s +0 +� +Rd +ˆφ1(x) � +(Tuφ1)(x) dx du ds = +c +(2π)dH2 +T +� T +0 +� s +0 +� +Rd +��� ˆφ1(x) +��� +2 +e−u|x|α dx du ds += +c +(2π)dH2 +T +� T +0 +� +Rd +��� ˆφ1(x) +��� +2 1 − e−|x|α +|x|α +dx ds ≤ +c +(2π)dT 1+γ−d/α +� +Rd +��� ˆφ1(x) +��� +2 +|x|α +dx. +It follows that +lim +T→∞ I1(T) = 0 +for +α < d < α(1 + γ). +(39) +Assume now that αγ < d ≤ α. Recall that 1 − e−x ≤ xδ for 0 < δ ≤ 1 and x ≥ 0. Moreover, the function +χ is bounded by c. Letting δ = 1 − d +α + γ +2 we get 0 < δ ≤ 1. From (38) it follows that +I1(T) +≤ +1 +H2 +T +� +Rd +� T +0 +φ1(x)χT (u) +� T−u +0 +Tsφ1(x)χT (u + s) ds du dx += +1 +H2 +T +� T +0 +� T−u +0 +� +Rd | ˆφ1(x)|2e−s|x|αχT (u)χT (u + s) dx ds du += +T 2 +H2 +T +� 1 +0 +� 1−u +0 +� +Rd | ˆφ1(x)|2e−Ts|x|αχ(u)χ(u + s) dx ds du +≤ +c2T 2 +H2 +T +� 1 +0 +� 1−u +0 +� +Rd | ˆφ1(x)|2e−Ts|x|α dx ds du += +c2 +T γ−d/α +� 1 +0 +� s +0 +� +Rd | ˆφ1(x)|2e−T(s−u)|x|α dx du ds += +c2 +T 1+γ− d +α +� +Rd +| ˆφ1(x)|2 +|x|α +� 1 +0 +� +1 − eTs|x|α� +ds dx ≤ +c2 +T 1+γ− d +α +� +Rd +| ˆφ1(x)|2 +|x|α +� 1 +0 +(Ts|x|α)δ ds dx +≤ +c2T δ +T 1+γ− d +α +� +Rd +| ˆφ1(x)|2 +|x|α(1−δ) dx +� 1 +0 +sδ ds ≤ +c2 +T +γ +2 +� +Rd +| ˆφ1(x)|2 +|x|d− αγ +2 dx. +16 + +Since 0 < d − (αγ)/2 < d, the last integral above is finite. Hence, +lim +T→∞ I1(T) = 0 +for +αγ < d ≤ α. +(40) +Putting together (39) and (40) gives +lim +T→∞ I1(T) = 0 +for +αγ < d < α(1 + γ). +(41) +Step 3. Proof of (35) for i = 4. By performing the change of variables v = T − r − u in (34) we obtain +−I4(T) = +� +Rd +� T +0 +� T−r +0 +Tr +� +ΨT(·, r)E· +� +e− � T −r−v +0 +ΨT (ξs,r+s) dsg(vΨT (ξ· +T−r−v, T − v, v)) +�� +(x) U(T − v) dv dr dx, +where, due to (15) and the fact that k ≥ 0, +vΨ(x, r, s) += +1 − f(x, r, s) − +� s +0 +k(x, r, s − v) U(s − v) dv ≤ 1 − f(x, r, s) += +� s +0 +Tu [Ψ(·, r + s)f(·, r + s, s − u)] (x) du ≤ +� s +0 +Ts−uΨ(·, r + s − u)(x) du +since |f| ≤ 1 due to (11), where the second equality follows from (12). Hence, +vΨT (x, T − v, v) ≤ +� v +0 +Tv−l [ΨT (·, T − l)] (x) dl = +� v +0 +Tl [ΨT(·, T − v + l)] (x) dl. +(42) +Using that g(s) = s2 +2 and the fact that ΨT ≥ 0, we get +−I4(T) +≤ +1 +2 +� +Rd +� T +0 +� T−r +0 +Tr +� +ΨT (·, r)TT−r−v +� � v +0 +TlΨT(·, T − v + l) dl +�2� +(x) U(T − v) dv dr dx, +where +� v +0 +TlΨT(·, T − v + l)(x) dl += +� v +0 +Tl +� 1 +FT +Ψ +� +·, T − v + l +T +�� +(x) dl += +1 +HT +� v +0 +Tl(φ1)(x) +�� 1 +T −s+u +T +φ2(t) +� +dt dl ≤ +C +HT +� v +0 +Tl(φ1)(x) dl +for 0 ≤ l ≤ v ≤ T. Now, for v ≤ 1, we get +� v +0 TlΨT (·, T +l−v)(x) dl ≤ CMH−1 +T , where M > 0 is a constant +such that |φ1(x)| ≤ M. On the other hand, for s > 1, we have +� v +0 +TlΨT(·, T + l − v)(x) dl +≤ +CM +HT ++ +� v +1 +TlΨT (·, T + l − v)(x) dl +≤ +CM +HT ++ C +HT +� v +1 +� +Rd φ1(y)pl(y − x) dy dl +where pl(·) denotes the density of Tl. By scaling properties of stable densities there exists a constant Cα > 0 +such that pl(x) ≤ Cαl− d +α , x ∈ Rd, l > 0. Since d > α, we obtain +C +HT +� v +1 +� +Rd φ1(y)pl(y − x) dy dl ≤ C(α, d) +HT +� +Rd φ1(x) dx. +17 + +Therefore, there exists a positive constant c(α, d, φ) such that +� v +0 +TlΨT (·, T + l − v)(x) dl ≤ c(α, d, φ) +HT +for all x ∈ Rd and T ≥ v > 0. +Hence, using invariance of Lebesgue measure for {Tt, t ≥ 0} and symmetry of stable densities, we get +−I4(T) +≤ +cα,d,φ U(T) +2HT +� +Rd +� T +0 +� T−r +0 +Tr +� +ΨT (·, r)TT−r−v +�� v +0 +TlΨT (·, T − v + l) dl +�� +(x) dv dr dx += +cα,d,φ U(T) +2HT +� T +0 +� T−r +0 +� v +0 +� +Rd TT−r−vΨT (·, r)(x)TlΨT (·, T − v + l)(x) dx dl dv dr +≤ +c1 +α,d,φ U(T) +2H3 +T +� T +0 +� T−r +0 +� v +0 +� +Rd e−(T−r−v)|x|α ˆφ1(x)e−l|x|α ˆφ1(x) dx dl dv dr += +c1 +α,d,φ U(T) +2H3 +T +� T +0 +� T−r +0 +� +Rd | ˆφ1(x)|2e−(T−r−v)|x|α 1 − e−v|x|α +|x|α +dv dr. +Changing the order of integration yields +−I4(T) +≤ +c1 +α,d,φ, U(T) +2H3 +T +� +Rd | ˆφ1(x)|2 +� T +0 +� +1 − e−(T−v)|x|α� � +1 − e−v|x|α� +|x|2α +dv dx +≤ +c2 +α,d,φ U(T) +H3 +T +� +Rd +� T +0 +� +1 − e−(T−v)|x|α� � +1 − e−v|x|α� +|x|2α +dv dx. +Since e−(T−v)|x|α ≥ e−T|x|α and e−v|x|α ≥ e−T|x|α for all 0 ≤ v ≤ T, we obtain +−I4(T) ≤ +c2 +α,d,φU(T) +H3 +T +� +Rd +� +1 − e−T|x|α +|x|α +�2 +dx = +c2 +α,d,φU(T) +T 1+ 3γ +2 − d +2α +� +Rd +� +1 − e−|z|α +|z|α +�2 +dz, +where the integral with respect to z is finite due to d < α(1 + γ) < 2α. Now, as T → ∞, the factor +c2 +α,d,φU(T) +T 1+ 3γ +2 − d +2α +grows like T −1− γ +2 + d +2α , which goes to zero as long as d < α(1 + γ) + α. This finishes the proof of (35) for +i = 4. +Step 4. Proof of (35) for i = 2. This part can be proved in a similar way as in [11] p. 512-515. Notice +that inequality (42) implies +�� s +0 TuΨT (·, T + u − s)(x)du +�2 − (vΨT (x, T − s, s))2 ≥ 0. Since g(s) = s2 +2 , it +follows that +0 ≤ −2I2(T) = +� T +0 +� +Rd +��� s +0 +TuΨT (·, T + u − s)(x)du +�2 +− (vΨT (x, T − s, s))2 +� +U(T − s) dx ds. +We will prove that, as T → ∞, +� T +0 +� +Rd +��� s +0 +TuΨT (·, T + u − s)du +�2 +− vΨT (x, T − s, s)2 +� +U(T − s) dx ds → 0. +(43) +18 + +Indeed, from (16) obtain +−vΨT (x, T − s, s) += +f(x, T − s, s) − 1 + 1 +2 +� s +0 +Tuv2 +ΨT (x, T − s + u, s − u) dU(u) − +� s +0 +� s−z +0 +Tz(ΨT (x, T − s + z)) +× Ex +� +e− +� u +0 ΨT (ξx +w,T−s+z+w)dwΦ(vΨT (ξx +u, T − s + u + z, s − z − u)) +� +U(u + z) du dz +≤ − +� s +0 +TuΦT (·, T − s + u)f(·, T − s + u, s − u)(x) du + 1 +2 +� s +0 +Tuv2 +ΨT (x, T − s + u, s − u) dU(u). +Therefore, +0 +≤ +� s +0 +TuΨT (·, T + u − s)(x)du − vΨT (x, T − s, s) +≤ +� s +0 +TuΨT (·, T − s + u) (1 − f(·, T − s + u, s − u)(x)) du ++1 +2 +� s +0 +Tuv2 +ΨT (x, T − s + u, s − u) dU(u). +(44) +Also, +1 − f(·, T − s + u, s − u) ≤ +� s−u +0 +TwΨT (·, T − s + u + w) dw +(45) +and +v2 +ΨT (·, T − s + u, s − u) ≤ +�� s−u +0 +TwΨT(·, T − s + u + w)dw +�2 +. +(46) +Thereby, +1 +2 +� s +0 +Tuv2 +ΨT (x, T − s + u, s − u)dU(u) ≤ 1 +2 +� s +0 +Tu +�� s−u +0 +TwΨT(x, T − s + u + w) dw +�2 +dU(u). +(47) +From (44), (45), (46) and (47), +0 +≤ +� s +0 +TuΨT (·, T + u − s)(x) du − vΨT (x, T − s, s) +≤ +� s +0 +Tu +� +ΨT (·, T + u − s) +� s−u +0 +TwΨT (·, T − s + u + w) +� +(x) dw du ++1 +2 +� s +0 +Tu +�� s−u +0 +TwΨT (x, T − s + u + w) dw +�2 +dU(u). +(48) +In addition, from the elementary relation 0 ≤ (a + b)2 − a2 − b2 = 2ab, for a, b ≥ 0 we have +0 ≤ +�� s +0 +TuΨT (·, T + u − s)(x) du − vΨT (x, T − s, s) + vΨT (x, T − s, s) +�2 +− v2 +ΨT (x, T − s, s) +− +�� s +0 +TuΨT (·, T + u − s)(x) du − vΨT (x, T − s, s) +�2 += 2 +�� s +0 +TuΨT (·, T + u − s)(x) du − vΨT (x, T − s, s) +� +vΨT (x, T − s, s). +19 + +Hence, +0 ≤ +�� s +0 +TuΨT(, T + u − s)(x)du +�2 +− v2 +ΨT (x, T − s, s) += +�� s +0 +TuΨT(·, T + u − s)(x)du − vΨT (x, T − s, s) +� +vΨT (x, T − s, s) ++ +�� s +0 +TuΨT(·, T + u − s)(x)du − vΨT (x, T − s, s) +�2 +, +where to due (48) and the inequality (a + b)2 ≤ 2(a2 + b2), a, b ≥ 0, +≤ 2 +�� s +0 +Tu +� +ΨT(·, T + u − s) +� s−u +0 +TwΨT (·, T − s + u + w)dw +� +(x) du +� s +0 +TuΨT (·, T + u − s)(x) du +� ++ 2 +� +1 +2 +� s +0 +Tu +�� s−u +0 +TwΨT(·, T − s + u + w) dw +�2 +(x) dU(u) +� s +0 +TuΨT (·, T − s + u)(x)du +� ++ 2 +�� s +0 +Tu +� +ΨT (·, T + u − s) +� s−u +0 +TwΨT (·, T − s + u + w) +� +(x)du +�2 ++ 2 +� +1 +2 +� s +0 +Tu +�� s−u +0 +TwΨ(·, T − s + u + w)dw +�2 +(x)dU(u) +�2 +. +We define +R1(T) += +� +Rd +� T +0 +�� s +0 +Tu +� +ΨT (·, T + u − s) +� s−u +0 +TwΨT (·, T − s + w + u) dw +� +(x)du +�2 +U(T − s) ds dx, +R2(T) += +� +Rd +� T +0 +�� s +0 +Tu +�� s−u +0 +TwΨT(·, T − s + u + w) dw +�2 +dU(u) +�2 +U(T − s) ds dx. +Then, by the Cauchy-Schwarz inequality applied to the measure +� +Rd +� T +0 U(T − s) ds dx it follows that +� +Rd +� T +0 +�� s +0 +TuΨT (T + u − s)(x))2 − v2 +ΨT (x, T − s, s) +� +U(T − s) ds dx +≤ C1(R1(T) + R2(T)) + C2 +� +I(T)( +� +R1(T) + +� +R2(T)). +We need to show that R1(T) → 0 and R2(T) → 0 as T → ∞. Indeed, for R1(T), +R1(T) ≤ C T +H4 +T +� +Rd +� 1 +0 +�� Ts +0 +Tu +� +φ1(·) +� Ts−u +0 +Twφ1(·) dw +� +(x) du +�2 +U(T(1 − s)) ds dx +≤ C T 3 +H4 +T +� +Rd +� 1 +0 +�� s +0 +TTu +� +φ1(·) +� T(s−u) +0 +Twφ1(·)dw +� +(x) du +�2 +U(T(1 − s)) ds dx +≤ C T 5 +H4 +T +� +Rd +� 1 +0 +�� s +0 +TTu +� +φ1(·) +� s−u +0 +TTwφ1(·) dw +� +(x) du +�2 +U(T(1 − s)) ds dx +≤ C T 4U(T) +H4 +T +� +Rd +�� 1 +0 +TTu +� +φ1(·) +� 1 +0 +TTwφ1(·) dw +� +(x) du +�2 +dx, +20 + +and by the self-similarity property of (Tt)t≥0, +R1(T) ≤ C T 4U(T) +H4 +T +� +Rd +�� +R2d T − 2d +α +� 1 +0 +pu((x − y)T − 1 +α )φ1(y) × +� 1 +0 +pw((y − z)T − 1 +α )φ1(z) dw du dz dy +�2 +dx. +Making the changes of variables x′ = xT − 1 +α , y′ = yT − 1 +α and z′ = zT − 1 +α , +R1(T) ≤ C U(T) +T 2γ +� +Rd +�� +R2d +� 1 +0 +pu(x − y) du T +d +2α φ1(T +1 +α y) × +� 1 +0 +pw(y − z) dw T +d +α φ1(T +1 +α z) dz dy +�2 +dx. +Letting f(x) = +� 1 +0 pu(x) du, g1,T (x) = T +d +2α φ1(T +1 +α x) and g2,T (x) = T +d +α φ1(T +1 +α x), we get +R1(T) ≤ C U(T) +T 2γ ∥f ∗ (g1,T (f ∗ g2,T ))∥2 +2. +Also, d < 2α as d < α(1 + γ) and γ < 1. +Moreover, it can be shown, as in [11], that ∥f∥2 +2 < ∞, +∥g1,T ∥2 = ∥φ1∥2 and ∥g2,T ∥1 = ∥φ1∥1 < ∞. Applying consecutively the inequalities of Young, H¨older and +Young again, we arrive to +R1(T) ≤ C U(T) +T 2γ ∥f∥2 +2 ∥g1,T ∥2 +2 ∥f∥2 +2 ∥g2,T ∥2 +1, +which implies limT→∞ R1(T) = 0. +To work the term R2(T) we put f1,T (x) := +� 1 +0 pu,α(x)U(Tu) +T γ−1 du, hence ∥f1,T∥1 < ∞ and also f1,T→f1 as +T → ∞, where f1 := +� 1 +0 pu,α(x) γuγ−1 +Γ(1+γ) du. Making the change of variables s′ = s +T gives +R2(T) ≤ C T +H4 +T +� +Rd +� 1 +0 +�� Ts +0 +Tu +�� Ts−u +0 +Twφ1(·) dw +�2 +(x)U(u)du +�2 +U(T(1 − s)) ds dx. +Making the change of variables u′ = u +T yields +R2(T) ≤ C T 2+β +H4 +T +� +Rd +� 1 +0 + + +� s +0 +TTu +�� T(s−u) +0 +Twφ1(·) dw +�2 +(x)U(Tu) du + + +2 +U(T(1 − s)) ds dx. +Making the change of variables w′ = w +T renders +R2(T) +≤ +C T 2+1+(2)(2) +H4 +T +� +Rd +� 1 +0 +�� s +0 +TTu +�� s−u +0 +TTwφ1(·) dw +�2 +(x)U(Tu) du +�2 +U(T(1 − s)) ds dx +≤ +C T 2+1+(2)(2) +H4 +T +U(T) +T +� +Rd +�� 1 +0 +TTu +�� 1 +0 +TTwφ1(·) dw +�2 +(x)U(Tu) +T γ−1 du +�2 +dx. +By the self-similarity property of (Tu)u≥0 and making the changes of variables x′ = T − 1 +α , y′ = T − 1 +α , and +z′ = T − 1 +α z, +R2(T) +≤ +C U(T) +T +d +α +� +Rd +�� +Rd +� 1 +0 +pu,α(x − y)U(Tu) +T γ−1 du +�� +Rd +� 1 +0 +pw,α(y − z) dwT +d +α φ1(T +1 +α z) dz +�2 +dy +�2 +dx += +C U(T) +T +d +α +∥f1,T ∗ (f ∗ g2,T )2∥2 +2, +21 + +and since ∥f1,T ∥1 < ∞, +R2(T) ≤ C U(T) +T +d +α +∥f1,T ∥2 +1∥f ∗ g2,T ∥2 +2 ≤ C U(T) +T +d +α +∥f1,T ∥2 +1∥f∥4 +2∥g2,T ∥4 +1, +where we have used Young’s inequality. Again, as ∥g2,T ∥1 = ∥φ1∥1 and αγ < d < α(1 + γ), we will have +U(T) +T +d +α +T→∞ +→ +0 and ∥f1∥1, ∥f∥2 < ∞. Therefore, limT→∞ R2(T) = 0. +We have proved that both R1(T) and R2(T) tend to 0 as T → ∞. This proves Step 4 and shows that +(35) holds. +□ +Lemma 3.5 Under the assumptions in Proposition 3.4, the limit (30) can be written as +lim +T→∞ E +� +e−⟨Ψ, ˜ +JT ⟩� += + + + + + + + + + + + +exp +� +γ⟨φ1,λ⟩2 +Γ(γ+1)(2π)d(2− d +α) +� +Rd e−|y|αdy +� 1 +0 +� 1 +0 Q(w, z)φ2(w)φ2(z) dw dz +� +, +if d ̸= α, +exp +� +γ⟨φ1,λ⟩2 +2Γ(γ+1)(2π)d +� +Rd e−|y|αdy +� 1 +0 +� 1 +0 K(w, z)φ2(w)φ2(z) dw dz +� +, +if d = α. +(49) +where +Q(w, z) = +� d +α − 1 +�−1 � z∧w +0 +sγ−1 � +(w ∧ z − s)2− d +α + (w ∨ z − s)2− d +α − (w + z − 2s)2− d +α +� +ds +(50) +and +K(w, z) := 1 +2 +� z∧w +0 +sγ−1 +� +(w + z − 2s) ln(w + z − 2s) − (w − s) ln(w − s) − (z − s) ln(z − s) +� +ds. +Proof: We first deal with the case d ̸= α. Recall that χ(u) = +� 1 +u φ2(w) dw. Substituting this into the +22 + +triple integral in the right hand side of (30), and changing the order of integration we have +� 1 +0 +� u +0 +� v +0 +(u + v − 2s)− d +α sγ−1χ(u)χ(v) ds dv du += +� 1 +0 +� w +0 +� z +0 +� z +s +� u +s +(u + v − 2s)− d +α sγ−1φ2(w)φ2(z) dv du ds dw dz ++ +� 1 +0 +� w +0 +� z +0 +� w +z +� z +s +sγ−1(u + v − 2s)− d +α φ2(w)φ2(z) dv du ds dz dw ++ +� 1 +0 +� z +0 +� w +0 +� w +s +� u +s +(u + v − 2s)− d +α sγ−1φ2(w)φ2(z) dv du ds dw dz += +� 1 +0 +� w +0 +� z +0 +� z +s +sγ−1 +� +21− d +α − 1 +1 − d +α +� +(u − s)1− d +α φ2(w)φ2(z) du ds dz dw ++ +� 1 +0 +� w +0 +� z +0 +� w +z +sγ−1 +� +(u + z − 2s)1− d +α − (u − s)1− d +α +1 − d +α +� +φ2(w)φ2(z) du ds dz dw ++ +� 1 +0 +� z +0 +� w +0 +� w +s +sγ−1 +� +21− d +α − 1 +1 − d +α +� +(u − s)1− d +α φ2(w)φ2(z) ds dw dz += +� 1 +0 +� w +0 +� z +0 +sγ−1 +� +21− d +α − 1 +(1 − d +α)(2 − d +α) +� +(z − s)2− d +α φ2(w)φ2(z) ds dz dw ++ +� 1 +0 +� w +0 +� z +0 +sγ−1 +� +(w + z − 2s)2− d +α − (w − s)2− d +α + (1 − 22− d +α )(z − s)2− d +α +(1 − d +α)(2 − d +α) +� +φ2(w)φ2(z) ds dw dz ++ +� 1 +0 +� z +0 +� w +0 +sγ−1 +� +21− d +α − 1 +(1 − d +α)(2 − d +α) +� +(w − s)2− d +α φ2(w)φ2(z) ds dw dz. +Due to symmetry of the function (w, z) �→ φ2(w)φ2(z) we conclude that +� 1 +0 +� u +0 +� v +0 +sγ−1(u + v − 2s)− d +α χ(u)χ(v)dsdvdu = +1 +2(2 − d +α) +� 1 +0 +� 1 +0 +Q(w, z)φ2(w)φ2(z) dw dz, +(51) +where +Q(w, z) = +� d +α − 1 +�−1 � z∧w +0 +sγ−1 +� +(w − s)2− d +α + (z − s)2− d +α − (w + z − 2s)2− d +α +� +ds. +For the case d = α, similarly as above we can show that +� 1 +0 +� u +0 +� v +0 +sγ−1(u + v − 2s)− d +α χ(u)χ(v)dsdvdu = 1 +2 +� 1 +0 +� 1 +0 +K(w, z)φ2(w)φ2(z) dw dz, +(52) +where +K(w, z) := 1 +2 +� z∧w +0 +sγ−1 +� +(w + z − 2s) ln(w + z − 2s) − (w − s) ln(w − s) − (z − s) ln(z − s) +� +ds. +The proof is finished because the expressions (51) and (52) are equivalent to (30) for d ̸= α and d = α +respectively. +□ +23 + +Proof of Theorem 3.1. Proposition 3.3 gives the tightness and Proposition 3.4 identifies uniquely any +limit point of {JT , T ≥ 0} for non-negative test functions. For general test functions the proof can be +done as in [8, page 9]. For the sake of brevity we omit the details. +□ +Proof of Theorem 3.2. The proof of this result can be done following the same lines as the proof of +Theorem 3.1 but using the fact that, in the case of lifetimes with finite mean µ, the renewal measure is +such that +U(Tr) +T +→ r +µ +as T → ∞ for all r > 0. +Formally, this can be thought as putting γ = 1 in all the preceding computations. +□ +4 +Weighted sub-fractional Brownian motion +This section is dedicated to investigate under which conditions on the parameters a and b, the function +Qa,b given in (5) is a covariance function. Moreover, when Qa,b is a covariance we prove several properties +of the associated centered Gaussian process. We start with the following theorem. +Theorem 4.1 For a > −1 and 0 ≤ b ≤ 2 with b ̸= 1, the function +Qa,b(w, z) := +1 +1 − b +� z∧w +0 +sa � +(z − s)b + (w − s)b − (w + z − 2s)b� +ds, +w, z ≥ 0, +(53) +is positive definite. +Remark 4.2 Let us mention several known instances of Qa,b given in (53): +(a) Theorem 3.1 (i) yields that the function +(w, z) �→ +� z∧w +0 +sa � +(z − s)b + (w − s)b − (w + z − 2s)b� +ds, +w, z ≥ 0, +(54) +with a = γ − 1 and b = 2 − d/α, appears as the temporal structure of the covariance function of the +rescaled occupation time fluctuation limit for a branching particle system in Rd with α-stable motions +and lifetimes having a Pareto tail distribution (1). +(b) In [10] Bojdecki et al. investigated the limit fluctuations of a rescaled occupation time process of a +branching particle system with particles moving according to d-dimensional α-stable motion, starting +with an inhomogeneous Poisson population with intensity measure dx/(1+|x|γ), where γ > 0. In this +case, for γ < d < α (hence d = 1) and normalization T 1−(d+γ)/2α, the limit of the occupation time +fluctuations is a Gaussian process whose temporal structure is determined by the covariance function +Ca,b(w, z) := +� z∧w +0 +sa � +(z − s)b + (w − s)b� +ds, +w, z ≥ 0, +(55) +24 + +for a = −γ/α and b = 1 − d/α; see [10, Thm. +2.2]. +Latter on, the same authors proved that +the maximal range of values of parameters a, b that makes (55) a covariance function is a > −1, +−1 < b ≤ 1 and |b| ≤ 1 + a. The authors named the centered Gaussian process with covariance +function (55), weighted fractional Brownian motion with parameters a and b, see [5]. Notice that both, +(54) and (55), are weighted covariance kernels, corresponding respectively to weighted sub-fractional +Brownian motion, and weighted fractional Brownian motion. +(c) From (53) it follows that +Q0,b(w, z) = +1 +(b + 1)(1 − b) +� +wb+1 + zb+1 − 1 +2 +� +(w + z)b+1 + |w − z|b+1�� +, +w, z ≥ 0, +which is (modulo a constant factor) the covariance function (3) of the sub-fractional Brownian motion +for 0 < b < 1. Recall that, in [8] the authors consider a Markovian binary branching particle system +and show that, for α < d < 2α, the limit of the rescaled occupation time fluctuations exists (see +Theorem 2.2 in [8]), and that the time structure in the covariance function of the limit process is that +of a sub-fractional Brownian motion. +Proof of Theorem 4.1. Note that, for b = 2 +Qa,2(w, z) += +2 +� z∧w +0 +sa(z − s)(w − s) ds, +which is a positive definite function and it is finite if and only if a > −1. For the case b = 0 we have that +Qa,0(w, z) = +1 +a + 1(s ∧ t)a+1, +which, for a > −1, corresponds to the covariance function of a time-changed Brownian motion. +Let us consider the case of a > −1 and 0 < b < 2, with b ̸= 1. Observe that, +Qa,b(w, z) += +2b +� z∧w +0 +� z∧w +s +� u +s +sa(u + v − 2s)b−2 dv du ds ++b +� z∧w +0 +� z∨w +z∧w +� z∧w +s +sa(u + v − 2s)b−2 dv du ds. +Letting C−1 +b +:= +� ∞ +0 +e−r +1 +2−b dr < ∞, for b < 2, we obtain +Qa,b(w, z) += +2bCb +� z∧w +0 +� z∧w +s +� u +s +� ∞ +0 +sae−(u−s)r +1 +2−b e−(v−s)r +1 +2−b dr dv du ds ++bCb +� z∧w +0 +� z∨w +z∧w +� z∧w +s +� ∞ +0 +sae−(u−s)r +1 +2−b e−(v−s)r +1 +2−b dr dv du ds += +bCb +� ∞ +0 +� z∧w +0 +sa e−2(z∧w−s)r +1 +2−b (e(z∧w−s)r +1 +2−b − 1)2 +r +2 +2−b +ds dr ++bCb +� ∞ +0 +� z∧w +0 +sa + +1 − e−(z∧w−s)r +1 +2−b +r +1 +2−b +· e−(z∧w−s)r +1 +2−b − e−(z∨w−s)r +1 +2−b +r +1 +2−b + + ds dr += +bCb +� ∞ +0 +� ∞ +0 +sa +r +2 +2−b +� +1(0,w)(s)(1 − e−(w−s)r +1 +2−b ) · 1(0,z)(s)(1 − e−(z−s)r +1 +2−b ) +� +ds dr. +25 + +It follows that (53) is a finite positive definite function for a > −1 and b ∈ (0, 2). +□ +The next result exhibits a range of parameters a and b for which the function Qa,b(·, ·) is not a covariance +function. +Lemma 4.3 The function Qa,b(·, ·) is not a covariance function in the following cases: +(i) a > −1 and −1 < b < 0, with a + b + 1 ≤ 0; +(ii) a > −1 and b > 2. +Notice that for a > −1 and b > −1, +Qa,b(t, t) = 2 − 2b +1 − b ta+b+1 +� 1 +0 +ua(1 − u)bdu ≡ 2 − 2b +1 − b B(a + 1, b + 1)ta+b+1. +(56) +The restrictions a > −1 and b > −1 are necessary for the integral above to be finite, and any a or b out of +this range is ruled out. Moreover, for 1 ≤ t, +Qa,b(1, t) = ta+b+1 +1 − b +� 1/t +0 +ua(1 − u)bdu + +1 +1 − bB(a + 1, b + 1) − ta+1 +1 − b +� 1/t +0 +ua(t + 1 − 2tu)bdu, +(57) +whereas for 1 ≥ t, +Qa,b(1, t) ≥ ta+b+1 +1 − b +� 1 +0 +ua(1 − u)bdu + +1 +1 − b +� t +0 +ua(1 − u)bdu. +(58) +Proof of Lemma 4.3 +(i) For a > −1 and −1 < b < 0, with a + b + 1 ≤ 0, the function Qa,b(·, ·) is not a covariance. In fact, +from (56) and (58), we see that for t ↓ 0, Qa,b(w, z) does not satisfy the inequality covariance +Qa,b(1, t) ≤ +� +Qa,b(1, 1)Qa,b(t, t). +(59) +(ii) Take a > −1 and b > 2. Assume first b > a + 1. From (56) we have that +� +Qa,b(1, 1)Qa,b(t, t) = 2b − 2 +(b − 1)B(α + 1, b + 1)t +a+b+1 +2 +. +(60) +On the other hand, from (57) it follows that for t ↑ ∞, the left-hand side of (59) is of order tb, whereas +the right-hand side of (59) is of order t +a+b+1 +2 +. Thus, Qa,b(·, ·) can not be a covariance function for +a > −1 and b > a + 1. +Assume now that a > −1 and 2 < b ≤ a + 1. Taking ǫ > 0 we can see that +Qa,b(1, 1 + ǫ) +≥ +1 +b − 1 +� 1 +0 +ua(2 + ǫ − 2u)bdu − +ǫb +(b − 1)(a + 1) += +2b +b − 1 +� 1 +0 +ua(1 + ǫ +2 − u)bdu − +ǫb +(b − 1)(a + 1), +26 + +where to get the inequality we have used that u �→ (1+ǫ−u)b +(1−u)b is decreasing, with minimum +ǫb at u = 1. Using arguments similar to those used to prove [5, (2.15)], it can be shown that +Qa,b(1, 1 + ǫ) − +� +Qa,b(1, 1)Qa,b(1 + ǫ, 1 + ǫ) ≥ Aǫ2 − Bǫb+1 − Cǫb, +where A, B and C are positive constants. Since we can take ǫ > 0 small enough to make the right- +hand side of the above inequality strictly positive, we conclude that Qa,b(·, ·) in not a covariance +function for a > −1 and 2 < b ≤ a + 1. +□ +On the basis of Theorem 4.1 and Lemma 4.3, we conjecture that Qa,b(·, ·) is also a finite positive definite +function when both conditions a > −1 and −1 < b < 0 with a + b + 1 > 0, hold. +Definition 4.4 A centered, real-valued Gaussian process ζ = {ζt, t ≥ 0} with covariance function (53), +whose parameters a and b satisfy the conditions given in Theorem 4.1, will be called weighted sub-fractional +Brownian motion. +Theorem 4.5 Let ζ be the weighted sub-fractional Brownian motion with parameters a and b. +(i) ζ is a self-similar process of index (a + b + 1)/2, i.e. for any c > 0, +(ζ(ct))t≥0 +d= +� +c(1+b+a)/2ζ(t) +� +t≥0 . +(ii) Assume that −1 < a ≤ 0 and b + a > 0. There exists a constant M > 0 such that +E +� +(ζ(t) − ζ(s))2� +≤ M|t − s|b+a+1, +0 ≤ s, t < ∞. +(61) +In particular, due to Kolmogorov’s continuity theorem, ζ possesses a continuous version with paths +which are locally-H¨older continuous with index δ, for any 0 < δ < (a + b + 1)/2. +(iii) For 0 ≤ r < v ≤ s < t there holds +Q(r, v, s, t) := E [(ζ(t) − ζ(s))(ζ(v) − ζ(r))] += +Qa,b(t, v) − Qa,b(t, r) − Qa,b(s, v) + Qa,b(s, r) += +1 +1 − b +�� v +r +ua � +(t − u)b − (s − u)b� +du + +� r +0 +ua � +(t + r − 2u)b − (s + r − 2u)b� +du +− +� v +0 +ua � +(t + v − 2u)b − (s + v − 2u)b� +du +� +. +(iv) (Long-range dependence) For b ∈ (0, 1) ∪ (1, 2) and 0 ≤ r < v ≤ s < t, +lim +T→∞ T 2−bQ(r, v, s + T, t + T) = +b +(a + 1)(a + 2)(t − s)(va+2 − ra+2). +(62) +(v) ζ is not a Markov process. +27 + +Proof: Since ζ is a Gaussian process, the proofs are based on properties of its covariance function Qa,b +given by 53. +(i) Let c be a positive constant and t ≥ 0. Then, +Qa,b(ct, ct) += +1 +1 − b +� ct +0 +sa � +(ct − s)b + (ct − s)b − (2ct − 2s)b� +ds = 2 − 2b +1 − b +� ct +0 +sa(ct − s)b ds += +cb+a+1Qa,b(t, t). +(ii) For 0 < s ≤ t we have +E +� +(ζ(t) − ζ(s))2� += +2 +� t +s +ua(t − u)bdu + 2 +� s +0 +ua(t + s − 2u)bdu +−2b +�� t +0 +ua(t − u)bdu + +� s +0 +ua(s − u)bdu +� +. +(63) +Again, we work each term in (63). Notice that +� t +s +ua(t − u)bdu ≤ (t − s)b +� t +s +uadu ≤ (t − s)b +�ta+1 − sa+1 +a + 1 +� +≤ ca(t − s)a+b+1, +(64) +where the last inequality is obtained using that t → ta+1 is (a + 1)-H¨older continuous and ca is a +positive constant. Also, +� s +0 +ua(t + s − 2u)bdu += +�1 +2 +�a+1 � 2s +0 +ua(t + s − u)b du = +�1 +2 +�a+1 +(t + s)a+b+1 +� +2s +t+s +0 +ua(1 − u)bdu +≤ +�1 +2 +�a+1 +(t + s)a+b+1B(a + 1, b + 1), +(65) +and +� t +0 +ua(t − u)bdu = ta+b+1 +� 1 +0 +ua(1 − u)bdu = ta+b+1B(a + 1, b + 1). +(66) +Since a+b+1 > 1, the mapping t → ta+b+1 is convex. Hence, (t+s +2 )a+b+1 ≤ ta+b+1+sa+b+1 +2 +. Therefore, +2−a(t + s)a+b+1B(a + 1, b + 1) ≤ 2b(ta+b+1 + ta+b+1)B(a + 1, b + 1). Finally, from (63)-(66) we obtain +that +E +� +(ζ(t) − ζ(s))2� +≤ M|t − s|a+b+1 +for some positive constant M. +(iii) Follows immediately from (53). +(iv) Let us first show that +lim +T→∞ T 1−bQ(r, v, s + T, t + T) = 0. +(67) +Using (iii), the equalities +T 1−b +�(t + T − u)b − (s + T − u)b +b +� += T 1−b +� t+T +s+T +(w − u)b−1 dw = +� t +s +�w + T − u +T +�b−1 +dw +28 + +and the bounded convergence theorem, we obtain +lim +T→∞ T 1−b +� v +r +ua � +(t − u)b − (s − u)b� +du = +b +a + 1(t − s) +� +va+1 − ra+1� +. +Similarly we get +lim +T→∞ T 1−b +� r +0 +ua � +(t + r − 2u)b − (s + r − 2u)b� +du = +b +a + 1(t − s)ra+1. +The limit (67) follows from +lim +T→∞ T 1−b +�� v +r +ua � +(t − u)b − (s − u)b� +du + +� r +0 +ua � +(t + r − 2u)b − (s + r − 2u)b� +du +− +� v +0 +ua � +(t + v − 2u)b − (s + v − 2u)b� +du +� += +b +a + 1(t − s) +� +va+1 − ra+1 + ra+1 − va+1� += 0. +Now, observe that +lim +T→∞ T 2−bQ(r, v, s + T, t + T) = lim +T→∞ +T 1−bQ(r, v, s + T, t + T) +T −1 +. +Due to (67) we can use L’Hospital’s theorem to calculate the last limit. Recall that, +(1 − b)Q(r, v, s + T, t + T) += +� v +r +ua � +(T + t − u)b − (T + s − u)b� +du +− +� v +0 +ua � +(T + t + v − 2u)b − (T + s + v − 2u)b� +du ++ +� r +0 +ua � +(T + t + r − 2u)b − (T + s + r − 2u)b� +du += +b +� v +r +ua +� t +s +(T + h − u)b−1dhdu − b +� v +0 +ua +� t +s +(T + h + v − 2u)b−1 dh du ++b +� r +0 +ua +� t +s +(T + h + r − 2u)b−1 dh du. +Therefore, +(1 − b)T 1−bQ(r, v, s + T, t + T) += +b +� v +r +ua +� t +s +� +1 + h − u +T +�b−1 +dh du − b +� v +0 +ua +� t +s +� +1 + h + v − 2u +T +�b−1 +dh du ++b +� r +0 +ua +� t +s +� +1 + h + r − 2u +T +�b−1 +dh du. +29 + +Applying L’Hospital’s rule we have that +lim +T→∞(1 − b)T 2−bQ(r, v, s + T, t + T) += +lim +T→∞ +� +−b(b − 1) +� v +r +ua +� t +s +� +1 + h − u +T +�b−2 +(h − u) dh du +� ++ lim +T→∞ +� +b(b − 1) +� v +0 +ua +� t +s +� +1 + h + v − 2u +T +�b−2 +(h + v − 2u) dh du +� +− lim +T→∞ +� +b(b − 1) +� r +0 +ua +� t +s +� +1 + h + r − 2u +T +�b−2 +(h + r − 2u) dh du +� +, +where +−b(b − 1) +� v +r +ua +� t +s +� +1 + h − u +T +�b−2 +(h − u) dh du +T→∞ +−−−−→ +−b(b − 1) +� v +r +ua +� t +s +(h − u) dh du += +−b(b − 1) +�t2 − s2 +2 +va+1 − ra+1 +a + 1 +− (t − s)va+2 − ra+2 +a + 2 +� +. +Similarly, one can see that +b(b − 1) +� v +0 +ua +� t +s +� +1 + h + v − 2u +T +�b−2 +(h + v − 2u) dh du +T→∞ +−−−−→ +b(b − 1) +�t2 − ss +2 +va+1 +a + 1 − a(t − s) +va+2 +(a + 1)(a + 2) +� +, +and +−b(b − 1) +� r +0 +ua +� t +s +� +1 + h + r − 2u +T +�b−2 +(h + r − 2u) dh du +T→∞ +−−−−→ +−b(b − 1) +�t2 − ss +2 +ra+1 +a + 1 − a(t − s) +ra+2 +(a + 1)(a + 2) +� +. +Putting all these limits together we obtain (62). +□ +(v) The result follows using [16, Proposition 13.7] (see also [13, Section III.8]) and the fact that the +function Qa,b given in (53) does not satisfy +Qa,b(s, t) = Qa,b(s, r)Qa,b(r, t) +Qa,b(r, r) +, +s ≤ r ≤ t. +□ +Theorem 4.6 Let b ∈ (0, 1)∪(1, 2], and {ζ(t), t ≥ 0} the weighted sub-fractional Brownian motion with pa- +rameters a and b. The process +� +T − a+b +2 (ζ(t + T) − ζ(T)), t ≥ 0 +� +converges in law to +� +2bbB(a+1,b) +b−1 +B(t), t ≥ 0 +� +as T → ∞, where {B(t), t ≥ 0} is a Brownian motion. +30 + +Proof Since ζ is a Gaussian process, to prove the desired convergence it suffices to show convergence of +the covariance function of the rescaled processes. Without loss of generality we assume that 0 < s ≤ t. +From (53) we have that +Qa,b(t + T, s + T) + Qa,b(T, T) − Qa,b(t + T, T) − Qa,b(T, s + T) += +1 +1 − b +�� s+T +0 +ua � +(T + t − u)b + (T + s − u)b − (2T + s + t − 2u)b� +du ++ +� T +0 +ua � +(T − u)b + (T − u)b − (2T − 2u)b� +du − +� T +0 +ua � +(T + t − u)b + (T − u)b − (2T + t − 2u)b� +du +− +� T +0 +ua � +(T + s − u)b + (T − u)b − (2T + s − 2u)b� +du +� += +1 +1 − b +�� T+s +T +ua � +(T + t − u)b + (T + s − u)b� +du − +� T+s +T +ua(2T + t + s − 2u)bdu ++ +� T +0 +ua � +(2T + s − 2u)b − (2T − 2u)b� +du − +� T +0 +ua � +(2T + t + s − 2u)b − (2T + t − 2u)b� +du +� +. +We are going to deal with each term separately. Changing variables u − T → u we get +T −a +� T+s +T +ua � +(T + t − u)b + (T + s − u)b� +du +(68) += +� s +0 +� +1 + u +T +�a � +(t − u)b + (s − u)b� +du T→∞ +→ +� s +0 +� +(t − u)b + (s − u)b� +du = tb+1 + sb+1 − (t − s)b+1 +b + 1 +, +and +T −a +� T+s +T +ua(2T + s + t − 2u)bdu = +� s +0 +(1 + u +T )a(s + t − 2u)bdu T→∞ +→ +(s + t)b+1 − (t − s)b+1 +2(b + 1) +. +(69) +Also, we have that (u = Tv) +T −a +� T +0 +ua � +(2T + s − 2u)b − (2T − 2u)b� +du += +T +� 1 +0 +ua � +(2T(1 − u) + s)b − (2T(1 − u))b� +du += +bT +� 1 +0 +ua +� s +0 +(2T(1 − u) + w)b−1 dw du += +T bb +� 1 +0 +ua +� s +0 +� +2(1 − u) + w +T +�b−1 +dw du. +(70) +Notice that +lim +T→∞ +� 1 +0 +ua +� s +0 +� +2(1 − u) + w +T +�b−1 +dw du = 2b−1B(a + 1, b)s. +(71) +Therefore, since b ∈ (0, 1) ∪ (1, 2], from (68), (69), (70) and (71) we conclude that +lim +T→∞ T −a−b +� +Qa,b(t + T, s + T) + Qa,b(T, T) − Qa,b(t + T, T) − Qa,b(T, s + T) +� += 2bbB(a + 1, b) +b − 1 +s +(72) +The proof finishes noticing that q(s, t) = t ∧ s is the covariance function of the standard Brownian motion. +□ +31 + +References +[1] Anderson, Kevin Karl. 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Appl. 43, No. 4, 620-632. +33 + diff --git a/rtFJT4oBgHgl3EQfbiyy/content/tmp_files/load_file.txt b/rtFJT4oBgHgl3EQfbiyy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bdac868b451387d5c0fb32996e2703e7788c7f1b --- /dev/null +++ b/rtFJT4oBgHgl3EQfbiyy/content/tmp_files/load_file.txt @@ -0,0 +1,1097 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf,len=1096 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='11540v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='PR] 27 Jan 2023 Occupation time fluctuations of an age-dependent critical binary branching particle system J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' L´opez-Mimbela∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Murillo-Salas† J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Ram´ırez-Gonz´alez‡ Abstract We study the limit fluctuations of the rescaled occupation time process of a branching particle system in Rd, where the particles are subject to symmetric α-stable migration (0 < α ≤ 2), critical binary branching, and general non-lattice lifetime distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' We focus on two different regimes: lifetime distributions having finite expectation, and Pareto-type lifetime distributions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' distributions belonging to the normal domain of attraction of a γ-stable law with γ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' In the latter case we show that, for dimensions αγ < d < α(1 + γ), the rescaled occupation time fluctuations converge weakly to a centered Gaussian process whose covariance function is explicitly calculated, and we call it weighted sub-fractional Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Moreover, in the case of lifetimes with finite mean, we show that for α < d < 2α the fluctuation limit turns out to be the same as in the case of exponentially distributed lifetimes studied by Bojdecki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' [7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' We also investigate the maximal parameter range allowing existence of the weighted sub-fractional Brownian motion and provide some of its fundamental properties, such as path continuity, long-range dependence, self-similarity and the lack of Markov property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Key words and phrases: branching particle systems, critical binary branching, Pareto-type tail lifetimes, occupation time fluctuations, sub-fractional motion, renewal theorem, long-range dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' MSC 2000 subject classifications: 60J80, 60E10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' 1 Introduction and main results Our aim in this paper is to investigate the occupation time fluctuations of a population in Rd which evolves as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' During its lifetime S, any given individual independently develops a spherically symmetric α-stable process with infinitesimal generator the fractional power ∆α := −(−∆)α/2 of the Laplacian, 0 < α ≤ 2, and at the end of its life it either disappears, or is replaced at the site where it died by two newborns, each event occurring with probability 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' The population starts off from a Poisson random ∗Centro de Investigaci´on en Matem´aticas, Guanajuato, Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' jalfredo@cimat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='mx †Departamento de Matem´aticas, Universidad de Guanajuato, Guanajuato, Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' amurillos@ugto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='mx ‡Centro de Investigaci´on en Matem´aticas, Guanajuato, Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' hermenegildo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='ramirez@cimat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='mx 1 field having the Lebesgue measure Λ as its intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Along with the usual independence assumptions in branching systems, we also assume that the particle lifetimes have a general non-lattice distribution, and that any individual in the initial population has age 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' We focus on two different regimes for the distribution of S: either S has finite mean µ > 0 or S has a distribution function F such that F(0) = 0, F(x) < 1 for all x ≥ 0, and 1 − F(t) ∼ 1 tγΓ(1 − γ) as t → ∞, (1) where 0 < γ < 1 and Γ denotes the usual Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Let Z(t) be the counting measure in Rd whose atoms are the positions of particles alive at time t, and let Z ≡ {Z(t), t ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Recall that the occupation time of the measure-valued process Z is again a measure-valued process J ≡ {J(t), t ≥ 0} which is given by ⟨ϕ, J(t)⟩ := � t 0 ⟨ϕ, Z(s)⟩ ds, t ≥ 0, for all bounded measurable functions ϕ : Rd → R+, where the notation ⟨ϕ, ν⟩ means � ϕ dν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Following [12] and [7], for each T > 0 we introduce the rescaled occupation time process JT (t) := J(Tt) defined by ⟨ϕ, JT (t)⟩ = � Tt 0 ⟨ϕ, Z(s)⟩ds = T � t 0 ⟨ϕ, Z(Ts)⟩ ds, t ≥ 0, and the rescaled occupation time fluctuation process {JT (t), t ≥ 0} given by ⟨ϕ, JT (t)⟩ := 1 HT � ⟨ϕ, JT (t)⟩ − E⟨ϕ, JT (t)⟩ � , t ≥ 0, where HT is a normalization factor such that HT → ∞ as T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' It was shown in [17] that, due to criticality of the branching and invariance of Λ for the α-stable semigroup, E⟨ϕ, JT (t)⟩ = Tt⟨ϕ, Λ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Hence, the rescaled occupation time fluctuation process takes the form ⟨ϕ, JT (t)⟩ := 1 HT � ⟨ϕ, JT (t)⟩ − Tt⟨ϕ, Λ⟩ � , t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' (2) The Markovian case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' the case of exponentially distributed particle lifetimes, has been thoroughly investigated by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Bojdecki, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Gorostiza and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Talarczyk in a series of seminal works, see [5, 7, 8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Among other results, they showed that when S possesses an exponential distribution and α < d < 2α, the occupation time fluctuation process, properly rescaled, converges weakly toward a Gaussian process in the space C([0, η], S′(Rd)) of continuous paths w : [0, η] → S′(Rd) for any η > 0, where S′(Rd) denotes the space of tempered distributions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' the strong dual of the space S(Rd) of rapidly decreasing smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' The limit process has a simple spatial structure whereas the temporal structure is characterized by that of sub-fractional Brownian motion (sub-fBm), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' a continuous centered Gaussian process {ζt, t ≥ 0} with covariance function C(s, t) := sh + th − 1 2 � (s + t)h + |s − t|h� , s, t ≥ 0, (3) with h = 3 − d/α (h ∈ (1, 2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' see [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' According to [7], sub-fBm exists for all h ∈ (0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' For h ̸= 1 this process does not have stationary increments, but possesses the so-called long-range dependence property, and for h = 1 it reduces to Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' 2 It is known [2] that the process Z fails to be Markovian if S does not have an exponential distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' There are relatively few publications on models related to non-Markovian spatial branching systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Laws of large numbers for the occupation times of Z have been investigated in [19] and [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Diffusion limit-type approximations for branching systems with non-exponential particle lifetimes were developed in [14] and [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Existence of a non-trivial equilibrium distribution for such kind of models was studied in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Assume that F is a general absolutely continuous function obeying (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' In this paper we prove that for dimensions satisfying αγ < d < α(1 + γ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' the occupation time fluctuation limit exists and is a centered Gaussian process whose covariance function has a simple spatial structure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' but its temporal structure is dictated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' for the case d ̸= α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' by a fractional noise with covariance function Q(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' t) := � d α − 1 �−1 � s∧t 0 rγ−1 � (s − r)2−d/α + (t − r)2−d/α − (t + s − 2r)2−d/α� dr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' t ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' (4) whereas for the case d = α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' the limit is a centered Gaussian process whose covariance function has a temporal structure determined by K(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' t) := � s∧t 0 rγ−1 � (s + t − 2r) ln(s + t − 2r) − (s − r) ln(s − r) − (t − r) ln(t − r) � dr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' The special but important case of particle lifetimes with finite mean is dealt with in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='2, where we show that for dimensions satisfying α < d < 2α the limit process is centered Gaussian, with covariance function of the form (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Hence, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='2 extends Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='2 in [8] to the case of non-exponential particle lifetimes with finite mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Moreover, in this case the effect of the lifetime distribution becomes apparent only through its mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' To obtain these results we follow the method of proof used in [8], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' the space-time random field weak convergence approach developed in [6], combined with the Feynman-Kac formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' However the adaptation to our case of such method is far from being straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Besides the lack of Markov property of Z, in our more general scenario the use of a Feynman-Kac formula is much more involved than in [8] due to the fact that the renewal function of F is in general nonlinear, in contrast to the linear renewal function of exponential lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Notice that the function (4) is a special case of the function Qa,b given by Qa,b(s, t) := 1 1 − b � s∧t 0 ra � (s − r)b + (t − r)b − (t + s − 2r)b� dr, s, t ≥ 0, a, b ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' (5) Several other interesting cases arise as special instances of (5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='2 bellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' This motivated our second goal in this paper, which is to determine suitable values of the parameters a, b ∈ R for which Qa,b is a covariance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' It turns out that, if the parameters a, b are restricted to the domains a > −1 and b ∈ [0, 2] with b ̸= 1, the function Qa,b is positive definite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' A centered real-valued Gaussian process with covariance function (5) will be called weighted sub-fractional Brownian motion, in analogy to the weighted fractional Brownian motion introduced in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' We recall that a weighted fractional 3 Brownian motion is a centered Gaussian process η := {η(t), t ≥ 0} with covariance function of the form � s∧t 0 ra � (s − r)b + (t − r)b� dr, s, t ≥ 0, for a > −1, |b| ≤ 1 and |b| < 1 + a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' In Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='5 we show that any weighted sub-fractional Brownian motion {ς(t), t ≥ 0} possesses long memory (also called long-range dependence), in the sense that E � (ς(t + T) − ς(s + T))(ς(v) − ς(r)) � ∼ T b−2 b (a + 1)(a + 2)(t − s)(va+2 − ra+2) as T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' It is worth to mention that the weighted fractional Brownian motion η also exhibits the long-range depen- dence property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' In this case, E � (η(t + T) − η(s + T))(η(v) − η(r)) � ∼ T b−1 b a + 1(t − s)(va+1 − ra+1) as T → ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' see [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' In Section 2 we prove a recursive relation for the Laplace functional of the branching particle system which we will need in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Section 3 is devoted to the proof of the main theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Finally, in Section 4 we investigate the maximal range of parameters a and b for which (5) becomes a covariance function, and provide several fundamental properties of weighted sub-fractional Brownian motion, such as path continuity, long-range dependence, self-similarity and the lack of Markov property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' In particular, the limit process obtained in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='1 (i) enjoys such properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' 2 Laplace functional In this section we will compute the Laplace functional of the occupation time process of Z in a general setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=', we only assume that the branching law is characterized by its probability generating function h(s) = �∞ k=0 pksk, |s| ≤ 1, and the particle lifetimes by a general distribution function F with support in [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' The symmetric α-stable motion in Rd will be denoted by ξ = {ξt, t ≥ 0} and by T = {Tt, t ≥ 0} its semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' By definition Zt(A) is the number of individuals living in A ∈ B(Rd) at time t ≥ 0, where B(Rd) denotes the system of Borel set in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Let {Sk, k ≥ 1} be a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' random variables with common distribution function F, and let Nt = ∞ � k=1 1{Wk≤t} and U(t) = ∞ � n=1 F ∗n(t), t ≥ 0, be the respective renewal process and renewal function, where the random sequence {Wk, k ≥ 0} is defined recursively by W0 = 0, Wk+1 = Wk + Sk, k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' 4 Define g(s) := h(1 − s) − (1 − s), |s| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Notice that in the case of critical binary branching h(s) = s + 1 2(1 − s)2 and g(s) = 1 2s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Now, for any nonnegative Ψ ∈ S(Rd+1), we define the function vΨ(x, r, t) := Ex � 1 − e− � t 0 ⟨Ψ(·,s+r),Zs⟩ ds� , x ∈ Rd, r, t ≥ 0, (6) where Ex denotes the expectation operator in a population starting with one particle of age 0, located at the position x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='1 The function vΨ(x, r, t) satisfies the integral equation vΨ(x, r, t) = Ex � 1 − e− � t 0 Ψ(ξs,r+s) ds� − � t 0 Ex � e− � u 0 Ψ(ξs,r+s) dsg � vΨ(ξu, r + u, t − u) �� dU(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' (7) Proof: Formula (7) obviously holds for t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Let t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' By conditioning on the first branching time we get, 1−vΨ(x, r, t) = Ex � e− � t 0 Ψ(ξs,r+s)ds1{S1>t} � +Ex � e− � S1 0 Ψ(ξs,r+s)dsh � 1 − vΨ � ξS1, r + S1, t − S1 �� 1{S1≤t} � , or equivalently, vΨ(x, r, t) =Ex �� 1 − e− � t 0 Ψ(ξs,r+s)ds � 1{S1>t} + � 1 − e− � S1 0 Ψ(ξs,r+s)ds � 1{S1≤t} − e− � S1 0 Ψ(ξs,r+s)dsg(vΨ(ξS1, r + S1, t − S1))1{S1≤t} � + Ex � e− � S1 0 Ψ(ξs,r+s)dsvΨ(ξS1, r + S1, t − S1)1{S1≤t} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' (8) Next, we consider the event [S1 ≤ t] and write ξx = {ξx s , s ≥ 0} for a symmetric α-stable motion starting in x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' Proceeding as above with r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' t and x replaced respectively by r+S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' t−S1 and ξS1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' and designating EξS1(·) the expected value starting with a particle at position ξS1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' given the σ−algebra σ((ξs)0≤s≤S1 ∪ S1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' we obtain vΨ(ξx S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' r + S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content=' t − S1)1{S1≤t} = EξS1 \uf8ee \uf8f0 \uf8eb \uf8ed1 − e − � t−S1 0 Ψ � ξ ξx S1 u ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFJT4oBgHgl3EQfbiyy/content/2301.11540v1.pdf'} +page_content='r+S1+u � du \uf8f6 \uf8f8 1{W1≤t 0) of particles. In the case of attraction, the problem of a finite number of +particles and their bound states has a physical meaning. In the classical limit, this +is modeled by rapidly decreasing boundary conditions. In the case of repulsion, the +problem corresponding to a gas of particles with a finite density is of interest. +It is well known that inverse scattering method for integration of the NLS equa- +tion is based on the so-called Zakharov-Shabat and Ablowitz-Kaup-Newell-Segur +scattering problem (see: [1, 16, 8]). +In 1971, V. Zakharov and A. Shabat [16] showed that the NLS equation can +be solved by means of the inverse scattering transform (IST) technique. The IST +as a method to solve the initial-value problem for the NLS equation has been +extensively studied in the literature, both in the focusing (χ = −1) and in the +defocusing (χ = 1) dispersion regimes. The IST for the defocusing NLS equation +with nonzero boundary conditions was first studied in 1973 by V. Zakharov and A. +Shabat [17] and a detailed study can be found in the monograph [8]. +In the work [4] a rigorous theory of the IST for the defocusing nonlinear Schr¨odinger +equation with nonvanishing boundary values u± = u0eiθ± as x → ±∞ is presented. +The IST theory for the defocusing NLS equation with nonzero boundary conditions +Key words and phrases. Defocusing nonlinear Schr¨odinger equation; Zakharov-Shabat system; +inverse scattering theory; nonzero boundary condition; self-consistent source. +1 +arXiv:2301.04588v1 [math.AP] 11 Jan 2023 + +2 +ANVAR REYIMBERGANOV +was studied by B. Gino, F. Emily and B. Prinari [2] and the focusing case has been +studied by G. Biondini, G. Kovaˇci´c [3], F. Demontis, B. Prinari, C. van der Mee, +F. Vitale [5]. +V.K. Melnikov [10, 11] showed that the NLS equation remains its integrability +by the inverse scattering method, if a source is added to them in the form of a +combination of eigenfunctions of the corresponding spectral problem. Namely, the +term “self-consistent source” was introduced in the works of V.K. Melnikov. +The NLS equation with the self-consistent sources in various classes of functions +were considered by A.B. Khasanov [9], I.D. Rakhimov [13], A.B. Yakhshimuratov +[15]. +In the matrix case, the inverse scattering theory for the matrix Zakharov-Shabat +system was investigated by P. Barbara, F. Demontis and C. Van der Mee [12, 6, 7] +and applied for the integration of the matrix NLS equation. In [14], the matrix +NLS equation with the self-consistent source was considered in the class of rapidly +decreasing matrix functions. +2. Formulation of the problem +We consider the following system of equations +iut − 2u |u|2 + uxx = −2i +N +� +n=1 +(f ∗ +1,ng∗ +2,n + f2,ng1,n) +(2) +∂f1,n +∂x +− u∗f2,n + iξnf1,n = ∂f2,n +∂x +− uf1,n − iξnf2,n = 0, n = 1, 2, ..., N, +(3) +∂g1,n +∂x +− ug2,n − iξng1,n = ∂g2,n +∂x +− u∗g1,n + iξng2,n = 0, n = 1, 2, ..., N +(4) +under the initial condition +(5) +u(x, 0) = u0(x), +x ∈ R. +Here the initial function u0(x), x ∈ R satisfies the following properties: +1) +(6) +� 0 +−∞ +(1 − x) +��u0(x) − ρeiα−�� dx + +� ∞ +0 +(1 + x) +��u0(x) − ρeiα+�� dx < ∞, +where ρ > 0 and 0 ≤ α± < 2π are arbitrary constants. +2) The system of equations (3) with coefficient u0(x) possesses exactly N eigen- +values ξ1(0), ξ2(0), ... , ξN(0). +We assume that the solution u(x, t) of the equation (2) is sufficiently smooth +and tends to its limits sufficiently rapidly as x → ±∞, i.e., for all t ≥ 0 satisfies +the condition +� 0 +−∞ +(1 − x) +���u(x, t) − ρeiα−−2iρ2t��� dx+ +� ∞ +0 +(1 + x) +���u(x, t) − ρeiα+−2iρ2t��� dx ++ +� ∞ +−∞ +2 +� +k=1 +���� +∂ku(x, t) +∂xk +���� dx < ∞. +(7) +It follows from this condition that the left-hand side of equation (2) for all t ≥ 0 +tends to zero as x → ±∞. +Taking this into account, the solutions Fn(x, t) = +(f1,n, f2,n)T and Gn(x, t) = (g1,n, g2,n)T of equations (3) and (4), respectively, are +to be chosen so that the expression in the right-hand side of equation (2) for all + +INTEGRATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION ... +3 +t ≥ 0 should tend to zero rapidly enough as x → ±∞. This can be done in the +following two ways: +(A) Let the functions Fn(x, t) and Gn(x, t) be the eigenfunctions of equations +(3) and (4) respectively, corresponding to the eigenvalues ξn(t). In this case, the +functions Fn(x, t) and Gn(x, t) belong to the L2(R) for all t ≥ 0. We assume that +(8) +� ∞ +−∞ +GT +n(s, t)Fn(s, t)ds = An(t), +t ≥ 0, +n = 1, 2, ..., N. +Where An(t) are given and the continuous functions of t. +(B) Let the function Fn(x, t) be the eigenfunctions of equations (3) corresponding +to the eigenvalues ξn(t) and let the function Gn(x, t) be unbounded solution of the +equation (4), satisfying equalities +(9) +f1,ng1,n − f2,ng2,n = Bn(t), +t ≥ 0, +n = 1, 2, ..., N. +Where the functions Bn(t) are given continuous real valued functions of t. +Our main goal is to obtain representations for solutions u(x, t), Fn(x, t), Gn(x, t), +n = 1, 2, ..., N of problem (2)-(7), within the framework of the inverse scattering +method. +3. Preliminaries +Let a function u(x, t) belong to the class of functions (7). In this section, we give +well known [8], necessary information concerning the theory of direct and inverse +scattering problems for the operator +L(t) = i +� +∂ +∂x +−u∗(x, t) +u(x, t) +− ∂ +∂x +� +, +t ≥ 0. +Consider the equation +(10) +(L(t) − ξI)f = 0 +with respect to an unknown 2 × 2 square matrix function f(x, ξ, t). Here ξ is a +complex spectral parameter. +We introduce a new spectral parameter p as follows p = +� +ξ2 − ρ2. The variable +p is then thought of as belonging to a Riemann surface Γ consisting of a sheet Γ+ +and a sheet Γ− which both coincide with the complex plane cut along the semi lines +Σ = (−∞, −ρ] ∪ [ρ, ∞) with its edges glued in such a way that p(ξ) is continuous +through the cut. The variable p is thought of as belonging to the complex plane +consisting of the upper half complex plane Γ+ and the lower half complex plane Γ− +glued together along the whole real line. For all ξ ∈ Σ, the branch of the square +root is fixed by the condition sign p(ξ) = sign ξ. +For ξ ∈ Σ, we define matrix Jost solutions f −(x, ξ, t) and f +(x, ξ, t) from the +right and the left, respectively as those square matrix solutions to (10) satisfying +asymptotics +(11) +f ± ∼ E±(x, ξ, t) +as +x → ±∞ +where +E±(x, ξ, t) = +� +1 +− i(ξ−p) +ρ +e−iα±+2iρ2t +i(ξ−p) +ρ +eiα±−2iρ2t +1 +� +e−ipσ3x. + +4 +ANVAR REYIMBERGANOV +Here and everywhere below we will use the standard Pauli matrices +σ1 = +� 0 +1 +1 +0 +� +, +σ2 = +� 0 +−i +i +0 +� +, +σ3 = +� 1 +0 +0 +−1 +� +. +If a function u(x, t) belongs to the class of functions (7), then such a solution to +equations (10) exists and is unique. +It can be shown that +(12) +d +dx det f ±(x, ξ, t) = 0. +From (12) and (11) it follows that +(13) +det f ±(x, ξ, t) = 2p(ξ − p) +ρ2 +. +The system (10) is invariant with respect to the involution +(14) +¯f ±(x, ξ, t) = σ1f ±(x, ξ, t)σ1, +ξ ∈ Σ. +We now call the columns of +f +(x, ξ, t) = +� ¯ψ(x, ξ, t) ψ(x, ξ, t) +� +, f −(x, ξ, t) = (ϕ(x, ξ, t) +¯ϕ(x, ξ, t)) +the Jost solutions from the right and the left, respectively. For the Jost solutions +we get the following asymptotic estimates +(15) +ψ(x, ξ, t) ∼ +� +− i(ξ−p) +ρ +e−iα++2iρ2t +1 +� +eipx, +x → ∞, +¯ψ(x, ξ, t) ∼ +� +1 +i(ξ−p) +ρ +eiα+−2iρ2t +� +e−ipx, +x → ∞, +(16) +ϕ(x, ξ, t) ∼ +� +1 +i(ξ−p) +ρ +eiα−−2iρ2t +� +e−ipx, +x → −∞, +¯ϕ(x, ξ, t) ∼ +� +− i(ξ−p) +ρ +e−iα−+2iρ2t +1 +� +eipx, +x → −∞, +Since f −(x, ξ, t) and f +(x, ξ, t) are square matrix solutions of the homogeneous +first order equation (10), we necessarily have for ξ ∈ Σ +(17) +f −(x, ξ, t) = f +(x, ξ, t)S(ξ, t) +where S(ξ, t) is the transition coefficient matrix. +From the involution property (14) for ξ ∈ Σ, it follows that +¯S(ξ, t) = σ1S(ξ, t)σ1. +Hence, we have +(18) +S(ξ, t) = +� +a(ξ, t) +¯b(ξ, t) +b(ξ, t) +¯a(ξ, t) +� +. +Coefficients a(ξ, t) and b(ξ, t) are called scattering coefficients. From relations (13) +and (18) we obtain +a(ξ, t)¯a(ξ, t) − b(ξ, t)¯b(ξ, t) = 1. + +INTEGRATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION ... +5 +By using (17) we can represent the scattering coefficients as +(19) +a(ξ, t) = +ρ2 +2p(ξ − p) det(ϕ(x, ξ, t), ψ(x, ξ, t)) +and +b(ξ, t) = +ρ2 +2p(ξ − p) det( ¯ψ(x, ξ, t), ϕ(x, ξ, t)). +If a function u(x, t) belongs to the class of functions (7), then for each x ∈ R +the Jost solutions ψ(x, ξ, t)e−ipx and ϕ(x, ξ, t)eipx are analytic for ξ ∈ Γ+ excluding +branch points ξ = ±ρ, there are asymptotes for |ξ| → ∞ +(20) +ϕ(x, ξ, t)eipx = +� +1 +i(ξ−p) +ρ +eiα−−2ip2t +� ++ O +�|1 + ξ − p| +|ξ| +� +, +(21) +ψ(x, ξ, t)e−ipx = +� +− i(ξ−p) +ρ +e−iα++2ip2t +1 +� ++ O +�|1 + ξ − p| +|ξ| +� +. +It follows from the analyticity properties of the Jost solutions and equality (19) +that the function a(ξ, t) can be analytically continued to the sheet Γ+ excluding +branch points ξ = ±ρ. +From (20) and (21) we obtain that for |ξ| → ∞, the function a(ξ, t) has the +asymptotics +(22) +a(ξ, t) = 1 + O +� 1 +|ξ| +� +as Im ξ > 0 +and +(23) +a(ξ, t) = e−iθ + O +� 1 +|ξ| +� +as Im ξ < 0 +where we recall θ = α+ − α−. +Similarly, the function ¯a(ξ, t) can be analytically continued to the sheet Γ−, +excluding branch points ξ = ±ρ. +It follows from the analyticity with respect to the function a(ξ, t) on Γ+ and from +the asymptotics (22), (23) that the function a(ξ, t) can have only a finite number +of zeros on the sheet Γ+. These zeros will be denoted by ξ1, ξ2, ..., ξN. In [4] it is +shown that all zeros are simple and all belong to the (−ρ, ρ). +It is seen from representation (19) that ξ = ξn the functions ϕ(x, ξ, t) and +ψ(x, ξ, t) are proportional to each other +(24) +ϕn(x, t) = cn(t)ψn(x, t), +¯ϕn(x, t) = c∗ +n(t) ¯ψn(x, t), +n = 1, 2, ..., N. +Where ϕn(x, t) = ϕ(x, ξn, t), ψn(x, t) = ψ(x, ξn, t). +The zeros of a(ξ, t) correspond to the eigenvalues of the equation (10). +The +equation (10) is self-adjoint, so its eigenvalues and thus the zeros of the function +a(ξ, t) are real. +Note, the vector functions +(25) +hn(x, t) = +∂ +∂ξ (ϕ(x, ξ, t) − cnψ(x, ξ, t))|ξ=ξn +˙a(ξn, t) +, n = 1, 2, ..., N +are a solution to the equations (L(t)−ξnI)hn = 0. Where ˙a(ξn, t) = +∂ +∂ξ a(ξ, t)|ξ=ξn. + +6 +ANVAR REYIMBERGANOV +From equality (25) it follows that +hn(x, t) ∼ −cn(t) +� +− i(ξn−pn) +ρ +e−iα−+2ip2t +1 +� +eipnx as x → −∞, +hn(x, t) ∼ +� +1 +i(ξn−pn) +ρ +eiα+−2ip2t +� +e−ipnx as x → ∞, +(26) +where pn = i +� +ρ2 − ξ2n. In particular, we have +(27) +det(ϕn, hn) = −2pn(ξn − pn) +ρ2 +cn, n = 1, 2, ..., N. +Definition 1. The set {a(ξ, t), b(ξ, t), ξn(t), cn(t), n = 1, 2, ..., N} is called the +scattering data for equation (10). +The direct scattering problem is to find the +scattering data via the given potentials u(x, t) and the inverse scattering problem +is to find the potentials u(x, t) of the equation (10) via the given scattering data. +Before we proceed further solving the inverse problem, it is convenient to in- +troduce a uniformization variable z (see [4, 8]) defined by the conformal mapping: +z = z(ξ) = ξ + p(ξ). Inverse mapping given by +ξ = 1 +2 +� +z + ρ2 +z +� +, p = z − ξ = 1 +2 +� +z − ρ2 +z +� +. +With this mapping the sheets Γ+ and Γ− of the Riemann surface Γ are, respectively, +mapped onto the upper and lower complex half-planes Im z > 0 and Im z < 0 of +the complex z-plane. The cut Σ on the Riemann surface is mapped onto the real +z axis. The segments [−ρ, ρ] on Γ+ and Γ− are mapped onto the upper and lower +semicircles of radius ρ and center at the origin of the z-plane. The neighborhood +of the point ξ = ∞ on Γ± with the condition ±Imξ > 0 is mapped into the +neighborhood of the point z = ∞, and the neighborhood of the point ξ = ∞ on Γ± +with the condition ±Imξ < 0 is mapped into the neighborhood of the point z = 0. +In terms of variable z, relation (17) can be written when Imz = 0 following form +(28) +f −(x, z, t) = f +(x, z, t)S(z, t) +here f ±(x, z, t) = f ±(x, ξ(z), t), S(z, t) = S(ξ(z), t) and one can obtain the sym- +metries of the scattering coefficients: +(29) +a(z, t) = ¯a +�ρ2 +z , t +� +, Imz ≥ 0, +b(z, t) = −¯b +�ρ2 +z , t +� +, Imz = 0. +Equality (29), together with the self-adjointness of the equation (10), ensure +that the scattering coefficient a(z, t) (¯a(z, t)) can only have zeros at zn = ξn + ivn +(¯zn = ξn − ivn), with −ρ < ξn < ρ and vn = +� +ρ2 − ξ2n > 0. +Taking into account the analyticity properties of a(z, t) in the upper half plane +Imz > 0 we can obtain the following representation +a(z, t) = +N +� +n=1 +z − zn +z − z∗n +exp +� +− 1 +2πi +� ∞ +−∞ +log(1 − |r(ζ, t)|2) +ζ − z +dζ +� +. + +INTEGRATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION ... +7 +Where r(z, t) ≡ b(z,t) +a(z,t) is called reflection coefficient. According to (23), for z → 0 +we obtain that +e−iθ = +N +� +n=1 +zn +z∗n +exp +� +− 1 +2πi +� ∞ +−∞ +log(1 − |r(ζ, t)|2) +ζ +dζ +� +. +If ϕn = +� ϕ1,n +ϕ2,n +� +is an eigenfunction of the equation (10) corresponding to +zn, then we define ¯ϕn = +� ϕ∗ +2,n +ϕ∗ +1,n +� +to be the eigenfunction of the equation (10) +corresponding to ¯zn. +It is well known that the inverse scattering theory of (10) can be formulated in +terms of the Gelfand-Levitan-Marchenko equations. The Jost solution ψ(x, z, t) of +the equation (10) can be represented in the following form +(30) +ψ(x, z, t) = +� +− iρ +z e−iα++2iρ2t +1 +� +e(x, z) + +� ∞ +x +K(x, y, t) +� +− iρ +z e−iα++2iρ2t +1 +� +e(y, z)dy, +here e(x, z) = e +i +2 +� +z− ρ2 +z +� +x and K(x, y) is a 2 × 2 matrix function which has to satisfy +the following Gelfand-Levitan-Marchenko equation: +K(x, y, t) + F(x + y, t) + +� ∞ +x +K(x, s, t)F(s + y, t)ds = 0, y ≥ x, +where K(x, y, t) and F(x + y, t) are defined as +K(x, y, t) = +� K11(x, y, t) +K12(x, y, t) +K21(x, y, t) +K22(x, y, t) +� +, F(x+y, t) = +� F1(x + y, t) +F ∗ +2 (x + y, t) +F2(x + y, t) +F1(x + y, t) +� +with +F1(x, t) = ρeiα+−2iρ2t +4πi +� ∞ +−∞ +b(z, t) +za(z, t) · e(x, z)dz − 1 +2 +N +� +n=1 +cn(t)ρe−iα++2iρ2t +˙a(zn, t)zn +· e(x, zn), +F2(x, t) = 1 +4π +� ∞ +−∞ +b(z, t) +a(z, t) · e(x, z)dz − 1 +2 +N +� +n=1 +icn(t) +˙a(zn, t) · e(x, zn). +In representations (30), the component K21(x, x, t) of the matrix K(x, y, t) have +relations with the potential +2K21(x, x, t) = ρeiα+−2iρ2t − u(x, t). +In the work [8], it was proven the uniquely determining of the potential u(x, t) by +the scattering data. +4. Time evolution +The use of the inverse scattering method for integration of the problem (2)-(7) is +based on the following. Let the function u(x, t) be a solution of equation (2), from +the class of functions (6). Consider equation (10) with a potential u(x, t) and find +the evolution from the scattering data. +Assuming that +(31) +FN+n = +� f ∗ +2,n +f ∗ +1,n +� +, GN+n = +� g∗ +2,n +g∗ +1,n +� +, +n = 1, 2, ..., N, + +8 +ANVAR REYIMBERGANOV +equation (2) can be represented as an equality of operators in the class of smooth +functions f(x, ξ, t) satisfying the equation (10): +∂L +∂t + [L, A] = i +2N +� +n=1 +[σ3, FnGT +n]. +Where [L, A] = LA − AL and +A = +� i |u|2 + 2iξ2 +−iu∗ +x − 2ξu∗ +iux − 2ξu +−i |u|2 − 2iξ2 +� +. +Lemma 1. Let f(x, ξ, t) be solution of the equation (10) and let φn(x, ξ, t), n = +1, 2, ..., 2N be any functions, which satisfy the conditions +(32) +∂φn +∂x = GT +nf, +n = 1, 2, ..., 2N. +Then, the function Gn(x, t) satisfy the equalities +(33) +GT +nσ3f + i(ξ − ξn)φn = 0 , n = 1, 2, ..., 2N +and the function +(34) +H = ∂f +∂t − Af + +2N +� +n=1 +Fnφn +satisfies the equation (10) for any ξ ∈ Σ. +4.1. Evolution equation for the scattering data in the case of a source +satisfying the conditions (A). Let us take matrix Jost solutions f −(x, ξ, t) and +f +(x, ξ, t) for ξ ∈ Σ as the solution f(x, ξ, t) and ξ = ξn, n = 1, 2, ..., N are +eigenvalues of the equation (10). +According to the definition of eigenfunctions, +there are αn(t) and βn(t) such that the relations hold +(35) +Fn(x, t) = αn(t)ψn(x, t), +Gn(x, t) = βn(t)σ1ϕn(x, t), n = 1, 2, ..., N +According to these relations, due to the assumptions (31), we obtain +(36) +FN+n(x, t) = α∗ +n(t) ¯ψn(x, t), GN+n(x, t) = β∗ +n(t)σ1 ¯ϕn(x, t), n = 1, 2, ..., N. +By definition functions Gn(x, t), belong to the L2(R) for all t ≥ 0 and matrix Jost +solutions f −(x, ξ, t), f +(x, ξ, t) are bounded for all ξ ∈ Σ. Therefore φ− +n ∈ L2(R) +and φ+ +n ∈ L2(R) for all t ≥ 0 and ξ ∈ Σ. Hence, by virtue of (33) it follows that at +any ξ ∈ Σ and n = 1, 2, ..., 2N the asymptotics +(37) +φ− +n (x, ξ, t) → 0 as x → −∞, +φ+ +n (x, ξ, t) → 0 as x → ∞ +are valid. So, from (32) for n = 1, 2, ..., 2N we obtain the following expressions +(38) +φ− +n = +� x +−∞ +GT +n(s,t)f −(s, ξ, t)ds, φ+ +n = − +� ∞ +x +GT +n(s, t)f +(s, ξ, t)ds. +Using the matrix Jost solutions f + and f − of equation (10), we rewrite equality +(34) in the form +(39) +H− = ∂f − +∂t − Af − + +2N +� +n=1 +Fnφ− +n + +INTEGRATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION ... +9 +and +(40) +H+ = ∂f + +∂t − Af + + +2N +� +n=1 +Fnφ+ +n . +These functions satisfy the equation (10) for any ξ ∈ Σ. Therefore, H+ and H− +are linearly dependent on f + and f −, respectively, i.e., there exist such C− +0 (ξ, t) +and C+ +0 (ξ, t) that the following identities hold +H−(x, ξ, t) = f −(x, ξ, t)C− +0 (ξ, t), H+(x, ξ, t) = f +(x, ξ, t)C+ +0 (ξ, t). +By virtue of the definition of the matrix A, from relations (39), (40) and from +asymptotics (11), (37) we obtain +H−(x, ξ, t) → −(iρ2 + 2iξp)E−(x, ξ, t)σ3, x → −∞, +(41) +H+(x, ξ, t) → −(iρ2 + 2iξp)E+(x, ξ, t)σ3, x → ∞. +(42) +By the uniqueness of the Jost solutions we get +(43) +H−(x, ξ, t) = −(iρ2 + 2iξp)f −(x, ξ, t)σ3, +H+(x, ξ, t) = −(iρ2 + 2iξp)f +(x, ξ, t)σ3. +We introduce the function H in the following form +H = H−(x, ξ, t) − H+(x, ξ, t)S(ξ, t). +Based on equalities (17) and (43), the function H can be rewritten in the form +(44) +H = (iρ2 + 2iξp)f +(x, ξ, t)[σ3, S(ξ, t)]. +On the other hand, by virtue of (17), (39) and (40) the equality +H =H−(x, ξ, t) − H+(x, ξ, t)S(ξ, t) = f +(x, ξ, t)St(x, ξ, t)+ ++ +2N +� +n=1 +[Fn(x, t)φ− +n (x, ξ, t) − Fn(x, t)φ+ +n (x, ξ, t)S(ξ, t)] +(45) +holds. +Based on equality (33) this relation becomes +H = f +(x, ξ, t)St(ξ, t)+ ++ +2N +� +n=1 +i +ξ − ξn +[Fn(x, t)GT +n(x, t)σ3f −(x, ξ, t) − Fn(x, t)GT +n(x, t)σ3f +(x, ξ, t)S(ξ, t)]. +Finally, based on (17), we obtain +(46) +H = f +(x, ξ, t)St(ξ, t). +Comparing equalities (44) and (46) we have +(iρ2 + 2iξp)f +(x, ξ, t)[σ3, S(ξ, t)] = f +(x, ξ, t)St(ξ, t). +Therefore, for all ξ ∈ Σ we have the relation +St(ξ, t) − (iρ2 + 2iξp)[σ3, S(ξ, t)] = 0, +i.e. +d +dta(ξ, t) = 0, +d +dtb(ξ, t) = −2(iρ2 + 2iξp)b(ξ, t). +Since, the function a(ξ, t) does not depend on t, hence we conclude that its zeros +ξn also do not depend on t. + +10 +ANVAR REYIMBERGANOV +Based on identities (39) and (40), we write the following equalities +H− +1 (x, ξn, t) = ∂ϕm(x, t) +∂t +− A(x, ξn, t)ϕm(x, t) + +2N +� +n=1 +Fn(x, t)φ− +1,n(x, ξn, t) +(47) +H+ +2 (x, ξn, t) = ∂ψm(x, t) +∂t +− A(x, ξn, t)ψm(x, t) + +2N +� +n=1 +Fn(x, t)φ+ +2,n(x, ξn, t) +(48) +By virtue of the definition of the matrix A, from relations (47), (48) and from +asymptotics (15), (16) we obtain +(49) +H− +1 (x, ξm, t) = (−iρ2 − 2iξmpm)ϕm(x, t), +H+ +2 (x, ξm, t) = (iρ2 + 2iξmpm)ψm(x, t). +We now introduce the following functions +Hm = H− +1 (x, ξm, t) − cm(t)H+ +2 (x, ξm, t), m = 1, 2, ..., 2N. +Using equalities (24), (49) the function Hm can be rewritten in the form +(50) +Hm = (−2iρ2 − 4iξmpm)ϕm(x, t). +Substituting instead of φ− +1,n(x, ξ, t) and φ+ +2,n(x, ξ, t) the expressions from (38) into +equalities (47), (48) and using (24), we obtain +(51) +H− +1 (x, ξm, t)−cm(t)H+ +2 (x, ξm, t) = dcm(t) +dt +ψm(x, t)+ +2N +� +n=1 +Fn(x, t) +� ∞ +−∞ +GT +n(s, t)ϕm(s, t)ds +If ξm ̸= ξn, according to equation (33) we get +� ∞ +−∞ +GT +n(s, t)ϕm(s, t)ds = 0. +According to (35) and (36), equality (51) can be rewritten in the form +(52) +H− +1 (x, ξm, t) − cm(t)H+ +2 (x, ξm, t) = dcm(t) +dt +ψm(x, t)+ ++ +�� ∞ +−∞ +GT +m(s, t)Fm(s, t)ds + +� ∞ +−∞ +GT +N+m(s, t)FN+m(s, t)ds +� +ϕm(x, t). +Comparing equalities (50) and (52) we obtain +(−2iρ2 − 4iξmpm)ϕm(x, t) = += dcm(t) +dt +ψm(x, t)+ +�� ∞ +−∞ +GT +m(s, t)Fm(s, t)ds + +� ∞ +−∞ +GT +N+m(s, t)FN+m(s, t)ds +� +ϕm(x, t). +Finally, using these equalities and taking into account (8) and (24) we determine +dcm(t) +dt += (−2iρ2 − 4iξmpm − Am(t) − A∗ +m(t))cm(t). +Thus, we have proved the following theorem. +Theorem 1. If functions u(x, t), Fk(x, t), Gk(x, t), k = 1, 2, ..., N are the solu- +tions of the problem (2)-(7) in the case of a source satisfying the conditions (A), +then the scattering data for the equation (10) satisfy the following relations +a(ξ, t) = a(ξ, 0), +b(ξ, t) = b(ξ, 0) exp(−2iρ2t − 4iξpt) +for +ξ ∈ Σ, + +INTEGRATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION ... +11 +ξk(t) = ξk(0), +k = 1, 2, ..., N. +ck(t) = ck(0) exp(−2iρ2t − 4iξkpkt − +� t +0 +(Ak(τ) + A∗ +k(τ))dτ), +k = 1, 2, ..., N. +4.2. Evolution equation for the scattering data in the case of a source +satisfying the conditions (B). Let us take matrix Jost solutions f −(x, ξ, t) and +f +(x, ξ, t) for ξ ∈ Σ as the solution f(x, ξ, t) and ξ = ξn, n = 1, 2, ..., N are +eigenvalues of the equation (10). According to the definition of eigenfunctions of +equation (10), there are αn(t) such that the relations +(53) +Fn(x, t) = αn(t)ψn(x, t), +FN+n(x, t) = α∗ +n(t) ¯ψn(x, t), +n = 1, 2, ..., N. +Due to the assumptions (B) the functions Gn(x, t) are unbounded functions. So, +there are βn(t) such that which follow the equalities +(54) +Gn(x, t) = +βn(t) +˙a(ξn, t)σ1ϕn(x, t) + σ1hn(x, t), +GN+n(x, t) = +β∗ +n(t) +˙¯a(ξn, t)σ1 ¯ϕn(x, t) + σ1¯hn(x, t), n = 1, 2, ..., N. +One can easily see from (9) and (27), that the quantities αn(t) satisfy the fol- +lowing equalities +(55) αn(t) = − +ρ2 +2pn(ξn − pn)Bn(t), +α∗ +n(t) = +ρ2 +2pn(ξn + pn)Bn(t), n = 1, 2, ..., N. +Using equalities (24), (33), (53), (54) and asymptotics (15), (26) we can verify +that at any ξ ∈ Σ and when x → ∞ the following asymptotics are valid: +Fnφ+ +n ∼ iαn(t) +ξ − ξn +� +(ξn−pn)2 +ρ2 +− i(ξn−pn) +ρ +e−iα++2iρ2t +i(ξn−pn) +ρ +eiα+−2iρ2t +1 +� +σ3E+(x, ξ, t), +FN+nφ+ +N+n ∼ iα∗ +n(t) +ξ − ξn +� +1 +− i(ξn+pn) +ρ +e−iα++2iρ2t +i(ξn+pn) +ρ +eiα+−2iρ2t +(ξn+pn)2 +ρ2 +� +σ3E+(x, ξ, t). +Taking into account of equalities (24), (33), (53), (54) and asymptotics (15), (26) +we are convinced that at any ξ ∈ Σ and x → −∞ there hold the asymptotics +Fnφ− +n ∼ −iαn(t) +ξ − ξn +� +1 +− i(ξn−pn) +ρ +e−iα−+2iρ2t +i(ξn−pn) +ρ +eiα−−2iρ2t +(ξn−pn)2 +ρ2 +� +σ3E−(x, ξ, t), +FN+nφ− +N+n ∼ −iα∗ +n(t) +ξ − ξn +� +(ξn+pn)2 +ρ2 +− i(ξn+pn) +ρ +e−iα−+2iρ2t +i(ξn+pn) +ρ +eiα−−2iρ2t +1 +� +σ3E−(x, ξ, t). +Using equalities (55) one can easily verify that at any ξ ∈ Σ the asymptotic +Fnφ+ +n + FN+nφ+ +N+n → 0 +for +x → ∞, +and +Fnφ− +n + FN+nφ− +N+n → 0 +for +x → −∞ +are valid. +Hence, it follows that the quantities H−(x, ξ, t) and H+(x, ξ, t) determined by +(39) and (40) satisfy equalities +(56) +H−(x, ξ, t) = f −(x, ξ, t)(−iρ2 − 2iξp)σ3, +H+(x, ξ, t) = f +(x, ξ, t)(−iρ2 − 2iξp)σ3. + +12 +ANVAR REYIMBERGANOV +Now, consider the function Hm of the form +Hm = H−(x, ξ, t) − H+(x, ξ, t)S(ξ, t). +Taking into account (56) we find that +(57) +Hm = (iρ2 + 2iξp)f +(x, ξ, t)[σ3, S(ξ, t)]. +From equalities (39), (40) and (17) it is easy to get that +H−(x, ξ, t) − H+(x, ξ, t)S(ξ, t) = f +(x, ξ, t)St(x, ξ, t)+ +(58) ++ +2N +� +n=1 +[Fn(x, t)φ− +n (x, ξ, t) − Fn(x, t)φ+ +n (x, ξ, t)S(ξ, t)]. +Using equalities (33), we obtain +H−(x, ξ, t) − H+(x, ξ, t)S(ξ, t) = f +(x, ξ, t)St(ξ, t)+ ++ +2N +� +n=1 +i +ξ − ξn +[Fn(x, t)GT +n(x, t)σ3f −(x, ξ, t) − Fn(x, t)GT +n(x, t)σ3f +(x, ξ, t)S(ξ, t)]. +By virtue of (17), it follows that +(59) +H−(x, ξ, t) − H+(x, ξ, t)S(ξ, t) = f +(x, ξ, t)St(ξ, t) +Comparing equalities (57) and (59) we have +(iρ2 + 2iξp)f +(x, ξ, t)[σ3, S(ξ, t)] = f +(x, ξ, t)St(ξ, t). +Therefore, for all ξ ∈ Σ we have +St(ξ, t) − (iρ2 + 2iξp)[σ3, S(ξ, t)] = 0, +i.e. +d +dta(ξ, t) = 0, +d +dtb(ξ, t) = −2(iρ2 + 2iξp)b(ξ, t). +Thus, we conclude that the function a(ξ, t) does not depend on t, so the zeros ξn(t) +of function a(ξ, t) do not depend on t. +Let us now find the evolution of the normalizing constants cm(t), m = 1, 2, ..., N. +We now introduce the following functions +(60) +Hm = H− +1 (x, ξm, t) − cm(t)H+ +2 (x, ξm, t), m = 1, 2, ..., N, +where +H− +1 (x, ξm, t) = ∂ϕm(x, t) +∂t +− A(x, ξm, t)ϕm(x, t) + +2N +� +n=1 +Φn(x, t)ϕ− +1,n(x, ξm, t), +(61) +H+ +2 (x, ξm, t) = ∂ψm(x, t) +∂t +− A(x, ξm, t)ψm(x, t) + +2N +� +n=1 +Φn(x, t)ϕ+ +2,n(x, ξm, t). +(62) +It is easy to show that +(63) +H− +1 (x, ξm, t) = (−iρ2 − 2iξmpm)ϕm(x, t), H+ +2 (x, ξm, t) = (iρ2 + 2iξmpm)ψm(x, t). +Substituting (63) into (60) and using equalities (24), we get for m = 1, 2, ..., N +(64) +Hm = (−2iρ2 − 4iξmpm)ϕm(x, t). + +INTEGRATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION ... +13 +On the other hand, using equalities (61), (62) and (24) we obtain +H− +1 (x, ξm, t) − cm(t)H+ +2 (x, ξm, t) = dcm(t) +dt +ψm(x, t)+ ++ +2N +� +n=1 +n̸=m +i +ξm − ξn +[Fn(x, t)GT +n(x, t)σ3ϕm(x, t) − cm(t)F + +n (x, t)GT +n(x, t)σ3ψm(x, t)]+ ++iFm(x, t)GT +m(x, t)σ3 +∂ +∂ξ (ϕ(x, ξ, t) − cm(t)ψ(x, ξ, t))|ξ=ξm + ++iFN+m(x, t)GT +N+m(x, t)σ3 +∂ +∂ξ (ϕ(x, ξ, t) − cm(t)ψ(x, ξ, t))|ξ=ξm . +According to (24) and (25), this equation can be rewritten in the following form +H− +1 (x, ξm, t) − cm(t)H+ +2 (x, ξm, t) = dcm(t) +dt +ψm(x, t)+ ++i˙a(ξn, t)Fm(x, t)GT +m(x, t)σ3hm(x, t) + i˙a(ξn, t)FN+m(x, t)GT +N+m(x, t)σ3hm(x, t). +Further, by virtue (9), (53) and (54), we obtain the equality +H− +1 (x, ξm, t) − cm(t)H+ +2 (x, ξm, t) = dcm(t) +dt +ψm(x, t)− +− iβm(t)Bm(t)ϕm(x, t) + iβ∗ +m(t)Bm(t)ϕm(x, t). +(65) +Comparing equalities (64) and (65), we obtain +(−2iρ2−4iξmpm)ϕm(x, t) = dcm(t) +dt +ψm(x, t)−iβm(t)Bm(t)ϕm(x, t)+iβ∗ +m(t)Bm(t)ϕm(x, t) +hence, taking into account equalities (24), we find +dcm(t) +dt += (−2iρ2 − 4iξmpm + i(βm(t) − β∗ +m(t))Bm(t))cm(t). +Thus, we have proved the following theorem. +Theorem 2. If functions u(x, t), Fk(x, t), Gk(x, t), k = 1, 2, ..., N are the solu- +tions of the problem (2)-(7) in the case of a source satisfying the conditions (B), +then the scattering data for the equation (10) satisfy the following relations +a(ξ, t) = a(ξ, 0), +b(ξ, t) = b(ξ, 0) exp(−2iρ2t − 4iξpt) +for +ξ ∈ Σ, +ξk(t) = ξk(0), +k = 1, 2, ..., N. +ck(t) = ck(0) exp(−2iρ2 − 4iξkpk + i +� t +0 +(βk(τ) − β∗ +k(τ)) Bk(τ)dτ, k = 1, 2, ..., N. +We will illustrate inverse scattering method of constructing exact solutions to +the NLS equation with concrete example. +Example 1. Let the initial function u0(x) have the form +u0(x) = ρ · eiα+eνx + eiα−e−νx +eνx + ce−νx +. +Where α+, α−, ρ, ν, c are positive real numbers and ρ > ν. + +14 +ANVAR REYIMBERGANOV +In this case, solving the direct scattering problem for the equation (10), we obtain +a(ξ, 0) = ξ + p − ζ − iν +ξ + p − ζ + iν , ζ = +� +ρ2 − ν2, +b(ξ, 0) = 0, +ξ1(0) = ζ, +c1(0) = i(ζ − iν) +ρ +ceiα−. +Based on Theorem 1, we can show the evolution of the scattering data in the +following form +a(ξ, t) = ξ + p − ζ − iν +ξ + p − ζ + iν , ζ = +� +ρ2 − ν2, +b(ξ, t) = 0, +ξ1(t) = ζ, +c1(t) = i(ζ − iν) +ρ +· c · exp(iα− − 2iρ2t + 4ζνt − +� t +0 +(Ak(τ) + A∗ +k(τ))dτ). +Applying the procedure of the inverse scattering problem, we find +u(x, t) = ρe−2iρ2t · eiα+eνx + eiα−ce−νx+4ζνt−g(t) +eνx + ce−νx+4ζνt−g(t) +, +where g(t) = +� t +0(A1(τ) + A∗ +1(τ)) dτ. +Using representation (30) and conditions (8), we obtain +F1 = α1(t) · +� +− i(ζ−iν) +ρ +· e−iα++2iρ2t +1 +� +· +1 +eνx + ce−νx+4ζνt−g(t) , +G1 = νA1(t) +α1(t) · +� +i(ζ−iν) +ρ +· eiα−−2iρ2t +1 +� +· +ce4ζνt−g(t) +eνx + ce−νx+4ζνt−g(t) . +Analogously, in the case (B), using results of Theorem 2, we obtain +u(x, t) = ρe−2iρ2t · eiα+eνx + eiα−ce−νx+4ζνt−g(t) +eνx + ce−νx+4ζνt−g(t) +, +F1 = − +ρ2B1(t) +2iν(ζ − iν) · +� +− i(ζ−iν) +ρ +· e−iα++2iρ2t +1 +� +· +1 +eνx + ce−νx+4ζνt−g(t) , +G1 = −2ν +ζ + iν · (νβ1(t) − 2xζ + iσ3) · +� +i(ζ−iν) +ρ +· eiα−−2iρ2t +1 +� +· +ce4ζνt−g(t) +eνx + ce−νx+4ζνt−g(t) + ++ +� +i(ζ−iν) +ρ +· eiα+−2iρ2t +1 +� +· +e2νx +eνx + ce−νx+4ζνt−g(t) + ++ +� +− i(ζ−iν) +ρ +· eiα−−2iρ2t +eiθ +� +· +c2e−2νx+8ζνt−2g(t) +eνx + ce−νx+4ζνt−g(t) , +where g(t) = −i +� t +0 (β1(τ) − β∗ +1(τ)) B1(τ)dτ. +Acknowledgments +This research was supported by program “Short-term research internships of +young scientists in leading foreign scientific organizations” of the Ministry of Inno- +vative Development of the Republic of Uzbekistan. Endless gratitude to Professor +Rogrigo Lopez for the great support of my research visit to University of Santiago +de Compostela. + +INTEGRATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION ... +15 +References +[1] Ablowitz M.J., Kaup D.J., Newell A.C., Segur H., The inverse scattering transform-Fourier +analysis for nonlinear problems, Studies in Appl. Math. 53 (1974), 249–315. +[2] Biondini G., Fagerstrom E., Prinari B., Inverse scattering transform for the defocusing non- +linear Schr¨odinger equation with fully asymmetric non-zero boundary conditions, Phys. D +333 (2016), 117–136. +[3] Biondini G., Kovaˇciˇc G., Inverse scattering transform for the focusing nonlinear Schr¨odinger +equation with nonzero boundary conditions, J. Math. Phys. 55 (2014), 031506, 22. +[4] Demontis F., Prinari B., van der Mee C., Vitale F., The inverse scattering transform for the +defocusing nonlinear Schr¨odinger equations with nonzero boundary conditions, Stud. 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Mat. 37 (2021), 63–76. +[15] Yakhshimuratov A., The nonlinear Schr¨odinger equation with a self-consistent source in the +class of periodic functions, Math. Phys. Anal. Geom. 14 (2011), 153–169. +[16] Zakharov V.E., Shabat A.B., Exact theory of two-dimensional self-focusing and one- +dimensional self-modulation of waves in nonlinear media, Soviet Journal of Experimental +and Theoretical Physics 61 (1971), 118–134. +[17] Zakharov V.E., Shabat A.B., Interaction between solitons in a stable medium, Soviet Journal +of Experimental and Theoretical Physics 37 (1973), 823–828. +Urgench State University, Uzbekistan +Email address: anvar@urdu.uz +URL: http://www.urdu.uz + diff --git a/sdE3T4oBgHgl3EQfjgoO/content/tmp_files/load_file.txt b/sdE3T4oBgHgl3EQfjgoO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..40b1e2e594ba8a40cf76bd734d6c55b6aa34cbf6 --- /dev/null +++ b/sdE3T4oBgHgl3EQfjgoO/content/tmp_files/load_file.txt @@ -0,0 +1,540 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf,len=539 +page_content='INTEGRATION OF THE NONLINEAR SCHR¨ODINGER EQUATION WITH A SELF-CONSISTENT SOURCE AND NONZERO BOUNDARY CONDITIONS ANVAR REYIMBERGANOV Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' This paper is devoted to the study of the defocusing nonlinear Schr¨odinger equation with a self-consistent source and nonzero boundary con- ditions by the method of the inverse scattering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' In cases where the source consists of a combination of eigenfunctions of the corresponding spec- tral problem for the Zakharov-Shabat system, the complete integrability of the nonlinear Schr¨odinger equation is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Namely, the evolutions of the scattering data of the self-adjoint Zakharov-Shabat system, whose po- tential is a solution of the defocusing nonlinear Schr¨odinger equation with a self-consistent source, are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Introduction Nonlinear Schr¨odinger (NLS) equation (1) iut − 2χ |u|2 u + uxx = 0 with various boundary conditions models a wide class of nonlinear phenomena in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' In the work [16], V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Zakharov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Shabat showed that NLS equation can be applied in the study of optical self-focusing and splitting of optical beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' The sign of the coupling constant corresponds to the attraction (χ < 0) and repulsion (χ > 0) of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' In the case of attraction, the problem of a finite number of particles and their bound states has a physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' In the classical limit, this is modeled by rapidly decreasing boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' In the case of repulsion, the problem corresponding to a gas of particles with a finite density is of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' It is well known that inverse scattering method for integration of the NLS equa- tion is based on the so-called Zakharov-Shabat and Ablowitz-Kaup-Newell-Segur scattering problem (see: [1, 16, 8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' In 1971, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Zakharov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Shabat [16] showed that the NLS equation can be solved by means of the inverse scattering transform (IST) technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' The IST as a method to solve the initial-value problem for the NLS equation has been extensively studied in the literature, both in the focusing (χ = −1) and in the defocusing (χ = 1) dispersion regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' The IST for the defocusing NLS equation with nonzero boundary conditions was first studied in 1973 by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Zakharov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Shabat [17] and a detailed study can be found in the monograph [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' In the work [4] a rigorous theory of the IST for the defocusing nonlinear Schr¨odinger equation with nonvanishing boundary values u± = u0eiθ± as x → ±∞ is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' The IST theory for the defocusing NLS equation with nonzero boundary conditions Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Defocusing nonlinear Schr¨odinger equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Zakharov-Shabat system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' inverse scattering theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' nonzero boundary condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' self-consistent source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='04588v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='AP] 11 Jan 2023 2 ANVAR REYIMBERGANOV was studied by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Gino, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Emily and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Prinari [2] and the focusing case has been studied by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Biondini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Kovaˇci´c [3], F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Demontis, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Prinari, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' van der Mee, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Vitale [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Melnikov [10, 11] showed that the NLS equation remains its integrability by the inverse scattering method, if a source is added to them in the form of a combination of eigenfunctions of the corresponding spectral problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Namely, the term “self-consistent source” was introduced in the works of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Melnikov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' The NLS equation with the self-consistent sources in various classes of functions were considered by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Khasanov [9], I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Rakhimov [13], A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Yakhshimuratov [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' In the matrix case, the inverse scattering theory for the matrix Zakharov-Shabat system was investigated by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Barbara, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Demontis and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Van der Mee [12, 6, 7] and applied for the integration of the matrix NLS equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' In [14], the matrix NLS equation with the self-consistent source was considered in the class of rapidly decreasing matrix functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Formulation of the problem We consider the following system of equations iut − 2u |u|2 + uxx = −2i N � n=1 (f ∗ 1,ng∗ 2,n + f2,ng1,n) (2) ∂f1,n ∂x − u∗f2,n + iξnf1,n = ∂f2,n ∂x − uf1,n − iξnf2,n = 0, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N, (3) ∂g1,n ∂x − ug2,n − iξng1,n = ∂g2,n ∂x − u∗g1,n + iξng2,n = 0, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N (4) under the initial condition (5) u(x, 0) = u0(x), x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Here the initial function u0(x), x ∈ R satisfies the following properties: 1) (6) � 0 −∞ (1 − x) ��u0(x) − ρeiα−�� dx + � ∞ 0 (1 + x) ��u0(x) − ρeiα+�� dx < ∞, where ρ > 0 and 0 ≤ α± < 2π are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 2) The system of equations (3) with coefficient u0(x) possesses exactly N eigen- values ξ1(0), ξ2(0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' , ξN(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' We assume that the solution u(x, t) of the equation (2) is sufficiently smooth and tends to its limits sufficiently rapidly as x → ±∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', for all t ≥ 0 satisfies the condition � 0 −∞ (1 − x) ���u(x, t) − ρeiα−−2iρ2t��� dx+ � ∞ 0 (1 + x) ���u(x, t) − ρeiα+−2iρ2t��� dx + � ∞ −∞ 2 � k=1 ���� ∂ku(x, t) ∂xk ���� dx < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' (7) It follows from this condition that the left-hand side of equation (2) for all t ≥ 0 tends to zero as x → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Taking this into account, the solutions Fn(x, t) = (f1,n, f2,n)T and Gn(x, t) = (g1,n, g2,n)T of equations (3) and (4), respectively, are to be chosen so that the expression in the right-hand side of equation (2) for all INTEGRATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 3 t ≥ 0 should tend to zero rapidly enough as x → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' This can be done in the following two ways: (A) Let the functions Fn(x, t) and Gn(x, t) be the eigenfunctions of equations (3) and (4) respectively, corresponding to the eigenvalues ξn(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' In this case, the functions Fn(x, t) and Gn(x, t) belong to the L2(R) for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' We assume that (8) � ∞ −∞ GT n(s, t)Fn(s, t)ds = An(t), t ≥ 0, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Where An(t) are given and the continuous functions of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' (B) Let the function Fn(x, t) be the eigenfunctions of equations (3) corresponding to the eigenvalues ξn(t) and let the function Gn(x, t) be unbounded solution of the equation (4), satisfying equalities (9) f1,ng1,n − f2,ng2,n = Bn(t), t ≥ 0, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Where the functions Bn(t) are given continuous real valued functions of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Our main goal is to obtain representations for solutions u(x, t), Fn(x, t), Gn(x, t), n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N of problem (2)-(7), within the framework of the inverse scattering method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Preliminaries Let a function u(x, t) belong to the class of functions (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' In this section, we give well known [8], necessary information concerning the theory of direct and inverse scattering problems for the operator L(t) = i � ∂ ∂x −u∗(x, t) u(x, t) − ∂ ∂x � , t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Consider the equation (10) (L(t) − ξI)f = 0 with respect to an unknown 2 × 2 square matrix function f(x, ξ, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Here ξ is a complex spectral parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' We introduce a new spectral parameter p as follows p = � ξ2 − ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' The variable p is then thought of as belonging to a Riemann surface Γ consisting of a sheet Γ+ and a sheet Γ− which both coincide with the complex plane cut along the semi lines Σ = (−∞, −ρ] ∪ [ρ, ∞) with its edges glued in such a way that p(ξ) is continuous through the cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' The variable p is thought of as belonging to the complex plane consisting of the upper half complex plane Γ+ and the lower half complex plane Γ− glued together along the whole real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' For all ξ ∈ Σ, the branch of the square root is fixed by the condition sign p(ξ) = sign ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' For ξ ∈ Σ, we define matrix Jost solutions f −(x, ξ, t) and f +(x, ξ, t) from the right and the left, respectively as those square matrix solutions to (10) satisfying asymptotics (11) f ± ∼ E±(x, ξ, t) as x → ±∞ where E±(x, ξ, t) = � 1 − i(ξ−p) ρ e−iα±+2iρ2t i(ξ−p) ρ eiα±−2iρ2t 1 � e−ipσ3x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 4 ANVAR REYIMBERGANOV Here and everywhere below we will use the standard Pauli matrices σ1 = � 0 1 1 0 � , σ2 = � 0 −i i 0 � , σ3 = � 1 0 0 −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' If a function u(x, t) belongs to the class of functions (7), then such a solution to equations (10) exists and is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' It can be shown that (12) d dx det f ±(x, ξ, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' From (12) and (11) it follows that (13) det f ±(x, ξ, t) = 2p(ξ − p) ρ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' The system (10) is invariant with respect to the involution (14) ¯f ±(x, ξ, t) = σ1f ±(x, ξ, t)σ1, ξ ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' We now call the columns of f +(x, ξ, t) = � ¯ψ(x, ξ, t) ψ(x, ξ, t) � , f −(x, ξ, t) = (ϕ(x, ξ, t) ¯ϕ(x, ξ, t)) the Jost solutions from the right and the left, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' For the Jost solutions we get the following asymptotic estimates (15) ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) ∼ � − i(ξ−p) ρ e−iα++2iρ2t 1 � eipx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' x → ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' ¯ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) ∼ � 1 i(ξ−p) ρ eiα+−2iρ2t � e−ipx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' x → ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' (16) ϕ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) ∼ � 1 i(ξ−p) ρ eiα−−2iρ2t � e−ipx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' x → −∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' ¯ϕ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) ∼ � − i(ξ−p) ρ e−iα−+2iρ2t 1 � eipx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' x → −∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Since f −(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) and f +(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) are square matrix solutions of the homogeneous first order equation (10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' we necessarily have for ξ ∈ Σ (17) f −(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) = f +(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t)S(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) where S(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) is the transition coefficient matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' From the involution property (14) for ξ ∈ Σ, it follows that ¯S(ξ, t) = σ1S(ξ, t)σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Hence, we have (18) S(ξ, t) = � a(ξ, t) ¯b(ξ, t) b(ξ, t) ¯a(ξ, t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Coefficients a(ξ, t) and b(ξ, t) are called scattering coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' From relations (13) and (18) we obtain a(ξ, t)¯a(ξ, t) − b(ξ, t)¯b(ξ, t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' INTEGRATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 5 By using (17) we can represent the scattering coefficients as (19) a(ξ, t) = ρ2 2p(ξ − p) det(ϕ(x, ξ, t), ψ(x, ξ, t)) and b(ξ, t) = ρ2 2p(ξ − p) det( ¯ψ(x, ξ, t), ϕ(x, ξ, t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' If a function u(x, t) belongs to the class of functions (7), then for each x ∈ R the Jost solutions ψ(x, ξ, t)e−ipx and ϕ(x, ξ, t)eipx are analytic for ξ ∈ Γ+ excluding branch points ξ = ±ρ, there are asymptotes for |ξ| → ∞ (20) ϕ(x, ξ, t)eipx = � 1 i(ξ−p) ρ eiα−−2ip2t � + O �|1 + ξ − p| |ξ| � , (21) ψ(x, ξ, t)e−ipx = � − i(ξ−p) ρ e−iα++2ip2t 1 � + O �|1 + ξ − p| |ξ| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' It follows from the analyticity properties of the Jost solutions and equality (19) that the function a(ξ, t) can be analytically continued to the sheet Γ+ excluding branch points ξ = ±ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' From (20) and (21) we obtain that for |ξ| → ∞, the function a(ξ, t) has the asymptotics (22) a(ξ, t) = 1 + O � 1 |ξ| � as Im ξ > 0 and (23) a(ξ, t) = e−iθ + O � 1 |ξ| � as Im ξ < 0 where we recall θ = α+ − α−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Similarly, the function ¯a(ξ, t) can be analytically continued to the sheet Γ−, excluding branch points ξ = ±ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' It follows from the analyticity with respect to the function a(ξ, t) on Γ+ and from the asymptotics (22), (23) that the function a(ξ, t) can have only a finite number of zeros on the sheet Γ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' These zeros will be denoted by ξ1, ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', ξN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' In [4] it is shown that all zeros are simple and all belong to the (−ρ, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' It is seen from representation (19) that ξ = ξn the functions ϕ(x, ξ, t) and ψ(x, ξ, t) are proportional to each other (24) ϕn(x, t) = cn(t)ψn(x, t), ¯ϕn(x, t) = c∗ n(t) ¯ψn(x, t), n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Where ϕn(x, t) = ϕ(x, ξn, t), ψn(x, t) = ψ(x, ξn, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' The zeros of a(ξ, t) correspond to the eigenvalues of the equation (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' The equation (10) is self-adjoint, so its eigenvalues and thus the zeros of the function a(ξ, t) are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Note, the vector functions (25) hn(x, t) = ∂ ∂ξ (ϕ(x, ξ, t) − cnψ(x, ξ, t))|ξ=ξn ˙a(ξn, t) , n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N are a solution to the equations (L(t)−ξnI)hn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Where ˙a(ξn, t) = ∂ ∂ξ a(ξ, t)|ξ=ξn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 6 ANVAR REYIMBERGANOV From equality (25) it follows that hn(x, t) ∼ −cn(t) � − i(ξn−pn) ρ e−iα−+2ip2t 1 � eipnx as x → −∞, hn(x, t) ∼ � 1 i(ξn−pn) ρ eiα+−2ip2t � e−ipnx as x → ∞, (26) where pn = i � ρ2 − ξ2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' In particular, we have (27) det(ϕn, hn) = −2pn(ξn − pn) ρ2 cn, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' The set {a(ξ, t), b(ξ, t), ξn(t), cn(t), n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N} is called the scattering data for equation (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' The direct scattering problem is to find the scattering data via the given potentials u(x, t) and the inverse scattering problem is to find the potentials u(x, t) of the equation (10) via the given scattering data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Before we proceed further solving the inverse problem, it is convenient to in- troduce a uniformization variable z (see [4, 8]) defined by the conformal mapping: z = z(ξ) = ξ + p(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Inverse mapping given by ξ = 1 2 � z + ρ2 z � , p = z − ξ = 1 2 � z − ρ2 z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' With this mapping the sheets Γ+ and Γ− of the Riemann surface Γ are, respectively, mapped onto the upper and lower complex half-planes Im z > 0 and Im z < 0 of the complex z-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' The cut Σ on the Riemann surface is mapped onto the real z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' The segments [−ρ, ρ] on Γ+ and Γ− are mapped onto the upper and lower semicircles of radius ρ and center at the origin of the z-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' The neighborhood of the point ξ = ∞ on Γ± with the condition ±Imξ > 0 is mapped into the neighborhood of the point z = ∞, and the neighborhood of the point ξ = ∞ on Γ± with the condition ±Imξ < 0 is mapped into the neighborhood of the point z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' In terms of variable z, relation (17) can be written when Imz = 0 following form (28) f −(x, z, t) = f +(x, z, t)S(z, t) here f ±(x, z, t) = f ±(x, ξ(z), t), S(z, t) = S(ξ(z), t) and one can obtain the sym- metries of the scattering coefficients: (29) a(z, t) = ¯a �ρ2 z , t � , Imz ≥ 0, b(z, t) = −¯b �ρ2 z , t � , Imz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Equality (29), together with the self-adjointness of the equation (10), ensure that the scattering coefficient a(z, t) (¯a(z, t)) can only have zeros at zn = ξn + ivn (¯zn = ξn − ivn), with −ρ < ξn < ρ and vn = � ρ2 − ξ2n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Taking into account the analyticity properties of a(z, t) in the upper half plane Imz > 0 we can obtain the following representation a(z, t) = N � n=1 z − zn z − z∗n exp � − 1 2πi � ∞ −∞ log(1 − |r(ζ, t)|2) ζ − z dζ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' INTEGRATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 7 Where r(z, t) ≡ b(z,t) a(z,t) is called reflection coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' According to (23), for z → 0 we obtain that e−iθ = N � n=1 zn z∗n exp � − 1 2πi � ∞ −∞ log(1 − |r(ζ, t)|2) ζ dζ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' If ϕn = � ϕ1,n ϕ2,n � is an eigenfunction of the equation (10) corresponding to zn, then we define ¯ϕn = � ϕ∗ 2,n ϕ∗ 1,n � to be the eigenfunction of the equation (10) corresponding to ¯zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' It is well known that the inverse scattering theory of (10) can be formulated in terms of the Gelfand-Levitan-Marchenko equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' The Jost solution ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) of the equation (10) can be represented in the following form (30) ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) = � − iρ z e−iα++2iρ2t 1 � e(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' z) + � ∞ x K(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) � − iρ z e−iα++2iρ2t 1 � e(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' z)dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' here e(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' z) = e i 2 � z− ρ2 z � x and K(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' y) is a 2 × 2 matrix function which has to satisfy the following Gelfand-Levitan-Marchenko equation: K(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) + F(x + y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) + � ∞ x K(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t)F(s + y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t)ds = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' y ≥ x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' where K(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) and F(x + y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) are defined as K(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) = � K11(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) K12(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) K21(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) K22(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' F(x+y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) = � F1(x + y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) F ∗ 2 (x + y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) F2(x + y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) F1(x + y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) � with F1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) = ρeiα+−2iρ2t 4πi � ∞ −∞ b(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) za(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) · e(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' z)dz − 1 2 N � n=1 cn(t)ρe−iα++2iρ2t ˙a(zn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t)zn e(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' zn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' F2(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) = 1 4π � ∞ −∞ b(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) a(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) · e(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' z)dz − 1 2 N � n=1 icn(t) ˙a(zn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) · e(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' In representations (30), the component K21(x, x, t) of the matrix K(x, y, t) have relations with the potential 2K21(x, x, t) = ρeiα+−2iρ2t − u(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' In the work [8], it was proven the uniquely determining of the potential u(x, t) by the scattering data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Time evolution The use of the inverse scattering method for integration of the problem (2)-(7) is based on the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Let the function u(x, t) be a solution of equation (2), from the class of functions (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Consider equation (10) with a potential u(x, t) and find the evolution from the scattering data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Assuming that (31) FN+n = � f ∗ 2,n f ∗ 1,n � , GN+n = � g∗ 2,n g∗ 1,n � , n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N, 8 ANVAR REYIMBERGANOV equation (2) can be represented as an equality of operators in the class of smooth functions f(x, ξ, t) satisfying the equation (10): ∂L ∂t + [L, A] = i 2N � n=1 [σ3, FnGT n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Where [L, A] = LA − AL and A = � i |u|2 + 2iξ2 −iu∗ x − 2ξu∗ iux − 2ξu −i |u|2 − 2iξ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Let f(x, ξ, t) be solution of the equation (10) and let φn(x, ξ, t), n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', 2N be any functions, which satisfy the conditions (32) ∂φn ∂x = GT nf, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Then, the function Gn(x, t) satisfy the equalities (33) GT nσ3f + i(ξ − ξn)φn = 0 , n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', 2N and the function (34) H = ∂f ∂t − Af + 2N � n=1 Fnφn satisfies the equation (10) for any ξ ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Evolution equation for the scattering data in the case of a source satisfying the conditions (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Let us take matrix Jost solutions f −(x, ξ, t) and f +(x, ξ, t) for ξ ∈ Σ as the solution f(x, ξ, t) and ξ = ξn, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N are eigenvalues of the equation (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' According to the definition of eigenfunctions, there are αn(t) and βn(t) such that the relations hold (35) Fn(x, t) = αn(t)ψn(x, t), Gn(x, t) = βn(t)σ1ϕn(x, t), n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N According to these relations, due to the assumptions (31), we obtain (36) FN+n(x, t) = α∗ n(t) ¯ψn(x, t), GN+n(x, t) = β∗ n(t)σ1 ¯ϕn(x, t), n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' By definition functions Gn(x, t), belong to the L2(R) for all t ≥ 0 and matrix Jost solutions f −(x, ξ, t), f +(x, ξ, t) are bounded for all ξ ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Therefore φ− n ∈ L2(R) and φ+ n ∈ L2(R) for all t ≥ 0 and ξ ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Hence, by virtue of (33) it follows that at any ξ ∈ Σ and n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', 2N the asymptotics (37) φ− n (x, ξ, t) → 0 as x → −∞, φ+ n (x, ξ, t) → 0 as x → ∞ are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' So, from (32) for n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', 2N we obtain the following expressions (38) φ− n = � x −∞ GT n(s,t)f −(s, ξ, t)ds, φ+ n = − � ∞ x GT n(s, t)f +(s, ξ, t)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Using the matrix Jost solutions f + and f − of equation (10), we rewrite equality (34) in the form (39) H− = ∂f − ∂t − Af − + 2N � n=1 Fnφ− n INTEGRATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 9 and (40) H+ = ∂f + ∂t − Af + + 2N � n=1 Fnφ+ n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' These functions satisfy the equation (10) for any ξ ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Therefore, H+ and H− are linearly dependent on f + and f −, respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', there exist such C− 0 (ξ, t) and C+ 0 (ξ, t) that the following identities hold H−(x, ξ, t) = f −(x, ξ, t)C− 0 (ξ, t), H+(x, ξ, t) = f +(x, ξ, t)C+ 0 (ξ, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' By virtue of the definition of the matrix A, from relations (39), (40) and from asymptotics (11), (37) we obtain H−(x, ξ, t) → −(iρ2 + 2iξp)E−(x, ξ, t)σ3, x → −∞, (41) H+(x, ξ, t) → −(iρ2 + 2iξp)E+(x, ξ, t)σ3, x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' (42) By the uniqueness of the Jost solutions we get (43) H−(x, ξ, t) = −(iρ2 + 2iξp)f −(x, ξ, t)σ3, H+(x, ξ, t) = −(iρ2 + 2iξp)f +(x, ξ, t)σ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' We introduce the function H in the following form H = H−(x, ξ, t) − H+(x, ξ, t)S(ξ, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Based on equalities (17) and (43), the function H can be rewritten in the form (44) H = (iρ2 + 2iξp)f +(x, ξ, t)[σ3, S(ξ, t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' On the other hand, by virtue of (17), (39) and (40) the equality H =H−(x, ξ, t) − H+(x, ξ, t)S(ξ, t) = f +(x, ξ, t)St(x, ξ, t)+ + 2N � n=1 [Fn(x, t)φ− n (x, ξ, t) − Fn(x, t)φ+ n (x, ξ, t)S(ξ, t)] (45) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Based on equality (33) this relation becomes H = f +(x, ξ, t)St(ξ, t)+ + 2N � n=1 i ξ − ξn [Fn(x, t)GT n(x, t)σ3f −(x, ξ, t) − Fn(x, t)GT n(x, t)σ3f +(x, ξ, t)S(ξ, t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Finally, based on (17), we obtain (46) H = f +(x, ξ, t)St(ξ, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Comparing equalities (44) and (46) we have (iρ2 + 2iξp)f +(x, ξ, t)[σ3, S(ξ, t)] = f +(x, ξ, t)St(ξ, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Therefore, for all ξ ∈ Σ we have the relation St(ξ, t) − (iρ2 + 2iξp)[σ3, S(ξ, t)] = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' d dta(ξ, t) = 0, d dtb(ξ, t) = −2(iρ2 + 2iξp)b(ξ, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Since, the function a(ξ, t) does not depend on t, hence we conclude that its zeros ξn also do not depend on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 10 ANVAR REYIMBERGANOV Based on identities (39) and (40),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' we write the following equalities H− 1 (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' ξn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) = ∂ϕm(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) ∂t − A(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' ξn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t)ϕm(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) + 2N � n=1 Fn(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t)φ− 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='n(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' ξn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) (47) H+ 2 (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' ξn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) = ∂ψm(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) ∂t − A(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' ξn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t)ψm(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) + 2N � n=1 Fn(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t)φ+ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='n(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' ξn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) (48) By virtue of the definition of the matrix A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' from relations (47),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' (48) and from asymptotics (15),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' (16) we obtain (49) H− 1 (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' ξm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) = (−iρ2 − 2iξmpm)ϕm(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' H+ 2 (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' ξm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) = (iρ2 + 2iξmpm)ψm(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' We now introduce the following functions Hm = H− 1 (x, ξm, t) − cm(t)H+ 2 (x, ξm, t), m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Using equalities (24), (49) the function Hm can be rewritten in the form (50) Hm = (−2iρ2 − 4iξmpm)ϕm(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Substituting instead of φ− 1,n(x, ξ, t) and φ+ 2,n(x, ξ, t) the expressions from (38) into equalities (47), (48) and using (24), we obtain (51) H− 1 (x, ξm, t)−cm(t)H+ 2 (x, ξm, t) = dcm(t) dt ψm(x, t)+ 2N � n=1 Fn(x, t) � ∞ −∞ GT n(s, t)ϕm(s, t)ds If ξm ̸= ξn, according to equation (33) we get � ∞ −∞ GT n(s, t)ϕm(s, t)ds = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' According to (35) and (36), equality (51) can be rewritten in the form (52) H− 1 (x, ξm, t) − cm(t)H+ 2 (x, ξm, t) = dcm(t) dt ψm(x, t)+ + �� ∞ −∞ GT m(s, t)Fm(s, t)ds + � ∞ −∞ GT N+m(s, t)FN+m(s, t)ds � ϕm(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Comparing equalities (50) and (52) we obtain (−2iρ2 − 4iξmpm)ϕm(x, t) = = dcm(t) dt ψm(x, t)+ �� ∞ −∞ GT m(s, t)Fm(s, t)ds + � ∞ −∞ GT N+m(s, t)FN+m(s, t)ds � ϕm(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Finally, using these equalities and taking into account (8) and (24) we determine dcm(t) dt = (−2iρ2 − 4iξmpm − Am(t) − A∗ m(t))cm(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Thus, we have proved the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' If functions u(x, t), Fk(x, t), Gk(x, t), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N are the solu- tions of the problem (2)-(7) in the case of a source satisfying the conditions (A), then the scattering data for the equation (10) satisfy the following relations a(ξ, t) = a(ξ, 0), b(ξ, t) = b(ξ, 0) exp(−2iρ2t − 4iξpt) for ξ ∈ Σ, INTEGRATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 11 ξk(t) = ξk(0), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' ck(t) = ck(0) exp(−2iρ2t − 4iξkpkt − � t 0 (Ak(τ) + A∗ k(τ))dτ), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Evolution equation for the scattering data in the case of a source satisfying the conditions (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Let us take matrix Jost solutions f −(x, ξ, t) and f +(x, ξ, t) for ξ ∈ Σ as the solution f(x, ξ, t) and ξ = ξn, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N are eigenvalues of the equation (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' According to the definition of eigenfunctions of equation (10), there are αn(t) such that the relations (53) Fn(x, t) = αn(t)ψn(x, t), FN+n(x, t) = α∗ n(t) ¯ψn(x, t), n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Due to the assumptions (B) the functions Gn(x, t) are unbounded functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' So, there are βn(t) such that which follow the equalities (54) Gn(x, t) = βn(t) ˙a(ξn, t)σ1ϕn(x, t) + σ1hn(x, t), GN+n(x, t) = β∗ n(t) ˙¯a(ξn, t)σ1 ¯ϕn(x, t) + σ1¯hn(x, t), n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' One can easily see from (9) and (27), that the quantities αn(t) satisfy the fol- lowing equalities (55) αn(t) = − ρ2 2pn(ξn − pn)Bn(t), α∗ n(t) = ρ2 2pn(ξn + pn)Bn(t), n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Using equalities (24), (33), (53), (54) and asymptotics (15), (26) we can verify that at any ξ ∈ Σ and when x → ∞ the following asymptotics are valid: Fnφ+ n ∼ iαn(t) ξ − ξn � (ξn−pn)2 ρ2 − i(ξn−pn) ρ e−iα++2iρ2t i(ξn−pn) ρ eiα+−2iρ2t 1 � σ3E+(x, ξ, t), FN+nφ+ N+n ∼ iα∗ n(t) ξ − ξn � 1 − i(ξn+pn) ρ e−iα++2iρ2t i(ξn+pn) ρ eiα+−2iρ2t (ξn+pn)2 ρ2 � σ3E+(x, ξ, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Taking into account of equalities (24), (33), (53), (54) and asymptotics (15), (26) we are convinced that at any ξ ∈ Σ and x → −∞ there hold the asymptotics Fnφ− n ∼ −iαn(t) ξ − ξn � 1 − i(ξn−pn) ρ e−iα−+2iρ2t i(ξn−pn) ρ eiα−−2iρ2t (ξn−pn)2 ρ2 � σ3E−(x, ξ, t), FN+nφ− N+n ∼ −iα∗ n(t) ξ − ξn � (ξn+pn)2 ρ2 − i(ξn+pn) ρ e−iα−+2iρ2t i(ξn+pn) ρ eiα−−2iρ2t 1 � σ3E−(x, ξ, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Using equalities (55) one can easily verify that at any ξ ∈ Σ the asymptotic Fnφ+ n + FN+nφ+ N+n → 0 for x → ∞, and Fnφ− n + FN+nφ− N+n → 0 for x → −∞ are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Hence, it follows that the quantities H−(x, ξ, t) and H+(x, ξ, t) determined by (39) and (40) satisfy equalities (56) H−(x, ξ, t) = f −(x, ξ, t)(−iρ2 − 2iξp)σ3, H+(x, ξ, t) = f +(x, ξ, t)(−iρ2 − 2iξp)σ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 12 ANVAR REYIMBERGANOV Now, consider the function Hm of the form Hm = H−(x, ξ, t) − H+(x, ξ, t)S(ξ, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Taking into account (56) we find that (57) Hm = (iρ2 + 2iξp)f +(x, ξ, t)[σ3, S(ξ, t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' From equalities (39), (40) and (17) it is easy to get that H−(x, ξ, t) − H+(x, ξ, t)S(ξ, t) = f +(x, ξ, t)St(x, ξ, t)+ (58) + 2N � n=1 [Fn(x, t)φ− n (x, ξ, t) − Fn(x, t)φ+ n (x, ξ, t)S(ξ, t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Using equalities (33), we obtain H−(x, ξ, t) − H+(x, ξ, t)S(ξ, t) = f +(x, ξ, t)St(ξ, t)+ + 2N � n=1 i ξ − ξn [Fn(x, t)GT n(x, t)σ3f −(x, ξ, t) − Fn(x, t)GT n(x, t)σ3f +(x, ξ, t)S(ξ, t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' By virtue of (17), it follows that (59) H−(x, ξ, t) − H+(x, ξ, t)S(ξ, t) = f +(x, ξ, t)St(ξ, t) Comparing equalities (57) and (59) we have (iρ2 + 2iξp)f +(x, ξ, t)[σ3, S(ξ, t)] = f +(x, ξ, t)St(ξ, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Therefore, for all ξ ∈ Σ we have St(ξ, t) − (iρ2 + 2iξp)[σ3, S(ξ, t)] = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' d dta(ξ, t) = 0, d dtb(ξ, t) = −2(iρ2 + 2iξp)b(ξ, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Thus, we conclude that the function a(ξ, t) does not depend on t, so the zeros ξn(t) of function a(ξ, t) do not depend on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Let us now find the evolution of the normalizing constants cm(t), m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' We now introduce the following functions (60) Hm = H− 1 (x, ξm, t) − cm(t)H+ 2 (x, ξm, t), m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N, where H− 1 (x, ξm, t) = ∂ϕm(x, t) ∂t − A(x, ξm, t)ϕm(x, t) + 2N � n=1 Φn(x, t)ϕ− 1,n(x, ξm, t), (61) H+ 2 (x, ξm, t) = ∂ψm(x, t) ∂t − A(x, ξm, t)ψm(x, t) + 2N � n=1 Φn(x, t)ϕ+ 2,n(x, ξm, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' (62) It is easy to show that (63) H− 1 (x, ξm, t) = (−iρ2 − 2iξmpm)ϕm(x, t), H+ 2 (x, ξm, t) = (iρ2 + 2iξmpm)ψm(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Substituting (63) into (60) and using equalities (24), we get for m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N (64) Hm = (−2iρ2 − 4iξmpm)ϕm(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' INTEGRATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 13 On the other hand, using equalities (61), (62) and (24) we obtain H− 1 (x, ξm, t) − cm(t)H+ 2 (x, ξm, t) = dcm(t) dt ψm(x, t)+ + 2N � n=1 n̸=m i ξm − ξn [Fn(x, t)GT n(x, t)σ3ϕm(x, t) − cm(t)F + n (x, t)GT n(x, t)σ3ψm(x, t)]+ +iFm(x, t)GT m(x, t)σ3 ∂ ∂ξ (ϕ(x, ξ, t) − cm(t)ψ(x, ξ, t))|ξ=ξm + +iFN+m(x, t)GT N+m(x, t)σ3 ∂ ∂ξ (ϕ(x, ξ, t) − cm(t)ψ(x, ξ, t))|ξ=ξm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' According to (24) and (25), this equation can be rewritten in the following form H− 1 (x, ξm, t) − cm(t)H+ 2 (x, ξm, t) = dcm(t) dt ψm(x, t)+ +i˙a(ξn, t)Fm(x, t)GT m(x, t)σ3hm(x, t) + i˙a(ξn, t)FN+m(x, t)GT N+m(x, t)σ3hm(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Further, by virtue (9), (53) and (54), we obtain the equality H− 1 (x, ξm, t) − cm(t)H+ 2 (x, ξm, t) = dcm(t) dt ψm(x, t)− − iβm(t)Bm(t)ϕm(x, t) + iβ∗ m(t)Bm(t)ϕm(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' (65) Comparing equalities (64) and (65), we obtain (−2iρ2−4iξmpm)ϕm(x, t) = dcm(t) dt ψm(x, t)−iβm(t)Bm(t)ϕm(x, t)+iβ∗ m(t)Bm(t)ϕm(x, t) hence, taking into account equalities (24), we find dcm(t) dt = (−2iρ2 − 4iξmpm + i(βm(t) − β∗ m(t))Bm(t))cm(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Thus, we have proved the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' If functions u(x, t), Fk(x, t), Gk(x, t), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N are the solu- tions of the problem (2)-(7) in the case of a source satisfying the conditions (B), then the scattering data for the equation (10) satisfy the following relations a(ξ, t) = a(ξ, 0), b(ξ, t) = b(ξ, 0) exp(−2iρ2t − 4iξpt) for ξ ∈ Σ, ξk(t) = ξk(0), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' ck(t) = ck(0) exp(−2iρ2 − 4iξkpk + i � t 0 (βk(τ) − β∗ k(τ)) Bk(τ)dτ, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' We will illustrate inverse scattering method of constructing exact solutions to the NLS equation with concrete example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Let the initial function u0(x) have the form u0(x) = ρ · eiα+eνx + eiα−e−νx eνx + ce−νx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Where α+, α−, ρ, ν, c are positive real numbers and ρ > ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 14 ANVAR REYIMBERGANOV In this case, solving the direct scattering problem for the equation (10), we obtain a(ξ, 0) = ξ + p − ζ − iν ξ + p − ζ + iν , ζ = � ρ2 − ν2, b(ξ, 0) = 0, ξ1(0) = ζ, c1(0) = i(ζ − iν) ρ ceiα−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Based on Theorem 1, we can show the evolution of the scattering data in the following form a(ξ, t) = ξ + p − ζ − iν ξ + p − ζ + iν , ζ = � ρ2 − ν2, b(ξ, t) = 0, ξ1(t) = ζ, c1(t) = i(ζ − iν) ρ c · exp(iα− − 2iρ2t + 4ζνt − � t 0 (Ak(τ) + A∗ k(τ))dτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Applying the procedure of the inverse scattering problem, we find u(x, t) = ρe−2iρ2t · eiα+eνx + eiα−ce−νx+4ζνt−g(t) eνx + ce−νx+4ζνt−g(t) , where g(t) = � t 0(A1(τ) + A∗ 1(τ)) dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Using representation (30) and conditions (8), we obtain F1 = α1(t) · � − i(ζ−iν) ρ e−iα++2iρ2t 1 � 1 eνx + ce−νx+4ζνt−g(t) , G1 = νA1(t) α1(t) · � i(ζ−iν) ρ eiα−−2iρ2t 1 � ce4ζνt−g(t) eνx + ce−νx+4ζνt−g(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Analogously,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' in the case (B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' using results of Theorem 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' we obtain u(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' t) = ρe−2iρ2t · eiα+eνx + eiα−ce−νx+4ζνt−g(t) eνx + ce−νx+4ζνt−g(t) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' F1 = − ρ2B1(t) 2iν(ζ − iν) · � − i(ζ−iν) ρ e−iα++2iρ2t 1 � 1 eνx + ce−νx+4ζνt−g(t) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' G1 = −2ν ζ + iν · (νβ1(t) − 2xζ + iσ3) · � i(ζ−iν) ρ eiα−−2iρ2t 1 � ce4ζνt−g(t) eνx + ce−νx+4ζνt−g(t) + + � i(ζ−iν) ρ eiα+−2iρ2t 1 � e2νx eνx + ce−νx+4ζνt−g(t) + + � − i(ζ−iν) ρ eiα−−2iρ2t eiθ � c2e−2νx+8ζνt−2g(t) eνx + ce−νx+4ζνt−g(t) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' where g(t) = −i � t 0 (β1(τ) − β∗ 1(τ)) B1(τ)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Acknowledgments This research was supported by program “Short-term research internships of young scientists in leading foreign scientific organizations” of the Ministry of Inno- vative Development of the Republic of Uzbekistan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Endless gratitude to Professor Rogrigo Lopez for the great support of my research visit to University of Santiago de Compostela.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' INTEGRATION OF THE NONLINEAR SCHR ¨ODINGER EQUATION .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 15 References [1] Ablowitz M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', Kaup D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', Newell A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', Segur H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', The inverse scattering transform-Fourier analysis for nonlinear problems, Studies in Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' 53 (1974), 249–315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' [2] Biondini G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', Fagerstrom E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', Prinari B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=', Inverse scattering transform for the defocusing non- linear Schr¨odinger 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content=' Urgench State University, Uzbekistan Email address: anvar@urdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='uz URL: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='urdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} +page_content='uz' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfjgoO/content/2301.04588v1.pdf'} diff --git a/tNE2T4oBgHgl3EQf1wiA/content/tmp_files/2301.04154v1.pdf.txt b/tNE2T4oBgHgl3EQf1wiA/content/tmp_files/2301.04154v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..60302142cb662949aea43113545d3473d8b302cf --- /dev/null +++ b/tNE2T4oBgHgl3EQf1wiA/content/tmp_files/2301.04154v1.pdf.txt @@ -0,0 +1,656 @@ +MNRAS 000, 1–5 (2023) +Preprint 12 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Accelerated phase-mixing in the stellar halo due to a rotating bar +Elliot Y. Davies +1★, Adam M. Dillamore +1, Eugene Vasiliev +1 and Vasily Belokurov +1. +1Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +In a galaxy merger, the stars tidally stripped from the satellite and accreted onto the host galaxy undergo phase mixing and +form finely-grained structures in the phase space. However, these fragile structures may be destroyed in the subsequent galaxy +evolution, in particular, by a rotating bar that appears well after the merger is completed. In this work, we investigate the +survivability of phase-space structures in the presence of a bar. We find that a bar with amplitude and pattern speed similar to +those of the Milky Way would blur and destroy a substantial amount of the substructure that consists of particles with pericentre +radii comparable to the bar length. While this appears to be in tension with the recent discovery of phase-space chevrons in Gaia +DR3 data, the most prominent chevrons in our simulations can still be recovered when applying the same analysis procedure as +in observations. Moreover, the smoothing effect is less pronounced in the population of stars whose angular momenta have the +opposite sign to the bar pattern speed. +Key words: Galaxy: halo – Galaxy: kinematics and dynamics – Galaxy: centre +1 INTRODUCTION +Debris stripped from a satellite galaxy onto its host during a merger +will evolve continuously in phase space, even in a static host potential +where the integrals of motion remain unchanged. This phase mixing +process results in the formation of fine substructure in phase space. +In the case of an eccentric merger, this phase space substructure +manifests as a series of chevron-like features in (𝑣𝑟, 𝑟) space (e.g. +Fillmore & Goldreich 1984; Sanderson & Helmi 2013; Dong-Páez +et al. 2022). While initially compact in phase space, the satellite de- +bris is stretched according to Liouville’s theorem as the progenitor +falls into the host potential. The merged debris then continuously +winds up into finer and finer substructure in phase space as it com- +pletes radial orbits. In the case of a relatively large satellite, not all +of the debris will be stripped in one go, and so several populations +of wound substructure will form. This continuous evolution of sub- +structure allows one to count the chevrons as if they were tree rings, +giving a hint to the age of the merger as well as to the properties of +the host potential, or the trajectory of the satellite. However, as the +debris is perturbed following the phase mixing of a large merger, this +substructure may not be preserved (e.g. Davies et al. 2022). One such +perturber could be a rotating bar. +It has long been established that the Milky Way (MW), like up +to two-thirds of all disc galaxies (Aguerri et al. 2009), has a stellar +bar at its centre. While originally found by gas kinematics (Peters +1975; Binney et al. 1991) and near-infrared emission (Blitz & Spergel +1991), evidence for the Galactic bar is now also strongly supported +by stellar kinematic data (e.g. Howard et al. 2009; Shen et al. 2010; +Debattista et al. 2017). +Recently, the discovery of a high radial anisotropy population of +★ E-mail: eyd20@cam.ac.uk +MW halo stars in the inner stellar halo (Belokurov et al. 2018; Helmi +et al. 2018) has led to the conclusion that a dwarf galaxy, dubbed +Gaia Sausage Enceladus (GSE), merged with the MW beginning +about 8 – 11 Gyrs ago. The GSE likely dominates much of the inner +stellar halo (Iorio & Belokurov 2019; Lancaster et al. 2019) and is +thought to be the most recent massive merger the MW has undergone +(e.g. Deason et al. 2013; Evans et al. 2020). With a total stellar mass +estimated to be about 108 – 109 M⊙, the GSE may account for up to +2/3 of the stellar mass in the MW halo (Fattahi et al. 2019; Dillamore +et al. 2022; Naidu et al. 2021). Conveniently, the debris from the last +major merger may be uniquely situated to studying the kinematics +of the bar, which extends no more than a few kpc from the Galactic +centre (e.g. Wegg et al. 2015; Lucey et al. 2022), as the GSE provides +a population of high eccentricity, low pericentre stars. +Observational studies into the dynamics and formation history +of the MW have been revolutionised by the Gaia mission (Gaia +Collaboration 2016), with its third data release (hereafter Gaia DR3) +providing tens of millions of stars with full 6d position and velocity +data (Vallenari et al. 2022). Motivated by the apparent detection +of chevron-shaped phase space structures in the Gaia DR3 data +(Belokurov et al. 2022), we investigate the behaviour and survival +of these chevrons in the presence of a rotating bar. Specifically, +we assume a quadrupole bar (Dehnen 2000; Monari et al. 2016) +that formed long enough after the most massive merger so that the +deposited debris was able to phase-mix sufficiently. +The outline of this work is as follows. In Section 2 we briefly +outline the details of our simulations. We present the results of these +simulations in Section 3, and compare them with Gaia DR3 data in +Section 4. Lastly, we summarise our results in Section 5. +© 2023 The Authors +arXiv:2301.04154v1 [astro-ph.GA] 10 Jan 2023 + +2 +E. Y. Davies et al. +2 SIMULATION METHOD +Initial Merger Simulation +We run an initial 𝑁-body merger simulation which is exactly the +same as the one described in Section 4.1 of Davies et al. (2022). +The merger simulation runs for a total of 5 Gyr, thereby allowing the +satellite to substantially phase mix before the introduction of a bar +potential. This 𝑁-body simulation aims to produce merged debris +with properties that approximate a GSE-like merger. The 2 × 1011 +M⊙ mass satellite galaxy is represented by 2×105 stellar and 8×105 +dark matter particles, and the 5 × 1011 M⊙ host has twice as many +particles. After evolving the 𝑁-body simulation for 5 Gyr with the +gyrfalcON code (Dehnen 2002), we create a static axisymmetric +potential from the final 𝑁-body snapshot represented by a multipole +expansion, and save the final positions and velocities of the satellite’s +stellar particles. We integrate the orbits of these stellar particles in +the static potential of the merger remnant (host plus satellite) plus +a rotating bar potential with a time-dependent amplitude, using the +Agama code (Vasiliev 2019). +Bar Potential +To represent the bar, we introduce a purely quadrupole potential with +zero total mass (Dehnen 2000) described by eqs. 1–3 in Monari +et al. (2016).The amplitude of the bar smoothly grows from zero to +a constant value 𝐴 𝑓 between times 𝑡0 and 𝑡1 as described by eq. 4 +in Dehnen (2000), with the final amplitude equivalently expressed +in terms of dimensionless parameter 𝛼 (eq. 7 in the same paper, in +which we set 𝑅0 = 8 kpc and 𝑣0 = 230 km/s). +We run two grids (one positive pattern speed, one negative pattern +speed) of 15 primary bar simulations each where the above values of +𝑅0 and 𝑣0 are always fixed. Likewise, the bar’s final radius is always +𝑅𝑏 = 2 kpc, the bar’s time of growth is always 2 Gyr from 𝑡0 = 5 + 1 +Gyr to 𝑡1 = 5 + 3 Gyr. The only parameters which are varied from +simulation to simulation are the bar strength 𝛼 and the bar pattern +speed Ωb, which has units of km/s/kpc. We consider the same values +of 𝛼 = {0.007, 0.010, 0.013} as in Dehnen (2000), and values of +|Ωb| = {36, 38, 40, 42, 44} km/s/kpc, covering a sensible range of +pattern speeds determined by observations (e.g. Sanders et al. 2019; +Lucey et al. 2022; Li et al. 2022; Leung et al. 2023). In addition +to these primary bar simulations, we run two additional sets of 15 +simulations with the same three values of alpha, but with |Ωb| = +{0, 5, 10, 20, 30} km/s/kpc, to ensure sensible behaviour down to +low pattern speed values, and to ensure any effect we see is actually +dependent on the rotation of the bar. +3 RESULTS +In this section we present our findings for how a rotating bar affects +the phase-mixed substructure, as a function of bar strength 𝛼, pattern +speed Ωb and whether the bar is co-rotating or counter-rotating with +the merged satellite debris. We retain only the satellite debris particles +with 𝐿𝑧 > 0 (∼ 2/3 of the total number), and denote the simulations +with Ωb > 0 as “co-rotating” and with Ωb < 0 as “counter-rotating”. +Note that for an eccentric GSE-like merger, there will always be a +mixture of co-rotating and counter-rotating debris. +After letting the merger debris evolve in the bar potential for a +total of 4 Gyrs, in which the bar is growing for the first 2 Gyrs, +we compare the orbital properties of the particles to a simulation in +which they evolve with no bar. We first examine the appearance of +the chevrons in the phase mixed satellite debris in the final snapshot +Figure 1. Final snapshot of the (𝑣𝑟 , 𝑟) space chevrons for all 2×105 satellite +debris particles in the preliminary simulation. The snapshot is taken after 5 +Gyrs of the initial 𝑁 body, plus 5 Gyrs of test particle evolution. Top row: +The left panel shows the evolution of the particles with no bar, whereas the +right panel shows the evolution of particles with a bar of strength 𝛼 = 0.01 +and pattern speed Ωb = 40 km/s/kpc which grows for 2 Gyr, after 6 Gyr of +no-bar phase mixing. Bottom row: The logarithm of the above density plots, +column-normalised and unsharp-masked. +of the simulation. The top row of Fig. 1 shows a comparison plot +between a simulation with no bar (left column), in which several +chevrons are clearly visible, and a simulation with a bar of strength +𝛼 = 0.01 and Ωb = 40 km/s/kpc (right column) , in which most of +the substructure disappears, except for an overdensity around 10 kpc. +However, a more sophisticated analysis procedure (identical to that +used in Belokurov et al. 2022) reveals further details. We column- +normalise the 2d density histogram and then subtract a smoothed +version of it (convolved with a Gaussian filter with width of 9 × 9 +pixels or 1.26 kpc × 60 km/s). The bottom row of Fig. 1 shows these +unsharp-masked density plots, in which finer structures disappear in +the presence of the bar, leaving only the largest chevron at 10 kpc +and a hint of another blurry one around 20 kpc. +One would expect that for a particle to have its orbital properties +changed by the presence of a bar, it must travel through (or close +to) the bar i.e. the particles pericentre 𝑟peri ≲ 𝑟bar. We confirm this +assumption in Fig. 2 by plotting the difference in energy for each +particle in the bar simulation with energy in the no-bar simulation, +𝐸 − 𝐸0. It is clear that particles with lower pericentres have a higher +change in energy. For reference, the bar’s radius (2 kpc) is shown by +the red dashed line. There is a more dramatic change as particles’ +pericentres get closer to the bar radius. We show this energy change +for all 2×105 stellar particles as a function of pericentre for the same +simulation with 𝛼 = 0.01 and Ωb = 40 km/s/kpc. Additionally, we +colour the data by median 𝐿𝑧 to reveal a dependence of the change +in energy on z-angular momentum. We note that particles whose +energies increased the most were those with high negative 𝐿𝑧. Few, +if any, particles with high negative 𝐿𝑧 had their energies decreased, +while many particles with positive 𝐿𝑧 did. This shows a tendency for +counter-rotating particles to increase their energy, and co-rotating +particles to have their energies decreased. We explore this effect +further in later plots. +Given the distinction between particles with low pericentre (i.e. +𝑟peri ≲ 𝑟bar) and with high pericentre (i.e. 𝑟peri ≳ 𝑟bar), it seems +sensible to visually inspect the chevrons when binned by pericentre. +In Fig. 3, we present the (𝑣𝑟, 𝑟) space for 3 different example simu- +lations, and split the particles up based on their pericentres. For this +sample we choose values of 𝛼 = 0.01 and |Ωb| = 40 km/s/kpc. More- +over, these particles are selected such that they only have 𝐿𝑧 > 0. The +MNRAS 000, 1–5 (2023) + +α=0.0 +400 +0=q +200 +Lkm/s. +0 +>-200 +-400α=0.01 +Qb=40400 +200 +Lkm/s +0 +>-200 +-400 +0 +10 +20 +30 +r[kpc]0 +10 +20 +30 +r[kpc]Accelerated phase-mixing due to a rotating bar +3 +400 +200 +0 +200 +400 +Lz [kpc km/s] +Figure 2. The energy change of each particle between the final snapshot +with a bar and the final snapshot without a bar, plotted against the particles’ +pericentre 1 Gyr before the bar is introduced. The dashed red vertical line +shows the bar radius of 2 kpc, and the plot is coloured by the median 𝐿𝑧. +Evidently, particles with small pericentres, comparable to the bar’s radius (2 +kpc), are affected the most. Moreover, there is a preference for particles with +large positive initial 𝐿𝑧 to have their energies decreases, and large negative +initial 𝐿𝑧 to have their energies increased. +left column shows particles with 𝑟peri < 𝑟bar and the right column +shows particles with 𝑟peri > 𝑟bar, where 𝑟bar = 2 kpc in all cases. +In both pericentres bins, the number of particles is comparable, with +𝑛 = 5.7 × 104 in the former and 𝑛 = 7.9 × 104 in the latter. From top +to bottom, we show the final snapshot of a simulation with no bar, +with a co-rotating bar, and with a counter-rotating bar. We also ran +a simulation with a non-rotating bar and found that the (𝑣𝑟, 𝑟) space +was unchanged from the no bar simulation. It is clear that the inclu- +sion of a rotating bar causes much of the substructure to be removed. +However, the co-rotating and counter-rotating bars differ in an inter- +esting way. While the particles with low 𝑟peri have their substructure +disturbed in both cases, albeit slightly less so in the counter-rotating +cases, the high 𝑟peri particles substructure is essentially unchanged +in the counter-rotating case. After reexamination of Fig. 2, this dif- +ference between co-rotating and counter-rotating could be related to +the fact that particles are more likely given an increase in energy in +the former case, and a decrease in energy in the latter. To explore +this in more detail, we examine the energy distribution of the debris +particles. +In Fig. 4 we show the 1d energy distributions in the final snap- +shots of two simulations: 𝛼 = 0.007 (0.013) in the top (bottom) rows. +Again we split the results into two bins based on pericentre, as well +as presenting them unbinned. Note that this pericentre binning also +splits the populations up into a low energy and a high energy group, +as expected. In all panels, we plot the final snapshot energies for the +simulations with no bar (grey filled histogram), for simulations with +a co-rotating bar (red line histogram), and for simulations with a +counter-rotating bar (blue line histogram). It is clear that the energy +distributions are changed more for the particles with lower pericen- +tres than for those with higher pericentres, with much more energy +substructure being retained in the high pericentre case. Moreover, we +find that the energy distribution substructures are better preserved in +the counter-rotating case. This is most easily seen in the simulation +with higher 𝛼, where the counter-rotating case still contains a large +peak, but the co-rotating case has become flatter in the middle. In +the leftmost column of Fig. 4 we no longer bin the distribution by +pericentres. Here, we show that there is an increased standard devia- +tion of energies 𝜎𝐸 for the co-rotating case, but a decreased 𝜎𝐸 for +the counter-rotating case, especially for the higher bar amplitude. In +Fig. 5, we illustrate the change in standard deviation for every sim- +ulation in the grid. Here we see that, in all cases, a counter-rotating +bar (Ωb < 0) causes the fractional change in standard deviation, +Figure 3. Final snapshot of (𝑣𝑟 , 𝑟) space, shown for four different simula- +tions. In all simulations shown (where a bar is present) the bar has a radius +of 𝑟bar = 2 kpc, and the only particles shown are those with positive initial +(1 Gyr before bar turn-on) z-angular momentum 𝐿𝑧. We cut on 𝐿𝑧 in order +to show the different effect of a co-rotating and a counter-rotating bar. We +have also split the particles into two groups; on the left we show those with +𝑟peri < 2 kpc and on the right we show those with 𝑟peri > 2 kpc. The top +row is for reference, and shows the final snapshot where no bar is present i.e. +𝛼 = 0.00 and Ω = 0 km/s/kpc. From top to bottom the next three rows show +a non-rotating bar (Ω = 0) with 𝛼 = 0.01, a rotating bar with the Ω = 40 +km/s/kpc and 𝛼 = 0.01, and finally a rotating bar with Ω = −40 km/s/kpc +and 𝛼 = 0.01. +𝛿𝜎𝐸 = (𝜎𝐸 − 𝜎𝐸0) / 𝜎𝐸0, to decrease. Moreover, the greater value +of 𝛼 causes a greater change in 𝜎𝐸. However, for the co-rotating case +(Ωb > 0), 𝜎𝐸 is always increased, and more so at larger values of +𝛼. The change in 𝜎𝐸 appears less dependent on the value of the the +pattern speed than on the bar strength. The top-left panel of Fig. 5 +shows that 𝛿𝜎𝐸 does increase as Ωb increases, but the change is from +highest to lowest Ω𝑏 is less than the change from highest to lowest +𝛼. Likewise, the bottom-left panel shows a reduced dependence on +Ωb in the counter-rotating case, for values nearer to observational +measured values of Ωb (plotted scatter points). +Lastly, we explore the change in the total energy across three sim- +ulations, this time fixing |Ωb| = 40 km/s/kpc, as the dependence on +pattern speed seems weaker than that of on bar strength. In Fig. 6 +we show the total energy of all particles (with 𝐿𝑧 > 0) as a func- +tion of time. The dotted lines shows counter-rotating simulations +and the solid lines show co-rotating simulations. The grey shaded +region indicates the time over which the bar grows from minimum +to maximum amplitude. Here we see clearly that the counter-rotating +bar causes an increase in the total energy of the particles, while the +co-rotating bar causes a total decrease of the energies. Again, these +effects are amplified when the bar strength is increased. +Combining the information learned from the results above, we +see that the effect of a co-rotating bar is to decrease the energies +of the low energy (low pericentre) population, while hardly chang- +ing the energies of the high-energy (high pericentre) populations. +This causes the energy spread to widen, and so some substructure +gets washed out. Moreover, since some particles lose energy, they +fall closer into the bar and become even more impacted. To confirm +MNRAS 000, 1–5 (2023) + +0.3 +[105 (km/s)2] +α=0.01 +0.2 +Ωb=40 +0.1 +0.0 +Eo +0.1 +-0.2 +-0.3 +2 +4 +6 +0 +8 +10 +rperi [kpc]4 +rperi rbar4 +Vr [102 km/s] +2 +0 +-2 +α=0.01 +Qb=40 +-44 +[102 km/s] +2 +0 +-2 +α=0.01 +Qb=-40 +0 +10 +20 +30 +40 +50 +r[kpc]0 +10 +20 +30 +40 +50 +r[kpc]4 +E. Y. Davies et al. +Figure 4. Histograms of energy distributions for two different bar simulations, +both with |Ωb | = 40 km/s/kpc. Again, we only show particles with initial +𝐿𝑧 > 0. The top row shows the final snapshot energy distributions for a +bar with 𝛼 = 0.007 and the bottom the distribution for 𝛼 = 0.013. Each +panel shows distributions for in a no-bar simulation in grey, a co-rotating bar +(Ωb = +40) in red, and a counter-rotating bar (Ωb = −40) in blue. In the left +column, we show particles of all pericentres, and indicate the corresponding +energy spreads, 𝜎𝐸. In the middle and right column we show the particles +binned by pericentre within the bar (𝑟peri < 2 kpc) radius and beyond the bar +radius (𝑟peri > 2 kpc), respectively. +Figure 5. The change in energy standard deviation 𝜎𝐸 of all 𝐿𝑧 > 0 particles +between simulations with a bar and simulations without a bar, plotted against +bar strength 𝛼 (right panel) and bar pattern speed Ωb (left panel). The top +panel shows simulations where the bar is co-rotating (Ωb > 0), and the bottom +panel shows simulations where the bar is counter-rotating (Ωb < 0). +this, we checked that the median Galactocentric radius of the debris +particles decreased in the co-rotating case. However, in a counter- +rotating bar, the low energy population has an injection of energy, +therefore narrowing the energy spread, since the high energy pop- +ulation remains essentially unchanged again. In this latter scenario, +more substructure is retained since the energy distribution doesn’t +get diluted in the same way, and the particles are not falling further +into the bar’s area of influence. The chevrons that do survive appear +to do so because they were initially overdense enough, and even with +dilution due to changing energies there remains enough particles +with similar enough energies to retain some substructure. +Figure 6. The total energy change of particles with 𝐿𝑧 > 0 in the stellar debris +as a function of time, for several simulations all with the same amplitude of +pattern speed, |Ωb | = 40 km/s/kpc. We consider both co-rotating (Ωb > 0) +and counter-rotating (Ωb < 0) bars, for bar strengths of 𝛼 = 0.007, 0.010 +and 0.013. The solid lines show the energy change for co-rotating bars, while +the dashed lines show the change for counter-rotating bars. The grey shaded +area indicates the time in which the bar is growing from an amplitude of zero +at 1 Gyr to its maximum amplitude at 3 Gyr. +4 DATA COMPARISON +From the results, we have learned that the bar’s influence on the +visual appearance of the (𝑣𝑟, 𝑟) chevrons is substantial for a range +of strengths and pattern speeds, but more so for particles with low +pericentres. With this is mind, it is important to reexamine the the +chevrons discovered in Gaia DR3. In Fig. 7 we present these data, +which has been cut down from the original 25 million with full +6-d information, in a similar same way as described in Belokurov +et al. (2022), ultimately providing us with about 1.5 × 105 halo- +like stars with |𝐿𝑧| < 500 kpc km/s. Here, we separate the data +into two pericentres bins. We calculate the pericentres using the +MilkyWayPotential from Gala (Price-Whelan 2017). The top +row shows the data for 𝐿𝑧 > 0, whereas the bottom rows shows the +data for 𝐿𝑧 < 0. The left column shows the data with 𝑟peri < 2 +kpc (quoted in the figure as 𝑟bar), and the right columns shows the +with 𝑟peri > 2 kpc. From this binning, we see that the chevron- +like features disappear when we consider only particles with larger +pericentres. Initially, this seems somewhat in tension with the work +above which suggests the bar should wipe smooth any substructure +with low pericentres. However, we point back to Fig. 1, where the +bottom row has been processed in the same was the Gaia DR3 data. +This figure illustrates that some phase-mixed substructure actually +remains recoverable even in the case of a rotating bar. However, the +existence of the chevrons in the data only within influence of the bar +points to the possibility that they are not the result of phase mixing +and instead are formed via resonance effects of the bar. We explore +this bar shepherding of the halo sub-structure further in an upcoming +companion paper by Dillamore et al. (in prep.). +5 SUMMARY +In this work we explore the impact of a rotating bar on the phase- +space substructure that results from the phase mixing of debris from +a large 𝑁-body merger. We allow the merger debris to phase mix +for 5 Gyr in a live 𝑁-body simulation, and then switch to a test- +particle integration of debris orbits in an axisymmetric potential of +the merger remnant, into which a bar component is gradually added +over the course of 2 Gyr. We run two sets of 30 bar test particle +simulations, where all parameters of the bar are fixed except for the +values of the bar strength 𝛼 = {0.007, 0.010, 0.013} and the values of +the pattern speed Ωb = {0, 5, 10, 20, 30, 36, 38, 40, 42, 44} km/s/kpc. +MNRAS 000, 1–5 (2023) + +co-rot. +5 +Oe = 1.70e4 +ct-rot. +α=0.007 +4 +O = 1.64e4 +[Qb|=40 +no bar +density +Oe = 1.65e4 +3 +2 +1all rperi +peri rbarco-rot. +5 +O = 1.80e4 +ct-rot. +α=0.013 +4 +Qe = 1.60e4 +[Qb|=40 +no bar +density +Qe = 1.65e4 +3 +2 +1 +0 +-1.0 +-0.6 +-0.2 +E[105 (km/s)21-1.0 +-0.6 +E[105 (km/s)2]-1.0 +-0.6 +-0.2 +E[105 (km/s)21b> 0 +0.10 +α= 0.013 +α=0.010 +α= 0.007 +0.08 +QEo) +0.06 +0.04 +L +6 +0.02 +0.00[Qb|= 44 +[Qb|= 42 +[Qb|=40 +IQbl = 38 +[Qb|=360.000 +o>q +-0.005 +0 +-0.010 +QEo +-0.015 +- +-0.020 +OE +-0.025 +-0.030 +5 +10 +203036 +538 +40 +42 +44 +pat. speed, Qb [km/s/kpc7 +10 +13 +strength, α/103-1.00 +[Qbl = 40 +bar growing +total E [1010 (km/s)2] +α=0.013 +α=0.010 +-1.01 +α= 0.007 +ct-rot. +-1.02 +co-rot. +-1.03 +-1.04 +-1.05 +2 +0 +1 +3 +4 +5 +t[Gyr]Accelerated phase-mixing due to a rotating bar +5 +Figure 7. The local GSE (𝑣𝑟 , 𝑟) phase space chevrons found using Gaia +DR3, binned by angular momentum and by pericentre. All panels show the +logarithm of column normalised density with smooth background removed. +The top row shows all stars with 𝐿𝑧 > 0, whereas the bottom row shows all +with 𝐿𝑧 < 0. The left column shows only data with 𝑟peri < 2 kpc, and the +right column shows only data with 𝑟peri < 2 kpc. Note how the chevrons are +no longer visible in the right column. +We show how the bar impacts the visual presentation of the (𝑣𝑟, 𝑟) +phase space substructure, as well as the distribution of energies. We +bin the results by pericentre, separating particles with 𝑟peri < 𝑟bar +from those with 𝑟peri > 𝑟bar. Additionally, we investigate the dif- +ference between a co-rotating bar and a counter-rotating bar. Lastly, +we compare our results to the data discovered in Gaia DR3 by (Be- +lokurov et al. 2022). Our findings can be summarised as follows: +(i) Particles with 𝑟peri < 𝑟bar have their orbital properties changed +more than those with 𝑟peri > 𝑟bar. +(ii) Satellite debris particles that are counter-rotating (co-rotating) +with the bar have their total energy increased (decreased) and their +energy spread decreased (increased). +(iii) In all simulations with bar pattern speed Ωb comparable to +that of the Milky Way, the chevrons in the (𝑣𝑟, 𝑟) phase space are +substantially blurred. However, some of the substructure is recov- +erable when an unsharp-masking filter is applied, as with the Gaia +DR3 data. +(iv) The visual appearance of the (𝑣𝑟, 𝑟) substructure is much less +impacted in the case of a counter-rotating bar than in the case of a +co-rotating bar. +(v) The chevrons found in the Gaia DR3 data seem to consist +entirely of particles with low pericentre, which may suggests that they +are the result of bar related phenomena, such as resonant interactions. +ACKNOWLEDGEMENTS +EYD thanks the Science and Technology Facilities Council (STFC) +for a PhD studentship (UKRI grant number 2605433). AMD thanks +STFC for a PhD studentship (UKRI grant number 2604986). +DATA AVAILABILITY +The simulations and analysis in this project can be reproduced with +publicly available software. We also make use of publicly available +Gaia DR3 data. +REFERENCES +Aguerri J. A. L., Méndez-Abreu J., Corsini E. M., 2009, A&A, 495, 491 +Belokurov V., Erkal D., Evans N. W., Koposov S. 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P., 2022, Astronomy & Astrophysics +Vasiliev E., 2019, MNRAS, 482, 1525 +Wegg C., Gerhard O., Portail M., 2015, MNRAS, 450, 4050 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–5 (2023) + + rbar400 +200 +VrLkm/s +0 +-200 +-400 +0 +2 +4 +6 +8 +10 +12 +14 +r「kpc]rperi > rbar +0 +2 +6 +8 +10 +12 +4 +14 +r[kpc] \ No newline at end of file diff --git a/tNE2T4oBgHgl3EQf1wiA/content/tmp_files/load_file.txt b/tNE2T4oBgHgl3EQf1wiA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e60fa02ee5784dc19cd2ece04b6783a14ba3555b --- /dev/null +++ b/tNE2T4oBgHgl3EQf1wiA/content/tmp_files/load_file.txt @@ -0,0 +1,545 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf,len=544 +page_content='MNRAS 000, 1–5 (2023) Preprint 12 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='0 Accelerated phase-mixing in the stellar halo due to a rotating bar Elliot Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Davies 1★, Adam M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Dillamore 1, Eugene Vasiliev 1 and Vasily Belokurov 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 1Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' in original form ZZZ ABSTRACT In a galaxy merger, the stars tidally stripped from the satellite and accreted onto the host galaxy undergo phase mixing and form finely-grained structures in the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' However, these fragile structures may be destroyed in the subsequent galaxy evolution, in particular, by a rotating bar that appears well after the merger is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' In this work, we investigate the survivability of phase-space structures in the presence of a bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We find that a bar with amplitude and pattern speed similar to those of the Milky Way would blur and destroy a substantial amount of the substructure that consists of particles with pericentre radii comparable to the bar length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' While this appears to be in tension with the recent discovery of phase-space chevrons in Gaia DR3 data, the most prominent chevrons in our simulations can still be recovered when applying the same analysis procedure as in observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Moreover, the smoothing effect is less pronounced in the population of stars whose angular momenta have the opposite sign to the bar pattern speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Key words: Galaxy: halo – Galaxy: kinematics and dynamics – Galaxy: centre 1 INTRODUCTION Debris stripped from a satellite galaxy onto its host during a merger will evolve continuously in phase space, even in a static host potential where the integrals of motion remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' This phase mixing process results in the formation of fine substructure in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' In the case of an eccentric merger, this phase space substructure manifests as a series of chevron-like features in (𝑣𝑟, 𝑟) space (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Fillmore & Goldreich 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Sanderson & Helmi 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Dong-Páez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' While initially compact in phase space, the satellite de- bris is stretched according to Liouville’s theorem as the progenitor falls into the host potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The merged debris then continuously winds up into finer and finer substructure in phase space as it com- pletes radial orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' In the case of a relatively large satellite, not all of the debris will be stripped in one go, and so several populations of wound substructure will form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' This continuous evolution of sub- structure allows one to count the chevrons as if they were tree rings, giving a hint to the age of the merger as well as to the properties of the host potential, or the trajectory of the satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' However, as the debris is perturbed following the phase mixing of a large merger, this substructure may not be preserved (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' One such perturber could be a rotating bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' It has long been established that the Milky Way (MW), like up to two-thirds of all disc galaxies (Aguerri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2009), has a stellar bar at its centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' While originally found by gas kinematics (Peters 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Binney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 1991) and near-infrared emission (Blitz & Spergel 1991), evidence for the Galactic bar is now also strongly supported by stellar kinematic data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Debattista et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Recently, the discovery of a high radial anisotropy population of ★ E-mail: eyd20@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='uk MW halo stars in the inner stellar halo (Belokurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Helmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2018) has led to the conclusion that a dwarf galaxy, dubbed Gaia Sausage Enceladus (GSE), merged with the MW beginning about 8 – 11 Gyrs ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The GSE likely dominates much of the inner stellar halo (Iorio & Belokurov 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Lancaster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2019) and is thought to be the most recent massive merger the MW has undergone (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Deason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' With a total stellar mass estimated to be about 108 – 109 M⊙, the GSE may account for up to 2/3 of the stellar mass in the MW halo (Fattahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Dillamore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Conveniently, the debris from the last major merger may be uniquely situated to studying the kinematics of the bar, which extends no more than a few kpc from the Galactic centre (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Wegg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Lucey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2022), as the GSE provides a population of high eccentricity, low pericentre stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Observational studies into the dynamics and formation history of the MW have been revolutionised by the Gaia mission (Gaia Collaboration 2016), with its third data release (hereafter Gaia DR3) providing tens of millions of stars with full 6d position and velocity data (Vallenari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Motivated by the apparent detection of chevron-shaped phase space structures in the Gaia DR3 data (Belokurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2022), we investigate the behaviour and survival of these chevrons in the presence of a rotating bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Specifically, we assume a quadrupole bar (Dehnen 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Monari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2016) that formed long enough after the most massive merger so that the deposited debris was able to phase-mix sufficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The outline of this work is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' In Section 2 we briefly outline the details of our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We present the results of these simulations in Section 3, and compare them with Gaia DR3 data in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Lastly, we summarise our results in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='04154v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='GA] 10 Jan 2023 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2 SIMULATION METHOD Initial Merger Simulation We run an initial 𝑁-body merger simulation which is exactly the same as the one described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='1 of Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The merger simulation runs for a total of 5 Gyr, thereby allowing the satellite to substantially phase mix before the introduction of a bar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' This 𝑁-body simulation aims to produce merged debris with properties that approximate a GSE-like merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The 2 × 1011 M⊙ mass satellite galaxy is represented by 2×105 stellar and 8×105 dark matter particles, and the 5 × 1011 M⊙ host has twice as many particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' After evolving the 𝑁-body simulation for 5 Gyr with the gyrfalcON code (Dehnen 2002), we create a static axisymmetric potential from the final 𝑁-body snapshot represented by a multipole expansion, and save the final positions and velocities of the satellite’s stellar particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We integrate the orbits of these stellar particles in the static potential of the merger remnant (host plus satellite) plus a rotating bar potential with a time-dependent amplitude, using the Agama code (Vasiliev 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Bar Potential To represent the bar, we introduce a purely quadrupole potential with zero total mass (Dehnen 2000) described by eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 1–3 in Monari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='The amplitude of the bar smoothly grows from zero to a constant value 𝐴 𝑓 between times 𝑡0 and 𝑡1 as described by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 4 in Dehnen (2000), with the final amplitude equivalently expressed in terms of dimensionless parameter 𝛼 (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 7 in the same paper, in which we set 𝑅0 = 8 kpc and 𝑣0 = 230 km/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We run two grids (one positive pattern speed, one negative pattern speed) of 15 primary bar simulations each where the above values of 𝑅0 and 𝑣0 are always fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Likewise, the bar’s final radius is always 𝑅𝑏 = 2 kpc, the bar’s time of growth is always 2 Gyr from 𝑡0 = 5 + 1 Gyr to 𝑡1 = 5 + 3 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The only parameters which are varied from simulation to simulation are the bar strength 𝛼 and the bar pattern speed Ωb, which has units of km/s/kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We consider the same values of 𝛼 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='007, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='010, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='013} as in Dehnen (2000), and values of |Ωb| = {36, 38, 40, 42, 44} km/s/kpc, covering a sensible range of pattern speeds determined by observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Lucey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' In addition to these primary bar simulations, we run two additional sets of 15 simulations with the same three values of alpha, but with |Ωb| = {0, 5, 10, 20, 30} km/s/kpc, to ensure sensible behaviour down to low pattern speed values, and to ensure any effect we see is actually dependent on the rotation of the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 3 RESULTS In this section we present our findings for how a rotating bar affects the phase-mixed substructure, as a function of bar strength 𝛼, pattern speed Ωb and whether the bar is co-rotating or counter-rotating with the merged satellite debris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We retain only the satellite debris particles with 𝐿𝑧 > 0 (∼ 2/3 of the total number), and denote the simulations with Ωb > 0 as “co-rotating” and with Ωb < 0 as “counter-rotating”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Note that for an eccentric GSE-like merger, there will always be a mixture of co-rotating and counter-rotating debris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' After letting the merger debris evolve in the bar potential for a total of 4 Gyrs, in which the bar is growing for the first 2 Gyrs, we compare the orbital properties of the particles to a simulation in which they evolve with no bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We first examine the appearance of the chevrons in the phase mixed satellite debris in the final snapshot Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Final snapshot of the (𝑣𝑟 , 𝑟) space chevrons for all 2×105 satellite debris particles in the preliminary simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The snapshot is taken after 5 Gyrs of the initial 𝑁 body, plus 5 Gyrs of test particle evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Top row: The left panel shows the evolution of the particles with no bar, whereas the right panel shows the evolution of particles with a bar of strength 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='01 and pattern speed Ωb = 40 km/s/kpc which grows for 2 Gyr, after 6 Gyr of no-bar phase mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Bottom row: The logarithm of the above density plots, column-normalised and unsharp-masked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 1 shows a comparison plot between a simulation with no bar (left column), in which several chevrons are clearly visible, and a simulation with a bar of strength 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='01 and Ωb = 40 km/s/kpc (right column) , in which most of the substructure disappears, except for an overdensity around 10 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' However, a more sophisticated analysis procedure (identical to that used in Belokurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2022) reveals further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We column- normalise the 2d density histogram and then subtract a smoothed version of it (convolved with a Gaussian filter with width of 9 × 9 pixels or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='26 kpc × 60 km/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The bottom row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 1 shows these unsharp-masked density plots, in which finer structures disappear in the presence of the bar, leaving only the largest chevron at 10 kpc and a hint of another blurry one around 20 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' One would expect that for a particle to have its orbital properties changed by the presence of a bar, it must travel through (or close to) the bar i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' the particles pericentre 𝑟peri ≲ 𝑟bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We confirm this assumption in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2 by plotting the difference in energy for each particle in the bar simulation with energy in the no-bar simulation, 𝐸 − 𝐸0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' It is clear that particles with lower pericentres have a higher change in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' For reference, the bar’s radius (2 kpc) is shown by the red dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' There is a more dramatic change as particles’ pericentres get closer to the bar radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We show this energy change for all 2×105 stellar particles as a function of pericentre for the same simulation with 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='01 and Ωb = 40 km/s/kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Additionally, we colour the data by median 𝐿𝑧 to reveal a dependence of the change in energy on z-angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We note that particles whose energies increased the most were those with high negative 𝐿𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Few, if any, particles with high negative 𝐿𝑧 had their energies decreased, while many particles with positive 𝐿𝑧 did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' This shows a tendency for counter-rotating particles to increase their energy, and co-rotating particles to have their energies decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We explore this effect further in later plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Given the distinction between particles with low pericentre (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 𝑟peri ≲ 𝑟bar) and with high pericentre (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 𝑟peri ≳ 𝑟bar), it seems sensible to visually inspect the chevrons when binned by pericentre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 3, we present the (𝑣𝑟, 𝑟) space for 3 different example simu- lations, and split the particles up based on their pericentres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' For this sample we choose values of 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='01 and |Ωb| = 40 km/s/kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' More- over, these particles are selected such that they only have 𝐿𝑧 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The MNRAS 000, 1–5 (2023) α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='0 400 0=q 200 Lkm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 0 >-200 400α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='01 Qb=40400 200 Lkm/s 0 >-200 400 0 10 20 30 r[kpc]0 10 20 30 r[kpc]Accelerated phase-mixing due to a rotating bar 3 400 200 0 200 400 Lz [kpc km/s] Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The energy change of each particle between the final snapshot with a bar and the final snapshot without a bar, plotted against the particles’ pericentre 1 Gyr before the bar is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The dashed red vertical line shows the bar radius of 2 kpc, and the plot is coloured by the median 𝐿𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Evidently, particles with small pericentres, comparable to the bar’s radius (2 kpc), are affected the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Moreover, there is a preference for particles with large positive initial 𝐿𝑧 to have their energies decreases, and large negative initial 𝐿𝑧 to have their energies increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' left column shows particles with 𝑟peri < 𝑟bar and the right column shows particles with 𝑟peri > 𝑟bar, where 𝑟bar = 2 kpc in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' In both pericentres bins, the number of particles is comparable, with 𝑛 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='7 × 104 in the former and 𝑛 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='9 × 104 in the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' From top to bottom, we show the final snapshot of a simulation with no bar, with a co-rotating bar, and with a counter-rotating bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We also ran a simulation with a non-rotating bar and found that the (𝑣𝑟, 𝑟) space was unchanged from the no bar simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' It is clear that the inclu- sion of a rotating bar causes much of the substructure to be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' However, the co-rotating and counter-rotating bars differ in an inter- esting way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' While the particles with low 𝑟peri have their substructure disturbed in both cases, albeit slightly less so in the counter-rotating cases, the high 𝑟peri particles substructure is essentially unchanged in the counter-rotating case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' After reexamination of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2, this dif- ference between co-rotating and counter-rotating could be related to the fact that particles are more likely given an increase in energy in the former case, and a decrease in energy in the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' To explore this in more detail, we examine the energy distribution of the debris particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 4 we show the 1d energy distributions in the final snap- shots of two simulations: 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='007 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='013) in the top (bottom) rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Again we split the results into two bins based on pericentre, as well as presenting them unbinned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Note that this pericentre binning also splits the populations up into a low energy and a high energy group, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' In all panels, we plot the final snapshot energies for the simulations with no bar (grey filled histogram), for simulations with a co-rotating bar (red line histogram), and for simulations with a counter-rotating bar (blue line histogram).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' It is clear that the energy distributions are changed more for the particles with lower pericen- tres than for those with higher pericentres, with much more energy substructure being retained in the high pericentre case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Moreover, we find that the energy distribution substructures are better preserved in the counter-rotating case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' This is most easily seen in the simulation with higher 𝛼, where the counter-rotating case still contains a large peak, but the co-rotating case has become flatter in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' In the leftmost column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 4 we no longer bin the distribution by pericentres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Here, we show that there is an increased standard devia- tion of energies 𝜎𝐸 for the co-rotating case, but a decreased 𝜎𝐸 for the counter-rotating case, especially for the higher bar amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 5, we illustrate the change in standard deviation for every sim- ulation in the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Here we see that, in all cases, a counter-rotating bar (Ωb < 0) causes the fractional change in standard deviation, Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Final snapshot of (𝑣𝑟 , 𝑟) space, shown for four different simula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' In all simulations shown (where a bar is present) the bar has a radius of 𝑟bar = 2 kpc, and the only particles shown are those with positive initial (1 Gyr before bar turn-on) z-angular momentum 𝐿𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We cut on 𝐿𝑧 in order to show the different effect of a co-rotating and a counter-rotating bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We have also split the particles into two groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' on the left we show those with 𝑟peri < 2 kpc and on the right we show those with 𝑟peri > 2 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The top row is for reference, and shows the final snapshot where no bar is present i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='00 and Ω = 0 km/s/kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' From top to bottom the next three rows show a non-rotating bar (Ω = 0) with 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='01, a rotating bar with the Ω = 40 km/s/kpc and 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='01, and finally a rotating bar with Ω = −40 km/s/kpc and 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 𝛿𝜎𝐸 = (𝜎𝐸 − 𝜎𝐸0) / 𝜎𝐸0, to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Moreover, the greater value of 𝛼 causes a greater change in 𝜎𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' However, for the co-rotating case (Ωb > 0), 𝜎𝐸 is always increased, and more so at larger values of 𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The change in 𝜎𝐸 appears less dependent on the value of the the pattern speed than on the bar strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The top-left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 5 shows that 𝛿𝜎𝐸 does increase as Ωb increases, but the change is from highest to lowest Ω𝑏 is less than the change from highest to lowest 𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Likewise, the bottom-left panel shows a reduced dependence on Ωb in the counter-rotating case, for values nearer to observational measured values of Ωb (plotted scatter points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Lastly, we explore the change in the total energy across three sim- ulations, this time fixing |Ωb| = 40 km/s/kpc, as the dependence on pattern speed seems weaker than that of on bar strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 6 we show the total energy of all particles (with 𝐿𝑧 > 0) as a func- tion of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The dotted lines shows counter-rotating simulations and the solid lines show co-rotating simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The grey shaded region indicates the time over which the bar grows from minimum to maximum amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Here we see clearly that the counter-rotating bar causes an increase in the total energy of the particles, while the co-rotating bar causes a total decrease of the energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Again, these effects are amplified when the bar strength is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Combining the information learned from the results above, we see that the effect of a co-rotating bar is to decrease the energies of the low energy (low pericentre) population, while hardly chang- ing the energies of the high-energy (high pericentre) populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' This causes the energy spread to widen, and so some substructure gets washed out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Moreover, since some particles lose energy, they fall closer into the bar and become even more impacted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' To confirm MNRAS 000, 1–5 (2023) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='3 [105 (km/s)2] α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='2 Ωb=40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='0 Eo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='3 2 4 6 0 8 10 rperi [kpc]4 rperi rbar4 Vr [102 km/s] 2 0 2 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='01 Qb=40 44 [102 km/s] 2 0 2 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='01 Qb=-40 0 10 20 30 40 50 r[kpc]0 10 20 30 40 50 r[kpc]4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Histograms of energy distributions for two different bar simulations, both with |Ωb | = 40 km/s/kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Again, we only show particles with initial 𝐿𝑧 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The top row shows the final snapshot energy distributions for a bar with 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='007 and the bottom the distribution for 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Each panel shows distributions for in a no-bar simulation in grey, a co-rotating bar (Ωb = +40) in red, and a counter-rotating bar (Ωb = −40) in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' In the left column, we show particles of all pericentres, and indicate the corresponding energy spreads, 𝜎𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' In the middle and right column we show the particles binned by pericentre within the bar (𝑟peri < 2 kpc) radius and beyond the bar radius (𝑟peri > 2 kpc), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The change in energy standard deviation 𝜎𝐸 of all 𝐿𝑧 > 0 particles between simulations with a bar and simulations without a bar, plotted against bar strength 𝛼 (right panel) and bar pattern speed Ωb (left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The top panel shows simulations where the bar is co-rotating (Ωb > 0), and the bottom panel shows simulations where the bar is counter-rotating (Ωb < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' this, we checked that the median Galactocentric radius of the debris particles decreased in the co-rotating case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' However, in a counter- rotating bar, the low energy population has an injection of energy, therefore narrowing the energy spread, since the high energy pop- ulation remains essentially unchanged again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' In this latter scenario, more substructure is retained since the energy distribution doesn’t get diluted in the same way, and the particles are not falling further into the bar’s area of influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The chevrons that do survive appear to do so because they were initially overdense enough, and even with dilution due to changing energies there remains enough particles with similar enough energies to retain some substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The total energy change of particles with 𝐿𝑧 > 0 in the stellar debris as a function of time, for several simulations all with the same amplitude of pattern speed, |Ωb | = 40 km/s/kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We consider both co-rotating (Ωb > 0) and counter-rotating (Ωb < 0) bars, for bar strengths of 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='007, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='010 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The solid lines show the energy change for co-rotating bars, while the dashed lines show the change for counter-rotating bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The grey shaded area indicates the time in which the bar is growing from an amplitude of zero at 1 Gyr to its maximum amplitude at 3 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 4 DATA COMPARISON From the results, we have learned that the bar’s influence on the visual appearance of the (𝑣𝑟, 𝑟) chevrons is substantial for a range of strengths and pattern speeds, but more so for particles with low pericentres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' With this is mind, it is important to reexamine the the chevrons discovered in Gaia DR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 7 we present these data, which has been cut down from the original 25 million with full 6-d information, in a similar same way as described in Belokurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' (2022), ultimately providing us with about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='5 × 105 halo- like stars with |𝐿𝑧| < 500 kpc km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Here, we separate the data into two pericentres bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We calculate the pericentres using the MilkyWayPotential from Gala (Price-Whelan 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The top row shows the data for 𝐿𝑧 > 0, whereas the bottom rows shows the data for 𝐿𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The left column shows the data with 𝑟peri < 2 kpc (quoted in the figure as 𝑟bar), and the right columns shows the with 𝑟peri > 2 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' From this binning, we see that the chevron- like features disappear when we consider only particles with larger pericentres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Initially, this seems somewhat in tension with the work above which suggests the bar should wipe smooth any substructure with low pericentres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' However, we point back to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 1, where the bottom row has been processed in the same was the Gaia DR3 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' This figure illustrates that some phase-mixed substructure actually remains recoverable even in the case of a rotating bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' However, the existence of the chevrons in the data only within influence of the bar points to the possibility that they are not the result of phase mixing and instead are formed via resonance effects of the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We explore this bar shepherding of the halo sub-structure further in an upcoming companion paper by Dillamore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' (in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 5 SUMMARY In this work we explore the impact of a rotating bar on the phase- space substructure that results from the phase mixing of debris from a large 𝑁-body merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We allow the merger debris to phase mix for 5 Gyr in a live 𝑁-body simulation, and then switch to a test- particle integration of debris orbits in an axisymmetric potential of the merger remnant, into which a bar component is gradually added over the course of 2 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We run two sets of 30 bar test particle simulations, where all parameters of the bar are fixed except for the values of the bar strength 𝛼 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='007, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='010, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='013} and the values of the pattern speed Ωb = {0, 5, 10, 20, 30, 36, 38, 40, 42, 44} km/s/kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' MNRAS 000, 1–5 (2023) co-rot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 5 Oe = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='70e4 ct-rot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='007 4 O = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='64e4 [Qb|=40 no bar density Oe = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='65e4 3 2 1all rperi peri rbarco-rot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 5 O = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='80e4 ct-rot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='013 4 Qe = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='60e4 [Qb|=40 no bar density Qe = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='65e4 3 2 1 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='2 E[105 (km/s)21-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='6 E[105 (km/s)2]-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='2 E[105 (km/s)21b> 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='10 α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='013 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='010 α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='08 QEo) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='04 L 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='00[Qb|= 44 [Qb|= 42 [Qb|=40 IQbl = 38 [Qb|=360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='000 o>q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='005 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='010 QEo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='020 OE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='030 5 10 203036 538 40 42 44 pat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' speed, Qb [km/s/kpc7 10 13 strength, α/103-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='00 [Qbl = 40 bar growing total E [1010 (km/s)2] α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='013 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='01 α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='007 ct-rot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='02 co-rot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content='05 2 0 1 3 4 5 t[Gyr]Accelerated phase-mixing due to a rotating bar 5 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The local GSE (𝑣𝑟 , 𝑟) phase space chevrons found using Gaia DR3, binned by angular momentum and by pericentre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' All panels show the logarithm of column normalised density with smooth background removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The top row shows all stars with 𝐿𝑧 > 0, whereas the bottom row shows all with 𝐿𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' The left column shows only data with 𝑟peri < 2 kpc, and the right column shows only data with 𝑟peri < 2 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Note how the chevrons are no longer visible in the right column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We show how the bar impacts the visual presentation of the (𝑣𝑟, 𝑟) phase space substructure, as well as the distribution of energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' We bin the results by pericentre, separating particles with 𝑟peri < 𝑟bar from those with 𝑟peri > 𝑟bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Additionally, we investigate the dif- ference between a co-rotating bar and a counter-rotating bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Lastly, we compare our results to the data discovered in Gaia DR3 by (Be- lokurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' Our findings can be summarised as follows: (i) Particles with 𝑟peri < 𝑟bar have their orbital properties changed more than those with 𝑟peri > 𝑟bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' (ii) Satellite debris particles that are counter-rotating (co-rotating) with the bar have their total energy increased (decreased) and their energy spread decreased (increased).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' (iii) In all simulations with bar pattern speed Ωb comparable to that of the Milky Way, the chevrons in the (𝑣𝑟, 𝑟) phase space are substantially blurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' However, some of the substructure is recov- erable when an unsharp-masking filter is applied, as with the Gaia DR3 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' (iv) The visual appearance of the (𝑣𝑟, 𝑟) substructure is much less impacted in the case of a counter-rotating bar than in the case of a co-rotating bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' (v) The chevrons found in the Gaia DR3 data seem to consist entirely of particles with low pericentre, which may suggests that they are the result of bar related phenomena, such as resonant interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' ACKNOWLEDGEMENTS EYD thanks the Science and Technology Facilities Council (STFC) for a PhD studentship (UKRI grant number 2605433).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' AMD thanks STFC for a PhD studentship (UKRI grant number 2604986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} +page_content=' DATA AVAILABILITY The simulations and analysis in this project can be reproduced with publicly available software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE2T4oBgHgl3EQf1wiA/content/2301.04154v1.pdf'} 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b/tNE4T4oBgHgl3EQfWAwH/content/tmp_files/2301.05028v1.pdf.txt @@ -0,0 +1,550 @@ +Available online at www.sciencedirect.com +Procedia CIRP 00 (2022) 000–000 +www.elsevier.com/locate/procedia +9th CIRP Conference on Assembly Technology and Systems +MotorFactory: A Blender Add-on for Large Dataset Generation +of Small Electric Motors +Chengzhi Wu *a, Kanran Zhoua, Jan-Philipp Kaiserb, Norbert Mitschkec, Jan-Felix Kleind, +Julius Pfrommere, J¨urgen Beyerera,e, Gisela Lanzab, Michael Heizmannc, Kai Furmansd +aInstitute for Anthropomatics and Robotics, Karlsruhe Institute of Technology, Kaiserstraße 12, 76131 Karlsruhe, Germany +bwbk Institute of Production Science, Karlsruhe Institute of Technology, Kaiserstraße 12, 76131 Karlsruhe, Germany +cInstitute of Industrial Information Technology, Karlsruhe Institute of Technology, Hertzstraße 16, 76187 Karlsruhe, Germany +dInstitute for Material Handling and Logistics, Karlsruhe Institute of Technology, Kaiserstraße 12, 76131 Karlsruhe, Germany +eFraunhofer Institute of Optronics, System Technologies and Image Exploitation IOSB, Fraunhoferstraße 1, 76131 Karlsruhe, Germany +* Corresponding author. Tel.: +49-(0)1523 8476995. E-mail address: chengzhi.wu@kit.edu +Abstract +To enable automatic disassembly of different product types with uncertain condition and degree of wear in remanufacturing, agile production +systems that can adapt dynamically to changing requirements are needed. Machine learning algorithms can be employed due to their generalization +capabilities of learning from various types and variants of products. However, in reality, datasets with a diversity of samples that can be used to +train models are difficult to obtain in the initial period. This may cause bad performances when the system tries to adapt to new unseen input data +in the future. In order to generate large datasets for different learning purposes, in our project, we present a Blender add-on named MotorFactory to +generate customized mesh models of various motor instances. MotorFactory allows to create mesh models which, complemented with additional +add-ons, can be further used to create synthetic RGB images, depth images, normal images, segmentation ground truth masks and 3D point cloud +datasets with point-wise semantic labels. The created synthetic datasets may be used for various tasks including motor type classification, object +detection for decentralized material transfer tasks, part segmentation for disassembly and handling tasks, or even reinforcement learning-based +robotics control or view-planning. +© 2022 The Authors. Published by Elsevier Ltd. +This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) +Peer-review under responsibility of the scientific committee of the 9th CIRP Conference on Assembly Technology and Systems. +Keywords: Remanufacturing; Blender Add-on; Machine learning; Large synthetic dataset generation +1. Introduction +Today’s industrial landscape is characterised by linear +economies. End-of-life (EOL) strategies for products such as +remanufacturing, in which used products are reprocessed, offer +the potential to decouple resource consumption from sustain- +able economic growth [25]. During the remanufacturing pro- +cess, the products are inspected and disassembled, then indi- +vidual parts are reworked or exchanged and again reassembled +into final products [24]. In contrast to related EOL-strategies, +remanufactured products ensure functionality and quality that +is equivalent or better compared to a new product [25]. Reman- +ufacturing systems face various challenges originating from un- +certain product states, inconsistent quality and fluctuating avail- +ability of products. Consequently, even today the vast majority +of processes in a remanufacturing system are carried out man- +ually [14]. In order to automate these processes, agile produc- +tion systems consisting of autonomously operating subsystem +are required which provide the highest possible flexibility and +adaptability. +In this paper, we consider the automated disassembly of dif- +ferent variants of end-of-life actuators which are commonly +used in vehicle manufacturing, e.g., as seat adjuster motors, +window lift motors or rear door motors. As shown in figure 1, +the considered subsystems are the intralogistics, the inspection, +as well as the disassembly which all heavily rely on product +information. However, in remanufacturing, each core is unique +since products have a high variance concerning their product +state. Therefore, handling, inspection and disassembly strate- +gies can not be defined in advance. Accordingly, the system +2212-8271 © 2022 The Authors. Published by Elsevier Ltd. +This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) +Peer-review under responsibility of the scientific committee of the 9th CIRP Conference on Assembly Technology and Systems. +arXiv:2301.05028v1 [cs.RO] 11 Jan 2023 + +NON +SOLUS +ELSEVIERIRPChengzhi Wu et al. / Procedia CIRP 00 (2022) 000–000 +2 +Fig. 1: Product-based relationships and considered applications between intralogistics, inspection and disassembly in an automated remanufacturing system. +must derive and execute these strategies during runtime based +on the actuator at hand. +Machine learning methods, and especially deep learning +methods, may be the key to achieve the necessary robustness to +deal with the high degree of variability. By learning the internal +structure on part level, (e.g. gear container, pole pot, electri- +cal connection), processes on unseen variants which have sim- +ilarities to the known population of actuators become feasible. +However, a major disadvantage of machine learning methods is +the required amount of data. Allocating and annotating a large +amount of data is time-consuming or even impossible in reality. +In the recent past, the synthetic generation of training data for +machine learning methods has become a popular alternative. +By leveraging transfer learning, it allows to train models that +rely less on elaborately labeled real-world data. We therefore +present an approach for generating synthetic training dataset as +well as its relevant applications for the handling-, inspection-, +and disassembly processes. +The remainder of this paper is structured as follows: Section +2 summarizes the state of the art of 3D synthetic dataset creation +and application. Section 3 describes technical details on the de- +veloped Blender add-on. Section 4 provides brief overviews on +applications build on top of the add-on while section 5 summa- +rizes presented outcome and discusses future work. +2. State of the art +Since the appearance of the iconic Stanford Bunny, gener- +ating synthetic datasets as training data for machine learning +purposes has already been widely discussed and used as a pos- +sible learning approach for various computer vision applica- +tions. Regarding synthetic dataset of 3D models, the Princeton +Shape Benchmark [23] provides a collection of 1,814 polygonal +models of objects from different categories as an early dataset. +ModelNet [30] contains 127,915 models for 3D object classifi- +cation and retrieval. More than three million annotated 3D mod- +els are collected in ShapeNet [2]. Its subsequent work PartNet +[17] additionally offers fine-grained semantic segmentation in- +formation for a subset of the models. By utilizing Thingiverse, +Thingi10K [32] provides a large dataset of 3D-printing mod- +els. More recently, the ABC dataset [12] collects over 1 mil- +lion CAD models, including lots of mechanical parts with sharp +edges and well defined surfaces, which are seldom included in +the previous synthetic datasets. +Regarding 3D scenes, [27] generates a synthetic dataset +for the segmentation and detection of objects in virtual street +scenes. Also [21] and [10] consider urban scenes and each pro- +vides a dataset for semantic segmentation in these environ- +ments. Other approaches in the image domain deal with the +generation of images from garden scenes [15], or specifically +for object detection and pose estimation [26]. There are also ap- +proaches for the generation of point clouds, such as that of [6], +in which, in contrast to the previously mentioned work, point +clouds of urban scenes are generated using Blender. Another +work using Blender deals with automatic generation of point +clouds of historical objects [18]. +However, in the production environment for industrial ap- +plications, approaches of generating synthetic dataset are sel- +dom employed. They may contribute in various applications in- +cluding product classification, segmentation of product compo- +nents, product tracking, and even determination of grasp points. +3. MotorFactory Add-on +The AgiProbot project aims for auto-detection, tracking and +disassembly of end-of-life products. To be specific, in our cur- +rent experimental setting, we work on small electric motors +used in vehicle manufacturing. Universal method needs to be +developed to deal with various types of motors, including the +ones with unseen specifications. However, we are only pro- +vided with a handful of motors with only few different product +specifications. The variance of data is actually not sufficient for +training with machine learning methods, especially those deep +learning-based ones. To deal with this problem, we created a +Blender add-on named MotorFactory, which can generate mo- +tor mesh models with a variety of specifications based on the +motor types we have currently. +As an open source software, Blender is a proven tool that +performs well in modeling shapes and creating highly cus- +tomizable add-ons. Our MotorFactory add-on is able to gen- +erate mesh models with various specifications and save them in +desired file formats. Each component of a generated motor can +2 + +Product +Intormation +4.1 Intralogistics +4.2 Inspection +4.3 Disassembly +Object Transfer and +View-Planning for Object +Point Cloud +Tracking +Inspection +Segmentation for object +Object Handling +Registration to aid Object +Disassembly +Inspection +Semantic Segmentation +of Armature ComponentsChengzhi Wu et al. / Procedia CIRP 00 (2022) 000–000 +3 +Fig. 2: Generated demo motors. Upper row: no textures added; bottom row: +textures added and rendered. Column 1/2/4: Type-A motors with two gears; +column 3/5: Type-B motors with one gear. +Fig. 3: An explosion figure of a motor generated with Version 2.0. The original +assembled motor model is also shown at the right most. +also be saved separately. Considering different requirements, +the add-on is implemented in two versions. Version 1.0 only +considers the components of motors that can be directly ob- +served from the appearance, while Version 2.0 further considers +the inner components. The generated models of both versions +contain the following components: (i) Pole Pot; (ii) Electric +Connection; (iii) Gear Container; (iv) Cover and (v) Bolts. Re- +garding inner components in version 2.0, the additionally gen- +erated parts are: (vi) Magnets; (vii) Armature; (viii) Lower Gear +and (ix) Upper Gear. To generate motors with various specifica- +tions, we provide lots of parameter options that control the type, +size, position and rotation of different parts of motor, e.g. bolt +position, gear size, or pole pot length. Additionally, both ver- +sions provide multiple bolt generation options to meet different +requirements. +To better design the building graph of different types of mo- +tors, we define two basic types based on the number of gears in +a motor. Type-A motor indicates the motors with two gears in- +side, while Type-B motor indicates the motors with only one +gear. Each motor type has different kind of gear containers. +Different gear containers further have different covers, which +come with different mounting points. Three options of exten- +sion shapes for covers are provided for Type-A motors, while +two options of are provided for Type-B motors. Figure 2 shows +(a) Demo of generated image dataset +(b) Demo of generated point cloud dataset +Fig. 4: Demos of generated dataset: (a) a rendered scene image with its cor- +responding depth image, normal image, and segmentation ground truth image; +(b) a simulated scene point cloud with its point-wise segmentation ground truth. +ten generated demo motors with different parameters. Figure 3 +shows an exploded view of a demo motor generated with Ver- +sion 2.0. All the individual components mentioned above are +modeled separately as illustrated. +The MotorFactory can be used to create a large amount of +different motor variants which are further used to generate syn- +thetic image and point cloud datasets. For example, to create +an image datasets, apart from the scene images rendered by +Blender directly, we can use BlenderProc [4] to generate the +corresponding depth images, normal images, and segmentation +ground truth images as illustrated in figure 4a. To create point +cloud datasets, we can use Blensor [7] to simulate the sensors. +Figure 4b gives a demo of generated scene point cloud with its +segmentation ground truth. The dataset generation setting and +pipeline depend on the actual needs of different tasks, hence +they may vary from task to task. +4. Application Use-Cases for the Motor Factory Add-On +The ability to generate synthetic motor data for computer +vision tasks enables the development or improvement of a vari- +ety of different use-cases inside the interdisciplinary AgiProbot +project. The presented use-cases follow an exemplary product +flow inside the AgiProbot demo-factory as illustrated in figure +1. The intralogistics system is responsible to transfer the prod- +uct to a station and handle it with a vision based manipulator +(4.1). Depending on the station type, inspection related tasks +(4.2) or disassembly procedures (4.3) are carried out. +3 + +Chengzhi Wu et al. / Procedia CIRP 00 (2022) 000–000 +4 +4.1. Intralogistics +Object Transfer and Tracking: The AgiProbot production +system follows the Fluid Automation framework to achieve the +required changeability needed for automated remanufacturing +[31]. Realizing the material flow between individual stations in +a matrix layout is the main task of the embedded intralogistics +system [11]. This includes the transfer of objects between an +autonomous mobile robot (AMR) and a transfer unit mounted +on a station module, as illustrated in figure 5a. From a mate- +rial flow perspective, objects are components such as products, +parts or assemblies or carrier devices such as boxes or work- +piece holders. The object transfer control workflow includes a +vision system mounted on the AMR which is used to detect and +track objects on the transfer units. Detecting objects and esti- +mating their pose is essential to realize the decentralized con- +trol between the AMR and the transfer unit. For known objects, +2D object detection algorithms can be used to estimate the ob- +ject’s bounding boxes and track them over time. However, well +performing deep learning based approaches, e.g. YOLO [1] and +its tracker DeepSort [29], require a vast amount of labeled in- +put data for the training process. In our case, by feeding gen- +erated motor models to Blender and using a BlenderProc-based +pipeline to generate synthetic image data, one can create high +quality rendered images with accurate ground truth annotations +[4]. Enriched with labeled real-world images, one can create +a dataset of reasonable size to train well performing real-time +detection and tracking network models. +Object Handling: Object handling describes vision-based +robotic pick-and place tasks, which are commonly required to +meet necessary preconditions to perform value-adding tasks +like manufacturing, assembly or disassembly. A robotic grasp- +ing system commonly consists of a robotic manipulator, an +RGB-D camera, a gripper and a target object. The robotic +grasping problem can be generally divided into three sub- +problems, namely the grasp detection, the trajectory planning +and the execution [13]. +Hereby, the grasp detection system is the key entry point and +includes the target object localization, the object pose estima- +tion, and the grasp estimation [5]. To solve the localization and +pose estimation problem for known objects, simple bounding- +boxes, as described in the previous use-case, are not sophisti- +cated enough. However, since the MotorFactory generates mesh +models on part level, annotated instance segmentation masks +can be generated. Training a segmentation network like Mask +R-CNN [9] allows to detect pixel-level segmentation instances +of classes. Being able to detect parts individually allows to han- +dle sub-assemblies and deriving pre-defined grasping points on +part level. Figure 5b shows the segmentation results on a real- +world image with a network model solely trained on synthetic +data. Trained an instance segmentation network on many ar- +tificially generated product variants increases its robustness in +detecting individual parts when a new variant is exposed to the +system. In an adaptive vision system, compared to objects that +are categorized to be unknown which only rely on salient detec- +tion, a never seen motor variant can therefore be detected with +the help of more sophisticated methods . +(a) Object transfer scenario +(b) Instance segmentation +Fig. 5: Object transfer and handling, using a motor of Type-A as an example. +4.2. Inspection +View-Planning for Object Inspection: Since wear and tear +or defects can occur anywhere on a returning product, it is usu- +ally necessary to inspect the entire surface of the product. The +formalized problem of completing product surface coverage us- +ing a robot-based optical system can be expressed as a view- +planning problem. In the literature so far, there are model-based +and non-model-based solution approaches [22]. Model-based +approaches determine robot poses for sensor sensing by pre- +planning based on a prior model. Non-model-based approaches +determine the next best robot pose for acquisition at runtime +through maximizing certain criteria. However, in many cases, +independent remanufacturers do not have product information +such as mesh models of the product at hand due to confiden- +tiality issues [8]. This hinders efficient pre-planning of visual +strategies for inspection. +With the help of this add-on, an important research gap can +be approached. In remanufacturing, the product variants consid- +ered are mostly different in detail but similar in principle. By us- +ing the generated dataset and machine learning methods such as +reinforcement learning, inspection strategies can be learned in +the simulation and transferred to previously unknown but sim- +ilar product variants under sufficient generalization capability. +In this case, the reinforcement learning agent generates an ac- +tion (pose of the robot’s end effector in space) based on a state +(currently recorded point cloud of the product) and receives a +reward (e.g. based on the relative information gain in the form +of new detected surfaces). The reinforcement learning agent is +trained in the simulation until its performance is satisfactory. +In theory, the amount of data required for validation in reality +then becomes smaller and the inspection agent can be used to +fulfil inspection goals defined by the reward signal for a large +number of different but similar variants of electric motors. +Registration to aid Object Inspection: The requirements +of completing surface coverage for objects can usually not +be satisfied with one-time application of view-planning ap- +proaches. This is due to the fact that during inspection some +surface areas of a product are permanently occluded by grip- +4 + +Motor99% +Gear77%Vision SystemChengzhi Wu et al. / Procedia CIRP 00 (2022) 000–000 +5 +(a) First clamping setup with the gear con- +tainer facing upwards +(b) Second clamping setup with the pole pot +facing upwards +Fig. 6: Visualization of an exemplary inspection routine with two clamping +setups to enable full target object inspection. +ping or clamping units, or simply because the measured object +is placed on a surface. After a successful product acquisition, +for example with the approach described in view planning, a +repositioning as well as the acquisition of the previously hid- +den surfaces is necessary. Since the repositioning changes the +position of the object, a registration is necessary to merge the +point clouds of all capturing steps in order to determine the en- +tire object surface captured so far as well as to plan the next +view based on. +Random sample consensus (RANSAC)-based approaches +are widely used for solving the registration problem, but they +are model-free, computationally expensive and do not take the +specific product properties into account [33]. Here, the pre- +sented add-on offers the possibility to apply data-driven reg- +istration approaches with deep learning-based methods, which +have proven successful compared to classical approaches [3]. +Synthetic point clouds of a product model can be generated +from different views of a sensor, whose pose is determined in +the form of the position and the orientation. This also allows +the transformation matrix between the acquired point clouds to +be determined and can be used as a target variable for registra- +tion procedures to learn. In this way, product-specific properties +can be taken into account during the training of the virtual reg- +istration process in order to achieve a faster and more precise +registration result for real-world data. +4.3. Disassembly +Point Cloud Segmentation for Object Disassembly: Com- +pared to images, point clouds contain more 3D information and +is another widely used data representation in industrial applica- +tions. Point cloud segmentation provides an alternative way to +segment the different components of a product to provide im- +portant information to robots. In recent years, methods of ap- +plying deep learning techniques on 3D point clouds have been +intensively investigated. PointNet [19] pioneered on this topic +by designing a symmetric function for unordered point input, +followed by works like PointNet++ [20] or DGCNN [28] which +focused on gathering local information. Apart from classifica- +tion, those methods also focus on point cloud segmentation in- +cluding part segmentation on 3D shapes and scene segmenta- +tion for indoor scenes. +(a) Synthetic point cloud +(b) Real-world point cloud +Fig. 7: Comparison between (a) a synthetic point cloud with labels generated +automatically, and (b) a real-world point cloud from the Zivid camera with +points annotated manually. Using a motor of Type-B as an example. +In our case, the target object is set fixed in a fixture when it is +transferred to the disassembly station. Then a real-world point +cloud is obtained with Zivid 3D Camera. However, the variance +of different taken point clouds, especially regarding the motors, +is not sufficient for learning purposes. In this case, we create a +synthetic dataset using the proposed add-on. A demo is given +in Figure 4b. Apart from the station table, the scene contains a +fixure with a random motor placed in a relatively fixed position. +To solve the problem of batches are not representative enough +due to the large size of raw point clouds, we cut a cuboid of sub- +point cloud that only considers the target object and its neigh- +bor points after generating them. Further data augmentation is +needed to enhance the generalization ability of the network, e.g. +adding random small mask panels, adding random jitter noise, +shifting fixure components, or rotating virtual cameras. Figure 7 +gives a comparison between a synthetic point cloud and a real- +world point cloud. Their scene settings are slightly different, +but the synthetic one is already good enough to use as training +data. After the network is pre-trained with the synthetic dataset, +transfer learning may be applied with a few real-world data to +finetune the model. Further important specification information +may also be obtained with additional post-processing and sub- +sequently provided to robots for other tasks. +Semantic Segmentation of Armature Components: The +armature represents a functionally relevant part of the motor +that requires a separate inspection in the given remanufacturing +task. Semantic segmentation is required for the recognition of +important surface parts of a armature. In a first proof-of-work +approach, a small dataset of armatures has already been cre- +ated to train a U-Net [16] for the segmentation task. However, +the dataset contains only a small variety of armatures with 75 +training images, most of them are taken from a similar perspec- +tive and are even partially inaccurately or inconsistently anno- +tated. Hence, the proposed Blender add-on can be used to gen- +erate additional training data to supplement or replace the given +dataset. Different armature types and wear conditions can be +obtained through suitable parameterization. Arbitrary perspec- +tives on the armature can also be generated. Furthermore, the +accurate annotation of the synthetic dataset improves the seg- +mentation performance and also increases the meaningfulness +of validation metrics. +5 + +Acquisition 2 +Acquisition1Acguisition 4 +Acguisition 3Chengzhi Wu et al. / Procedia CIRP 00 (2022) 000–000 +6 +5. Conclusion and Outlook +In this paper, we present MotorFactory, a blender add-on +which allows the end-user to automatically create a variety of +different small electric motor models and components. Based +on the generated motors, synthetic datasets can be created for +different computer vision tasks. Being able to generate and an- +notate product models easily may further increase the level of +automation in challenging remanufacturing environments. In- +side the interdisciplinary AgiProbot project, we identified sev- +eral use-cases from the intralogistics, inspection, and disassem- +bly domains. All of the use-cases benefit greatly from having +access to synthetic product models. +In the future, an open-source benchmark dataset for object +attribute estimation based on the add-on will be created and +published. New versions of the presented add-on will include +a greater amount of other electric motor types and variants, e.g. +starters, or even other common remanufacturing products. Tex- +tures, which increase the level of realism, e.g. worn-out screw +heads, rust or discolouration, are also intended to be added. The +elaborated use-cases will be fully implemented and evaluated. +Feedback based on the evaluation can then be used to further +improve the add-on and its features. +Acknowledgements +The AgiProbot project is funded by the Carl-Zeiss Founda- +tion. +References +[1] Bochkovskiy, A., Wang, C.Y., Liao, H., 2020. Yolov4: Optimal speed and +accuracy of object detection. ArXiv abs/2004.10934. +[2] Chang, A.X., Funkhouser, T., Guibas, L., Hanrahan, P., Huang, Q., Li, +Z., Savarese, S., Savva, M., Song, S., Su, H., Xiao, J., Yi, L., Yu, +F., 2015. +Shapenet: An information-rich 3d model repository. +ArXiv +abs/1512.03012. +[3] Choy, C., Dong, W., Koltun, V., 2020. Deep global registration. CVPR , +2511–2520. +[4] Denninger, M., Sundermeyer, M., Winkelbauer, D., Zidan, Y., Olefir, D., +Elbadrawy, M., Lodhi, A., Katam, H., 2019. Blenderproc. arXiv preprint +arXiv:1911.01911 . +[5] Du, G., Wang, K., Lian, S., Zhao, K., 2021. 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Fast global registration, in: ECCV, +pp. 766–782. +6 + diff --git a/tNE4T4oBgHgl3EQfWAwH/content/tmp_files/load_file.txt b/tNE4T4oBgHgl3EQfWAwH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..def13b5e0b0f19292cfd288fc3a920291af19981 --- /dev/null +++ b/tNE4T4oBgHgl3EQfWAwH/content/tmp_files/load_file.txt @@ -0,0 +1,549 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf,len=548 +page_content='Available online at www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='com Procedia CIRP 00 (2022) 000–000 www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='elsevier.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Karlsruhe Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Kaiserstraße 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' 76131 Karlsruhe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Germany eFraunhofer Institute of Optronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' System Technologies and Image Exploitation IOSB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Fraunhoferstraße 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' 76131 Karlsruhe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Germany Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' : +49-(0)1523 8476995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' E-mail address: chengzhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='wu@kit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='edu Abstract To enable automatic disassembly of different product types with uncertain condition and degree of wear in remanufacturing, agile production systems that can adapt dynamically to changing requirements are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Machine learning algorithms can be employed due to their generalization capabilities of learning from various types and variants of products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' However, in reality, datasets with a diversity of samples that can be used to train models are difficult to obtain in the initial period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' This may cause bad performances when the system tries to adapt to new unseen input data in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' In order to generate large datasets for different learning purposes, in our project, we present a Blender add-on named MotorFactory to generate customized mesh models of various motor instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' MotorFactory allows to create mesh models which, complemented with additional add-ons, can be further used to create synthetic RGB images, depth images, normal images, segmentation ground truth masks and 3D point cloud datasets with point-wise semantic labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' The created synthetic datasets may be used for various tasks including motor type classification, object detection for decentralized material transfer tasks, part segmentation for disassembly and handling tasks, or even reinforcement learning-based robotics control or view-planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' © 2022 The Authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Published by Elsevier Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' This is an open access article under the CC BY-NC-ND license (http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='org/licenses/by-nc-nd/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='0/) Peer-review under responsibility of the scientific committee of the 9th CIRP Conference on Assembly Technology and Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Keywords: Remanufacturing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Blender Add-on;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Machine learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Large synthetic dataset generation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Introduction Today’s industrial landscape is characterised by linear economies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' End-of-life (EOL) strategies for products such as remanufacturing, in which used products are reprocessed, offer the potential to decouple resource consumption from sustain- able economic growth [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' During the remanufacturing pro- cess, the products are inspected and disassembled, then indi- vidual parts are reworked or exchanged and again reassembled into final products [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' In contrast to related EOL-strategies, remanufactured products ensure functionality and quality that is equivalent or better compared to a new product [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Reman- ufacturing systems face various challenges originating from un- certain product states, inconsistent quality and fluctuating avail- ability of products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Consequently, even today the vast majority of processes in a remanufacturing system are carried out man- ually [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' In order to automate these processes, agile produc- tion systems consisting of autonomously operating subsystem are required which provide the highest possible flexibility and adaptability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' In this paper, we consider the automated disassembly of dif- ferent variants of end-of-life actuators which are commonly used in vehicle manufacturing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=', as seat adjuster motors, window lift motors or rear door motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' As shown in figure 1, the considered subsystems are the intralogistics, the inspection, as well as the disassembly which all heavily rely on product information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' However, in remanufacturing, each core is unique since products have a high variance concerning their product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Therefore, handling, inspection and disassembly strate- gies can not be defined in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Accordingly, the system 2212-8271 © 2022 The Authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Published by Elsevier Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' This is an open access article under the CC BY-NC-ND license (http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='org/licenses/by-nc-nd/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='0/) Peer-review under responsibility of the scientific committee of the 9th CIRP Conference on Assembly Technology and Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='05028v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='RO] 11 Jan 2023 NON SOLUS ELSEVIERIRPChengzhi Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' / Procedia CIRP 00 (2022) 000–000 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' 1: Product-based relationships and considered applications between intralogistics, inspection and disassembly in an automated remanufacturing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' must derive and execute these strategies during runtime based on the actuator at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Machine learning methods, and especially deep learning methods, may be the key to achieve the necessary robustness to deal with the high degree of variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' By learning the internal structure on part level, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' gear container, pole pot, electri- cal connection), processes on unseen variants which have sim- ilarities to the known population of actuators become feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' However, a major disadvantage of machine learning methods is the required amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Allocating and annotating a large amount of data is time-consuming or even impossible in reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' In the recent past, the synthetic generation of training data for machine learning methods has become a popular alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' By leveraging transfer learning, it allows to train models that rely less on elaborately labeled real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' We therefore present an approach for generating synthetic training dataset as well as its relevant applications for the handling-, inspection-, and disassembly processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' The remainder of this paper is structured as follows: Section 2 summarizes the state of the art of 3D synthetic dataset creation and application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Section 3 describes technical details on the de- veloped Blender add-on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Section 4 provides brief overviews on applications build on top of the add-on while section 5 summa- rizes presented outcome and discusses future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' State of the art Since the appearance of the iconic Stanford Bunny, gener- ating synthetic datasets as training data for machine learning purposes has already been widely discussed and used as a pos- sible learning approach for various computer vision applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Regarding synthetic dataset of 3D models, the Princeton Shape Benchmark [23] provides a collection of 1,814 polygonal models of objects from different categories as an early dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' ModelNet [30] contains 127,915 models for 3D object classifi- cation and retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' More than three million annotated 3D mod- els are collected in ShapeNet [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Its subsequent work PartNet [17] additionally offers fine-grained semantic segmentation in- formation for a subset of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' By utilizing Thingiverse, Thingi10K [32] provides a large dataset of 3D-printing mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' More recently, the ABC dataset [12] collects over 1 mil- lion CAD models, including lots of mechanical parts with sharp edges and well defined surfaces, which are seldom included in the previous synthetic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Regarding 3D scenes, [27] generates a synthetic dataset for the segmentation and detection of objects in virtual street scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Also [21] and [10] consider urban scenes and each pro- vides a dataset for semantic segmentation in these environ- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Other approaches in the image domain deal with the generation of images from garden scenes [15], or specifically for object detection and pose estimation [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' There are also ap- proaches for the generation of point clouds, such as that of [6], in which, in contrast to the previously mentioned work, point clouds of urban scenes are generated using Blender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Another work using Blender deals with automatic generation of point clouds of historical objects [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' However, in the production environment for industrial ap- plications, approaches of generating synthetic dataset are sel- dom employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' They may contribute in various applications in- cluding product classification, segmentation of product compo- nents, product tracking, and even determination of grasp points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' MotorFactory Add-on The AgiProbot project aims for auto-detection, tracking and disassembly of end-of-life products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' To be specific, in our cur- rent experimental setting, we work on small electric motors used in vehicle manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Universal method needs to be developed to deal with various types of motors, including the ones with unseen specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' However, we are only pro- vided with a handful of motors with only few different product specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' The variance of data is actually not sufficient for training with machine learning methods, especially those deep learning-based ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' To deal with this problem, we created a Blender add-on named MotorFactory, which can generate mo- tor mesh models with a variety of specifications based on the motor types we have currently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' As an open source software, Blender is a proven tool that performs well in modeling shapes and creating highly cus- tomizable add-ons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Our MotorFactory add-on is able to gen- erate mesh models with various specifications and save them in desired file formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Each component of a generated motor can 2 Product Intormation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='1 Intralogistics 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='2 Inspection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='3 Disassembly Object Transfer and View-Planning for Object Point Cloud Tracking Inspection Segmentation for object Object Handling Registration to aid Object Disassembly Inspection Semantic Segmentation of Armature ComponentsChengzhi Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' / Procedia CIRP 00 (2022) 000–000 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' 2: Generated demo motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Upper row: no textures added;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' bottom row: textures added and rendered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Column 1/2/4: Type-A motors with two gears;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' column 3/5: Type-B motors with one gear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' 3: An explosion figure of a motor generated with Version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' The original assembled motor model is also shown at the right most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' also be saved separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Considering different requirements, the add-on is implemented in two versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='0 only considers the components of motors that can be directly ob- served from the appearance, while Version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='0 further considers the inner components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' The generated models of both versions contain the following components: (i) Pole Pot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' (ii) Electric Connection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' (iii) Gear Container;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' (iv) Cover and (v) Bolts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Re- garding inner components in version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='0, the additionally gen- erated parts are: (vi) Magnets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' (vii) Armature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' (viii) Lower Gear and (ix) Upper Gear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' To generate motors with various specifica- tions, we provide lots of parameter options that control the type, size, position and rotation of different parts of motor, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' bolt position, gear size, or pole pot length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Additionally, both ver- sions provide multiple bolt generation options to meet different requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' To better design the building graph of different types of mo- tors, we define two basic types based on the number of gears in a motor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Type-A motor indicates the motors with two gears in- side, while Type-B motor indicates the motors with only one gear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Each motor type has different kind of gear containers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Different gear containers further have different covers, which come with different mounting points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Three options of exten- sion shapes for covers are provided for Type-A motors, while two options of are provided for Type-B motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Figure 2 shows (a) Demo of generated image dataset (b) Demo of generated point cloud dataset Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' 4: Demos of generated dataset: (a) a rendered scene image with its cor- responding depth image, normal image, and segmentation ground truth image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' (b) a simulated scene point cloud with its point-wise segmentation ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' ten generated demo motors with different parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Figure 3 shows an exploded view of a demo motor generated with Ver- sion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' All the individual components mentioned above are modeled separately as illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' The MotorFactory can be used to create a large amount of different motor variants which are further used to generate syn- thetic image and point cloud datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' For example, to create an image datasets, apart from the scene images rendered by Blender directly, we can use BlenderProc [4] to generate the corresponding depth images, normal images, and segmentation ground truth images as illustrated in figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' To create point cloud datasets, we can use Blensor [7] to simulate the sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Figure 4b gives a demo of generated scene point cloud with its segmentation ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' The dataset generation setting and pipeline depend on the actual needs of different tasks, hence they may vary from task to task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Application Use-Cases for the Motor Factory Add-On The ability to generate synthetic motor data for computer vision tasks enables the development or improvement of a vari- ety of different use-cases inside the interdisciplinary AgiProbot project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' The presented use-cases follow an exemplary product flow inside the AgiProbot demo-factory as illustrated in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' The intralogistics system is responsible to transfer the prod- uct to a station and handle it with a vision based manipulator (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Depending on the station type, inspection related tasks (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='2) or disassembly procedures (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='3) are carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' 3 Chengzhi Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' / Procedia CIRP 00 (2022) 000–000 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Intralogistics Object Transfer and Tracking: The AgiProbot production system follows the Fluid Automation framework to achieve the required changeability needed for automated remanufacturing [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Realizing the material flow between individual stations in a matrix layout is the main task of the embedded intralogistics system [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' This includes the transfer of objects between an autonomous mobile robot (AMR) and a transfer unit mounted on a station module, as illustrated in figure 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' From a mate- rial flow perspective, objects are components such as products, parts or assemblies or carrier devices such as boxes or work- piece holders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' The object transfer control workflow includes a vision system mounted on the AMR which is used to detect and track objects on the transfer units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Detecting objects and esti- mating their pose is essential to realize the decentralized con- trol between the AMR and the transfer unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' For known objects, 2D object detection algorithms can be used to estimate the ob- ject’s bounding boxes and track them over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' However, well performing deep learning based approaches, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' YOLO [1] and its tracker DeepSort [29], require a vast amount of labeled in- put data for the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' In our case, by feeding gen- erated motor models to Blender and using a BlenderProc-based pipeline to generate synthetic image data, one can create high quality rendered images with accurate ground truth annotations [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Enriched with labeled real-world images, one can create a dataset of reasonable size to train well performing real-time detection and tracking network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Object Handling: Object handling describes vision-based robotic pick-and place tasks, which are commonly required to meet necessary preconditions to perform value-adding tasks like manufacturing, assembly or disassembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' A robotic grasp- ing system commonly consists of a robotic manipulator, an RGB-D camera, a gripper and a target object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' The robotic grasping problem can be generally divided into three sub- problems, namely the grasp detection, the trajectory planning and the execution [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Hereby, the grasp detection system is the key entry point and includes the target object localization, the object pose estima- tion, and the grasp estimation [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' To solve the localization and pose estimation problem for known objects, simple bounding- boxes, as described in the previous use-case, are not sophisti- cated enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' However, since the MotorFactory generates mesh models on part level, annotated instance segmentation masks can be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Training a segmentation network like Mask R-CNN [9] allows to detect pixel-level segmentation instances of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Being able to detect parts individually allows to han- dle sub-assemblies and deriving pre-defined grasping points on part level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Figure 5b shows the segmentation results on a real- world image with a network model solely trained on synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Trained an instance segmentation network on many ar- tificially generated product variants increases its robustness in detecting individual parts when a new variant is exposed to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' In an adaptive vision system, compared to objects that are categorized to be unknown which only rely on salient detec- tion, a never seen motor variant can therefore be detected with the help of more sophisticated methods .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' (a) Object transfer scenario (b) Instance segmentation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' 5: Object transfer and handling, using a motor of Type-A as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Inspection View-Planning for Object Inspection: Since wear and tear or defects can occur anywhere on a returning product, it is usu- ally necessary to inspect the entire surface of the product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' The formalized problem of completing product surface coverage us- ing a robot-based optical system can be expressed as a view- planning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' In the literature so far, there are model-based and non-model-based solution approaches [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Model-based approaches determine robot poses for sensor sensing by pre- planning based on a prior model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Non-model-based approaches determine the next best robot pose for acquisition at runtime through maximizing certain criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' However, in many cases, independent remanufacturers do not have product information such as mesh models of the product at hand due to confiden- tiality issues [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' This hinders efficient pre-planning of visual strategies for inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' With the help of this add-on, an important research gap can be approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' In remanufacturing, the product variants consid- ered are mostly different in detail but similar in principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' By us- ing the generated dataset and machine learning methods such as reinforcement learning, inspection strategies can be learned in the simulation and transferred to previously unknown but sim- ilar product variants under sufficient generalization capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' In this case, the reinforcement learning agent generates an ac- tion (pose of the robot’s end effector in space) based on a state (currently recorded point cloud of the product) and receives a reward (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' based on the relative information gain in the form of new detected surfaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' The reinforcement learning agent is trained in the simulation until its performance is satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' In theory, the amount of data required for validation in reality then becomes smaller and the inspection agent can be used to fulfil inspection goals defined by the reward signal for a large number of different but similar variants of electric motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Registration to aid Object Inspection: The requirements of completing surface coverage for objects can usually not be satisfied with one-time application of view-planning ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' This is due to the fact that during inspection some surface areas of a product are permanently occluded by grip- 4 Motor99% Gear77%Vision SystemChengzhi Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' / Procedia CIRP 00 (2022) 000–000 5 (a) First clamping setup with the gear con- tainer facing upwards (b) Second clamping setup with the pole pot facing upwards Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' 6: Visualization of an exemplary inspection routine with two clamping setups to enable full target object inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' ping or clamping units, or simply because the measured object is placed on a surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' After a successful product acquisition, for example with the approach described in view planning, a repositioning as well as the acquisition of the previously hid- den surfaces is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Since the repositioning changes the position of the object, a registration is necessary to merge the point clouds of all capturing steps in order to determine the en- tire object surface captured so far as well as to plan the next view based on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Random sample consensus (RANSAC)-based approaches are widely used for solving the registration problem, but they are model-free, computationally expensive and do not take the specific product properties into account [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Here, the pre- sented add-on offers the possibility to apply data-driven reg- istration approaches with deep learning-based methods, which have proven successful compared to classical approaches [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Synthetic point clouds of a product model can be generated from different views of a sensor, whose pose is determined in the form of the position and the orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' This also allows the transformation matrix between the acquired point clouds to be determined and can be used as a target variable for registra- tion procedures to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' In this way, product-specific properties can be taken into account during the training of the virtual reg- istration process in order to achieve a faster and more precise registration result for real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Disassembly Point Cloud Segmentation for Object Disassembly: Com- pared to images, point clouds contain more 3D information and is another widely used data representation in industrial applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Point cloud segmentation provides an alternative way to segment the different components of a product to provide im- portant information to robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' In recent years, methods of ap- plying deep learning techniques on 3D point clouds have been intensively investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' PointNet [19] pioneered on this topic by designing a symmetric function for unordered point input, followed by works like PointNet++ [20] or DGCNN [28] which focused on gathering local information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Apart from classifica- tion, those methods also focus on point cloud segmentation in- cluding part segmentation on 3D shapes and scene segmenta- tion for indoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' (a) Synthetic point cloud (b) Real-world point cloud Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' 7: Comparison between (a) a synthetic point cloud with labels generated automatically, and (b) a real-world point cloud from the Zivid camera with points annotated manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Using a motor of Type-B as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' In our case, the target object is set fixed in a fixture when it is transferred to the disassembly station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Then a real-world point cloud is obtained with Zivid 3D Camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' However, the variance of different taken point clouds, especially regarding the motors, is not sufficient for learning purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' In this case, we create a synthetic dataset using the proposed add-on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' A demo is given in Figure 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Apart from the station table, the scene contains a fixure with a random motor placed in a relatively fixed position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' To solve the problem of batches are not representative enough due to the large size of raw point clouds, we cut a cuboid of sub- point cloud that only considers the target object and its neigh- bor points after generating them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Further data augmentation is needed to enhance the generalization ability of the network, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' adding random small mask panels, adding random jitter noise, shifting fixure components, or rotating virtual cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Figure 7 gives a comparison between a synthetic point cloud and a real- world point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Their scene settings are slightly different, but the synthetic one is already good enough to use as training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' After the network is pre-trained with the synthetic dataset, transfer learning may be applied with a few real-world data to finetune the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Further important specification information may also be obtained with additional post-processing and sub- sequently provided to robots for other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Semantic Segmentation of Armature Components: The armature represents a functionally relevant part of the motor that requires a separate inspection in the given remanufacturing task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Semantic segmentation is required for the recognition of important surface parts of a armature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' In a first proof-of-work approach, a small dataset of armatures has already been cre- ated to train a U-Net [16] for the segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' However, the dataset contains only a small variety of armatures with 75 training images, most of them are taken from a similar perspec- tive and are even partially inaccurately or inconsistently anno- tated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Hence, the proposed Blender add-on can be used to gen- erate additional training data to supplement or replace the given dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Different armature types and wear conditions can be obtained through suitable parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Arbitrary perspec- tives on the armature can also be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Furthermore, the accurate annotation of the synthetic dataset improves the seg- mentation performance and also increases the meaningfulness of validation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' 5 Acquisition 2 Acquisition1Acguisition 4 Acguisition 3Chengzhi Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' / Procedia CIRP 00 (2022) 000–000 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Conclusion and Outlook In this paper, we present MotorFactory, a blender add-on which allows the end-user to automatically create a variety of different small electric motor models and components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Based on the generated motors, synthetic datasets can be created for different computer vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Being able to generate and an- notate product models easily may further increase the level of automation in challenging remanufacturing environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' In- side the interdisciplinary AgiProbot project, we identified sev- eral use-cases from the intralogistics, inspection, and disassem- bly domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' All of the use-cases benefit greatly from having access to synthetic product models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' In the future, an open-source benchmark dataset for object attribute estimation based on the add-on will be created and published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' New versions of the presented add-on will include a greater amount of other electric motor types and variants, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' starters, or even other common remanufacturing products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Tex- tures, which increase the level of realism, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' worn-out screw heads, rust or discolouration, are also intended to be added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' The elaborated use-cases will be fully implemented and evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Feedback based on the evaluation can then be used to further improve the add-on and its features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' Acknowledgements The AgiProbot project is funded by the Carl-Zeiss Founda- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=' References [1] Bochkovskiy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE4T4oBgHgl3EQfWAwH/content/2301.05028v1.pdf'} +page_content=', Wang, C.' metadata={'source': 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a/ydE4T4oBgHgl3EQfYgwr/content/tmp_files/2301.05049v1.pdf.txt b/ydE4T4oBgHgl3EQfYgwr/content/tmp_files/2301.05049v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..97bb903f40ed27e840a80dd70cf241c94c100c02 --- /dev/null +++ b/ydE4T4oBgHgl3EQfYgwr/content/tmp_files/2301.05049v1.pdf.txt @@ -0,0 +1,534 @@ +On Voronoi visibility maps of 1.5D terrains +with multiple viewpoints⋆ +Vahideh Keikhaa,b, Maria Saumella,c,∗ +aThe Czech Academy of Sciences, Institute of Computer Science, Czech Republic. +bFaculty of Mathematics and Physics, Department of Applied Mathematics, Charles University, Prague, Czech Republic. +cDepartment of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Czech Republic. +Abstract +Given an n-vertex 1.5D terrain T and a set P of m < n viewpoints, the Voronoi visibility map VorVis(T , P) is a +partitioning of T into regions such that each region is assigned to the closest (in Euclidean distance) visible viewpoint. +The colored visibility map ColVis(T , P) is a partitioning of T into regions that have the same set of visible viewpoints. +In this paper, we propose an algorithm to compute VorVis(T , P) that runs in O(n + (m2 + kc) log n) time, where kc +and kv denote the total complexity of ColVis(T , P) and VorVis(T , P), respectively. This improves upon a previous +algorithm for this problem. We also generalize our algorithm to higher order Voronoi visibility maps, and to Voronoi +visibility maps with respect to other distances. Finally, we prove bounds relating kv to kc, and we show an application +of our algorithm to a problem on limited range of sight. +Keywords: Visibility, 1.5D terrains, Voronoi diagrams, multiple viewpoints. +1. Introduction +A 1.5D terrain T is an x-monotone polygonal chain of n vertices in R2. Two points on T are visible if the segment +connecting them does not contain any point strictly below T . +Visibility problems in terrains are fundamental in geographical information science and have many applications, +such as placing fireguard or telecommunication towers [4], identifying areas that are not visible from sensitive sites [15], +or solving problems related to sensor networks [17]. Although 2.5D terrains are more interesting for modelling and +forecasting, 1.5D terrains are easier to visualize and to analyze. They give insights into the difficulties of 2.5D terrains +in terrain analysis, and their proper understanding is seen as an essential step towards the ultimate goal of settling the +2.5D case. For this reason, visibility problems in 1.5D terrains have been intensively studied by the computational +geometry community during the last 15 years. +In this paper, we focus on the variant where a set P of m < n viewpoints are located on vertices of T (we refer to +the end of this section for a discussion on the assumption m < n). For each viewpoint p ∈ P, the viewshed of p is the +set of points of T that are visible from p (see Fig. 1 for an example). Our goal is to efficiently extract information +about the visibility of T with respect to P. We continue the work initiated in [12], where the following structures are +introduced. +The visibility map Vis(T , P) is a partitioning of T into a visible region (containing all portions of T that are visible +by at least one element in P) and an invisible region (containing the portions that are not visible by any element in +P). See Fig. 2a for an example. The visible region of the visibility map is equal to the union of the viewsheds of all +viewpoints in P. +⋆Supported by the Czech Science Foundation, grant number GJ19-06792Y, and with institutional support RVO:67985807. V.K was also partially +supported by Charles University project UNCE/SCI/004 and by the Czech Academy of Sciences (Praemium Academiae awarded to M. Paluˇs). +∗Corresponding author +Email addresses: keikha@cs.cas.cz (Vahideh Keikha), maria.saumell@fit.cvut.cz (Maria Saumell) +Preprint submitted to Elsevier +January 13, 2023 +arXiv:2301.05049v1 [cs.CG] 12 Jan 2023 + +p +Figure 1: The viewshed of p. +pk +pj +pi +(a) +pk +(b) +pj +pi +pk +pj +pi +(c) +Figure 2: (a) Vis(T , P) (the visible region is shown in gray). (b) ColVis(T , P). (c) VorVis(T , P). +The colored visibility map ColVis(T , P) is a partitioning of T into regions that have the same set of visible +viewpoints. See Fig. 2b for an example. +Finally, the Voronoi visibility map VorVis(T , P) is a partitioning of T into regions that have the same closest visible +viewpoint, where the distance used is the Euclidean distance (not the distance along the terrain). See Fig. 2c for an +example. +Algorithms to compute these structures for both 1.5D and 2.5D terrains are proposed in [12]. The algorithm to +obtain VorVis(T , P) of a 1.5D terrain runs in O(n + (m2 + kc) log n + kv(m + log n log m)) time, where kc and kv denote +the total complexity of ColVis(T , P) and VorVis(T , P), respectively. Both kc and kv have size O(mn), and this bound +is asymptotically tight [12]. The algorithm first computes ColVis(T , P), and then it spends Θ(m) time to find each +single region of VorVis(T , P). In this paper, we show that VorVis(T , P) can be extracted from ColVis(T , P) in a 1.5D +terrain much more efficiently, resulting in an O(n + (m2 + kc) log n)-time algorithm. We use an observation related to +intersections of the terrain with bisectors of pairs of viewpoints that also allows us to prove a relationship between kc +and kv. +Let us point out that, apart from the mentioned output-sensitive algorithm for VorVis(T , P) of a 1.5D terrain, the +authors of [12] also propose a divide-and-conquer algorithm running in O(mn log m) time, which is worst-case nearly +optimal (recall that the maximum complexity of VorVis(T , P) is Θ(mn)). Therefore, our new algorithm does not +represent an improvement in the worst-case instances, but in instances where the original output-sensitive algorithm +is faster than the divide-and-conquer one, and kvm is the dominant term in the running time. An example of such an +instance is m = Θ( √n), kc = Θ(n3/4) and kv = Θ(n3/4). +In this paper, we also provide generalizations of our algorithm to compute Voronoi visibility maps of higher order +(that is, containing the information about the k closest visible viewpoints, for some k > 1), and Voronoi visibility +maps with respect to two other distances: the Euclidean distance along the terrain and the link distance. All of these +generalizations have the same running time as the original algorithm. +Finally, the new algorithm for VorVis(T , P) also allows us to solve efficiently a problem related to limited range of +sight. These problems are motivated by the fact that, even though many visibility problems assume an infinite range +of visibility, the intensity of light, signals and other phenomena modelled with viewpoints decreases over distance in +realistic environments. In this spirit, the problem of illuminating a polygonal area with the minimum total energy was +introduced by O’Rourke [16], and studied in [7, 8]. We consider a related problem on terrains, namely, computing the +minimum value r∗ such that, if the viewpoints can only see objects within distance r∗, the obtained visibility map is the +same as Vis(T , P). We show that this problem can also be solved in O(n + (m2 + kc) log n) time. +Related Work. When there is only one viewpoint, computing the visibility map of a 1.5D terrain can be done in O(n) +time by converting the terrain into a simple polygon and applying the algorithm from [13]. One of the first results on +the variant with more than one viewpoint is an O((n + m) log m) time algorithm to detect if there are any visible pairs +2 + +pj +pi +q +bi,j +pj +q +bi,j +(a) +(b) +r +r +pi +Figure 3: Illustration of Lemma 1: (a) x(pi) > x(p j); (b) x(pi) < x(pj). +of viewpoints above a 1.5D terrain [2]. Later, a systematic study of Vis(T , P), VorVis(T , P) and ColVis(T , P) was +carried out in [12] for both 1.5D and 2.5D terrains. A problem that is very related to the construction of Vis(T , P) is +that of computing the total visibility index of the terrain, that is, the number of viewpoints that are visible from each of +the viewpoints. This problem can be solved in O(n log2 n) time [1]. +The situation where the locations of the viewpoints are unknown has been thoroughly studied. It is well-known +that computing the minimum number of viewpoints to keep a 1.5D terrain illuminated is NP-complete [9, 14], but the +problem admits a PTAS [9, 10, 11]. If the viewpoints are restricted to lie on a line, the same problem can be solved in +linear time [6]. +Assumptions. As in [12], we assume that no three vertices of T are aligned. For the sake of simplicity, we also assume +that no edge of T is contained in the bisector of two viewpoints in P, and that no point on T is at the same distance +from three or more viewpoints in P. +As mentioned earlier, we restrict to the case where the viewpoints lie on terrain vertices; the same assumption is +made in [12], and it has the implication that m ≤ n. Notice that no generality is lost because, if viewpoints are located +in the interior of terrain edges, we can simply add vertices to the terrain and apply our algorithms. Furthermore, placing +a superlinear number of viewpoints on the terrain does not seem to make much sense: If more than two viewpoints lie +on the same edge, it is easy to see that the union of the viewsheds of the leftmost and rightmost viewpoints contains the +viewshed of any other viewpoint on the edge. Therefore, all the intermediate viewpoints are somewhat irrelevant for +visibility purposes. +Finally let us mention that, in [12], kv and kc do not only include the number of points of T that are on the boundary +of two distinct regions of the respective diagrams, but also the total number of vertices of T , that is, n. For the sake of +consistency, we follow the same convention in this paper. +2. Complexity of the Voronoi visibility map +In [12], it is stated that the complexity of VorVis(T , P) can be higher than, lower than, or equal to that of +ColVis(T , P). In this section, we refine this statement. Recall that in both cases the complexity is O(mn), and this +bound is asymptotically tight [12]. +Let us introduce some terminology. The Voronoi viewshed WT (p, P) of p is the set of points in the viewshed of p +that are closer to p than to any other viewpoint that is visible from them. +Since we have assumed that no edge of T is contained in the bisector of two viewpoints, the shared boundary +between two consecutive regions of VorVis(T , P) is always a single point of T . We call such points event points +of VorVis(T , P). Event points of ColVis(T , P) are defined analogously, that is, as points on the boundary of two +consecutive regions of the map. +We denote by bi,j the perpendicular bisector of two viewpoints pi, p j. Additionally, we denote by qi,j an event point +of VorVis(T , P) such that a point infinitesimally to the left and right of qi, j belongs to WT (pi, P) and WT (pj, P), +respectively (notice that an event qi, j is different from an event qj,i). There are three (not mutually exclusive) possibilities: +(i) pi becomes invisible at qi, j1; (ii) pj becomes visible at qi,j; (iii) pi and p j are visible at qi, j, and qi, j is an intersection +1When we write that pi becomes invisible at qi,j, we mean that it is visible immediately to the left of qi,j and invisible immediately to its right. +We use the same rational when we write that p j becomes visible at qi,j. +3 + +pj +pi +q +bi,j +(a) +(b) +t +Figure 4: (a) Illustration of the charging scheme of events of VorVis(T , Pr) at the intersection of a bisector and T (proof of Theorem 1): The event q +is charged to pi because no point t to the right of q belongs to WT (pi, Pr). (b) An instance where kv = kc + 2m − 2: ColVis(T , P) consists of three +portions (a portion visible by all viewpoints surrounded by two portions not visible by any viewpoint), while in VorVis(T , P) (illustrated in the +figure) the visible portion is subdivided into 2m − 1 parts. +point between bi, j and T . +In the following lemma, we prove the key observation of this paper: Even though a bisector bi, j might intersect the +terrain Θ(n) times, only two such intersections are relevant and might produce events of type (iii). +Lemma 1. Let pi ∈ P be lower2 than pj ∈ P. Let q be an intersection point between bi, j and T to the left (respectively, +right) of pi. Then any point to the left (respectively, right) of q that is visible from pi is closer to p j than to pi. Hence, +there is no event qi,j or q j,i of type (iii) that lies to the left (respectively, right) of q. +Proof. Since pi is assumed to have a smaller y-coordinate than that of p j, the region of the plane closer to pi than to p j +is the one below bi,j. But any point r that is on bi, j or below it and to the left (respectively, right) of q is not visible from +pi because the line segment pir contains a point (specifically, the point vertically aligned with q) that lies strictly below +the terrain surface; see Fig. 3 for an illustration. +The second part of the statement follows because visibility from pi is one of the conditions of events of type +(iii). +To prove our bounds, we also use this well-known property of visibility in 1.5D terrains, known as order claim: +Lemma 2 (Claim 2.1 in [3]). Let a, b, c, and d be four points on T such that x(a) < x(b) < x(c) < x(d). If a sees c and +b sees d, then a sees d. +We denote by VorVis(T , Pℓ) and ColVis(T , Pℓ) the Voronoi and colored visibility maps of T assuming that +viewpoints can only see themselves and to their left. Further, we denote by WT (p, Pℓ) the Voronoi viewshed of p +under the same assumption. VorVis(T , Pr), ColVis(T , Pr) and WT (p, Pr) are defined analogously using visibility to +the right. We can now prove the following: +Theorem 1. Given a terrain T with n vertices and a set P of m viewpoints placed on vertices of T , the following +bound holds: +kv ≤ min{kc + m2, 2kc + 8m − 4}. +Proof. Since the vertices of T are counted in both kv and kc, we exclude them from our analysis. +We start by proving that kv ≤ kc + m2. Notice that events of type (i) and (ii) are also events of ColVis(T , P). Let us +prove that there are at most m2 events of type (iii). +Let pi, pj be a pair of viewpoints. If pi and p j are at the same height, bi, j is vertical and only intersects T once, so +there is at most one event of VorVis(T , P) on bi, j ∩ T . Otherwise, we assume without loss of generality that pi is lower +than pj. By Lemma 1 the only candidates for events qi,j or qj,i of type (iii) are the left-most intersection point of type +2We say that p is lower (respectively, higher) than q when it has a smaller (respectively, greater) y-coordinate than that of q. +4 + +bi, j ∩ T among all such points to the right of pi and the right-most one among all points to the left. Thus, every pair of +viewpoints creates at most two events of type (iii). +We next prove the second upper bound for kv. We denote by kℓ +v, kℓ +c, kr +v and kr +c the total complexity of all the regions +of VorVis(T , Pℓ), ColVis(T , Pℓ), VorVis(T , Pr) and ColVis(T , Pr), respectively. +Each event of ColVis(T , P) can be uniquely assigned to an event of either ColVis(T , Pℓ) or ColVis(T , Pr): If +the event concerns viewpoint pi (becoming visible or invisible) and it is to the left of pi, the same event appears in +ColVis(T , Pℓ) and it is assigned to it. If the event is to the right of pi, it is assigned to the same event in ColVis(T , Pr). +If the event is on pi, it is easy to see that pi is either the left-most point of T , in which case we assign it to the same +event in ColVis(T , Pr), or the right-most point of T , in which case we assign it to the same event in ColVis(T , Pℓ). +Each event of ColVis(T , Pℓ) or ColVis(T , Pr) that did not get any event of ColVis(T , P) assigned to it lies at the +same position of some viewpoint that is not the left-most or the right-most point of T . Indeed, if pi ∈ P is such a +viewpoint, then, at the position where it lies, pi becomes visible in ColVis(T , Pr) and invisible in ColVis(T , Pℓ)3, +but there are no such events in ColVis(T , P) (where there is a portion of T visible from pi containing pi not on its +boundary but in its interior). This proves that kℓ +c + kr +c ≤ kc + 2m. +Next, we show a relationship between kr +v and kr +c. Suppose that we traverse VorVis(T , Pr) from left to right, and we +stop at every event that is not an event of ColVis(T , Pr), that is, the event is at the intersection of a bisector bi, j and T . +Since in VorVis(T , Pr) viewpoints can only see themselves and to their right, pi and p j are to the left of the event q. +Without loss of generality, suppose that p j is to the left of pi. As in the proof of Lemma 1, if pi was higher than p j, +no point on bi, j to the right of pi would be visible from pj, contradicting the existence of the event at the intersection +of bi,j and T . Further, pi and pj are not at the same height because both are to the left of q. Hence, pi is lower than +p j. Let t be a point to the right of q that is visible from pi (see Fig. 4a). By the Lemma 1 with a = pj, b = pi, c = q +and d = t, t is also visible from p j. By Lemma 1, t is closer to pj than to pi. Therefore, t � WT (pi, Pr). This implies +that there is no portion of WT (pi, Pr) to the right of q and, in particular no more event caused by the intersection of a +bisector of pi and another viewpoint. We charge the event q to pi, and obtain that there are at most m − 1 events of this +type. Hence, kr +v ≤ kr +c + m − 1. +Finally, we derive a bound for kv based on kr +v and kℓ +v. +Let us take a continuous portion T ′ of T that belongs to the Voronoi viewshed of some viewpoint pi in +VorVis(T , Pr), and to the Voronoi viewshed of some viewpoint pj in VorVis(T , Pℓ). Let T ′ be maximal with +this property. Notice that pi is to the left of T ′, while p j is to its right. Furthermore, in VorVis(T , P), every point of +T ′ belongs to the Voronoi viewshed of pi or p j. We next show that bi, j intersects T ′ at most once. The claim is clear +when y(pi) = y(pj). Otherwise, we assume without loss of generality that pi is lower than p j. Let q be the left-most +intersection point (if any) between bi,j and T ′. We have that q is to the right of pi. Additionally, all points of T ′ to the +right of q are visible from pi because they belong to the Voronoi viewshed of pi in VorVis(T , Pr). By Lemma 1, all +points of T ′ to the right of q are closer to p j than to pi. In consequence, there is no intersection point between bi, j and +T ′ to the right of q, and bi, j intersects T ′ at most once. This implies that T ′ gets split into at most two portions of the +final diagram. +The situation where in at least one of VorVis(T , Pr) or VorVis(T , Pℓ) a portion does not have any visible viewpoint +is trivial. +Consequently, kv ≤ 2(kr +v + kℓ +v). +Putting everything together, +kv ≤ 2(kr +v + kℓ +v) +≤ 2(kr +c + kℓ +c + 2m − 2) +≤ 2(kc + 4m − 2) = 2kc + 8m − 4. +Regarding lower bounds, we show the following: +Example 1. There exists a terrain with n vertices and a set of m viewpoints placed on vertices of the terrain such that +kv = kc + 2m − 2. The construction is illustrated in Fig. 4b. +3Strictly speaking, in ColVis(T , Pℓ) pi becomes invisible immediately to its right. +5 + +3. Computation of the Voronoi visibility map +The algorithm we propose is simple: We sweep the terrain from left to right, and maintain the set of visible points +in a balanced binary search tree, where the key of every viewpoint is the Euclidean distance to the point of the terrain +currently swept by the sweep line; the relevant viewpoint is always the closest visible one. The algorithm is based +on the observation that maintaining the whole set of viewpoints sorted by distance to T might be expensive (since a +bisector of two viewpoints might intersect the terrain Θ(n) times), while, by Lemma 1, maintaining the set of the visible +ones is not (since, out of the potential Θ(n) intersections, the two viewpoints are visible in at most two). Thanks to this +observation, new events of VorVis(T , P) are found in O(log m) time rather than O(m). We next present the details. +The algorithm sweeps the terrain from left to right and stops at points that are candidates for event points. The +candidates for events of type (i) and (ii) are the events of ColVis(T , P). We explain in Section 3.2 which are the +candidates for events of type (iii). +3.1. Events of ColVis(T , P) +We compute ColVis(T , P) using the version of the algorithm from [12] that returns a doubly-linked list with the +vertices of ColVis(T , P) sorted from left to right, together with the visibility information provided as follows: The +visible viewpoints are specified for the first component of ColVis(T , P) and, for the other components, the algorithm +outputs the changes in the set of visible viewpoints with respect to the component immediately to the left. +3.2. Candidates for events of type (iii) +We next describe the candidates for events of type (iii) associated with a pair of viewpoints pi, pj ∈ P. +If pi and p j are at the same height, the only intersection point of bi,j with T lies between both viewpoints. If such +point is visible from both pi and pj, we add it as a candidate for event of type (iii). +Otherwise, we may assume, without loss of generality, that pi is lower than pj. By Lemma 1 the only candidates +for events of type (iii) involving pi and pj are the left-most intersection point of type bi, j ∩ T among all such points to +the right of pi and the right-most one among all points to the left. For the sake of simplicity, we first assume that bi, j +is not tangent to T at any of these intersection points. Then each of these intersection points is added to the list of +candidates for events swept by the line if and only if it is visible from both pi and pj. +Finally, let q be one of the two candidates for events of type (iii) involving pi and pj. Suppose that q is to the right +of pi (the other case is symmetric). If bi, j is tangent to T at q, points of T infinitesimally to the left or right of q are +closer to pi than to pj (while q is equidistant). Additionally, pi becomes invisible right after q. In consequence, it is not +needed to add q to the list of candidates for events of type (iii): Right before q, the algorithm knows that pi is closer +to the terrain than p j. At q, the algorithm processes that pi becomes invisible, and pj (if it is visible) automatically +gets higher priority than pi in the list of candidates for the “owner” of the current Voronoi visibility region. The key +argument (there are no more candidates for events of type (iii) to the right of q) also holds in this case. +3.3. Data structures +The algorithm uses the following data structures. +We maintain a balanced binary search tree H that contains the viewpoints that are visible at the current point of the +sweep. These viewpoints are sorted in the tree according to their corresponding key, which is the distance from the +viewpoint to the current intersection point between the sweep line and the terrain. The keys are not stored in the tree +because they change as the sweep line moves, but each of them can be computed when needed in constant time. The +algorithm always chooses as the “owner” of the current Voronoi visibility region the viewpoint of H with the minimum +key. +In H, we perform insertions and deletions when viewpoints become visible and invisible, respectively. During these +operations, when at some node of the tree we need to decide whether we move to its left or right subtree, we simply +compute the key associated to the viewpoint in that node, and compare it with the key of the viewpoint that we want to +insert or delete. Therefore, insertions and deletions can be performed in the standard way in O(log m) time. +When the sweep line encounters a candidate for an event of type (iii) (let us call it q), the relative order of two +visible viewpoints with respect to their current distance to the terrain changes (formally speaking, it changes right after +q). A possible way to reflect this in H is to delete from the tree one of the two viewpoints associated with q, and then +6 + +' +& +$ +% +Input: T , P, E +Output: VorVis(T , P) +1: H := ∅, tℓ := left-most point of T , and p∗ := ⊥ +2: while E � ∅ do +3: +extract the next element q of E +4: +if q is the last element of E then +5: +output ((tℓ, q), p∗) +6: +break +7: +else if some viewpoint v becomes visible at q then +8: +insert v in H +9: +else if some viewpoint v becomes invisible at q then +10: +delete v from H +11: +else if q is an intersection point between T and bi, j then +12: +update the positions of pi and p j in H +13: +end if +14: +update pmin +15: +if pmin � p∗ then +16: +output ((tℓ, q), p∗) +17: +tℓ := q, p∗ := pmin +18: +end if +19: end while +Figure 5: Computation of VorVis(T , P). E is the list of potential events, H is the tree containing the viewpoints that are currently visible, tℓ is the +left endpoint of the current portion of T , p∗ is the closest visible viewpoint in that portion, and pmin is the viewpoint in H with the minimum key. +insert it again using as keys the distances from the viewpoints to a point of T infinitesimally to the right of q (and still +to the left of the next event in the list). Thus, candidates for events of type (iii) can be processed in H in O(log m) time. +Additionally, we use a data structure that allows us to answer ray-shooting queries in T in O(log n) time [5]. Such +queries are used to decide whether a given pair of points are mutually visible, and to find the relevant intersections +between T and the bisector of a pair of viewpoints. +3.4. Description of the algorithm +Given q, r on T with x(q) < x(r), we denote by T (q, r) and T [q, r] the open and closed portion of the terrain +between q and r, respectively. +Our algorithm, outlined in Fig. 5, takes as input T , P and a list E of potential events sorted from left to right +containing all events of ColVis(T , P) together with the O(m2) candidates for events of type (iii). The list E also +contains an event at the right-most point of the terrain. +The algorithm outputs VorVis(T , P) as a list of pairs ((q, r), pi) such that pi is the closest visible viewpoint in +T (q, r) (if T (q, r) is not visible from any viewpoint, we output ((q, r), ⊥)). The variables tℓ and p∗ in the algorithm refer +to the left endpoint of the portion of T currently analyzed by the algorithm and the closest visible viewpoint in that +portion, respectively. The variable pmin refers to the viewpoint in H with the minimum key (if H is empty, pmin = ⊥). +Initially, H := ∅, tℓ := left-most point of T , and p∗ := ⊥. +We repeat the following procedure until E is empty: We extract the next element q from E, and proceed according +to four cases, corresponding to lines 4, 7, 9, and 11 of the pseudocode in Fig. 5. For the sake of simplicity, in the +description in Fig. 5 we deliberately ignore the situation where several events of distinct type occur at the same point +of T , which we tackle in the next paragraph. The cases in lines 4, 7 and 9 are clear. Regarding the case starting at +line 11, in line 12 we update the positions of pi and pj in H as explained in Section 3.3 (see the paragraph where we +discuss the case where the sweep line encounters a candidate for an event of type (iii)). We also point out that, if q is an +intersection point between T and more than one bisector of type bi,j, the bisectors can be processed in any order.4 +4By our general position assumptions, q is not equidistant from three or more viewpoints, so at most one of the bisectors through q might involve +7 + +It remains to explain how to deal with the situation where several events of distinct type occur at the same point of +T . In this case, we first perform the modifications in H triggered by all the events at that point (insertions of viewpoints +becoming visible, deletions of viewpoints becoming invisible and updates of the positions of pairs of viewpoints). After +updating H in this way, we update pmin; if pmin � p∗, we output ((tℓ, q), p∗), set tℓ := q, and set p∗ := pmin. +3.5. Correctness and running time +We first show that the algorithm for VorVis(T , P) always selects the closest visible viewpoint. Changes in the +visibility status of the viewpoints correspond to events of ColVis(T , P), which are added to E, so the set of visible +viewpoints contained in H is correct at any time of the sweep. Regarding the distances from the viewpoints to the +terrain, every time that a viewpoint is swept or becomes visible, it is inserted in H correctly (according to its current +distance to the terrain). Changes in the order of the visible viewpoints with respect to their distances to T coincide with +intersections of T with the bisectors among them. As argued in the proof of Theorem 1, for every pair of viewpoints it +happens at most twice that both viewpoints are visible at an intersection point between T and their bisector. Such an +event is precomputed and stored in E, and later processed by the algorithm. +We next analyze the complexity of the algorithm. +The map ColVis(T , P) can be computed in O(n + (m2 + kc) log n) time using the algorithm in [12]. This map has at +most kc regions; however, due to the fact that several viewpoints might become visible or invisible at the same time, +when sweeping ColVis(T , P) from left to right, the number of times that a viewpoint becomes visible or invisible, +added over all viewpoints, can be higher; an upper bound of kc + m2 is given in [12]. Each time that a viewpoint +changes its visibility status, we perform an insertion or a deletion in H, which takes O(log m) time. The algorithm +processes at most m2 intersections between the terrain and bisectors of endpoints in O(log m) time each. Consequently, +VorVis(T , P) can be extracted from ColVis(T , P) in O((m2 + kc) log m) time. The space complexity of the algorithm is +the space required to store the terrain, the events and the data structures, that is, O(n + m2 + kc). +We conclude with the following: +Theorem 2. The Voronoi visibility map of a 1.5D terrain can be constructed in O(n + (m2 + kc) log n) time and +O(n + m2 + kc) space. +4. Extensions +In this section, we present adaptations of the previous algorithm to compute related maps. +4.1. Higher order Voronoi visibility maps +We define the kth-order Voronoi visibility map VorVisk(T , P) as a partitioning of T into regions that have the same +set of ℓ closest visible viewpoints, where ℓ is the minimum of k and the number of visible viewpoints in the region. +Observe that the mth-order Voronoi visibility map is equal to ColVis(T , P). +We can easily compute VorVisk(T , P) by adapting the algorithm from Section 3. In this case, we need to maintain +two additional variables: the total number b of viewpoints that are visible at the point currently swept by the line, +and, from the current set of ℓ closest visible viewpoints, the furthest one, denoted pmax. Analogously to the algorithm +for ColVis(T , P), for space reasons our algorithm for VorVisk(T , P) returns a doubly-linked list with the vertices of +VorVisk(T , P) sorted from left to right, together with the following information: The set of ℓ closest visible viewpoints +is specified for the first component of VorVisk(T , P) and, for the other components, the algorithm outputs the changes +in the set of ℓ closest visible viewpoints with respect to the component immediately to the left. +Let q be the next element from the list of events E, computed as in the previous section. We explain in detail the +case where one or more viewpoints become visible at q, and leave the remaining cases to the interested reader. Let P′ +denote the set of viewpoints becoming visible at q. We update b. If, after this update, b ≤ k, we report vertex q together +with the set P′ (containing the new viewpoints in the set of ℓ closest visible viewpoints). We also insert the viewpoints +of P′ in H. Otherwise, let b′ and b be the number of visible viewpoints right before q and at q, respectively. If b′ < k, +we remove from P′ the set of k − b′ closest viewpoints to q (obtained after sorting the viewpoints of P′ according to +p∗. +8 + +their distance to q), we add these viewpoints to a set P′ +in, and we insert them in H. After possibly performing this +operation in P′, we proceed as follows: We extract the closest viewpoint to q of P′; if it is closer to q than pmax, we add +this viewpoint to P′ +in, we insert it in H, we add viewpoint pmax to Pout, and we update pmax. Notice that pmax can be +updated by finding the predecessor in H of the “old” pmax, that is, in O(log m) time. We repeat this process until P′ is +empty or the next element in P′ is farther to q than pmax. Then we insert the remaining viewpoints of P′ (if any) in +H. Finally, we report vertex q together with the set P′ +in (containing the new viewpoints in the set of ℓ closest visible +viewpoints) and the set Pout (containing the viewpoints that stop belonging to the set of ℓ closest visible viewpoints). +Clearly, every change in the visibility status of a viewpoint and every intersection of T with the bisector of two +visible viewpoints can be processed in O(log m + log n) time. Hence, we obtain: +Theorem 3. The kth-order Voronoi visibility map of a 1.5D terrain can be constructed in O(n + (m2 + kc) log n) time +and O(n + m2 + kc) space. +4.2. Other distances +Given q, r on T with x(q) < x(r), two other natural distances between q and r are the Euclidean length of the +portion T [q, r], which we will call Euclidean distance along the terrain, and the number of vertices in the portion +T (q, r), which we will call link distance.5 We may define the Voronoi visibility map of T based on these distances. +The relevant difference with respect to the standard case is the shape of the bisectors between two viewpoints pi +and pj. In the case of the Euclidean distance along the terrain, there is exactly one point of T that is equidistant to pi +and pj, and this point can be computed in O(log n) time after preprocessing T so that the Euclidean distance along the +terrain between any pair of vertices of T can be computed in O(1) time.6 Regarding the link distance, if there is an odd +number of vertices between pi and pj, there is exactly one vertex of T that is equidistant to pi and pj, and this vertex +can be computed in O(1) time. However, if there is an even number of vertices between pi and p j, there is an open +edge of T such that all of its points are at the same link distance from pi and pj. In this case, we must either allow the +border between two consecutive Voronoi regions to be 1-dimensional, or, if simplicity is more desirable, we might +(artificially) select an interior point of this edge as the intersection point between T and the bisector of pi and p j. +After adding the corresponding candidates for events of type (iii) based on the explanations in the previous +paragraph, the rest of the algorithm is equal to the one for the general case. The running time remains the same because, +given a pair of points on T , in both cases the distance between them can be computed in O(1) time. Therefore, we +conclude: +Theorem 4. The Voronoi visibility map of a 1.5D terrain with respect to the Euclidean distance along the terrain or to +the link distance can be constructed in O(n + (m2 + kc) log n) time and O(n + m2 + kc) space. +5. Computation of r∗ +We recall that r∗ is the minimum value of r such that, if the viewpoints can only see objects that are within distance +r, the visibility map of T does not change. +Let Pr denote the set of viewpoints P with the restriction that the visibility range of the viewpoints is r. We then +may define Vis(T , Pr), VorVis(T , Pr)...in the natural way. Notice that, for P∞, we obtain the same objects as in the +standard case. +Let d(x, y) denote the Euclidean distance between two points x, y ∈ R2. +Lemma 3. r∗ = max +i=1,...,m{ +sup +x∈WT (pi,P∞) +d(pi, x)}. +5For the link distance, we take the open portion of the terrain T (q, r) so that any two points on the same edge (including the endpoints) are at +(link) distance zero. +6If we store, for every vertex q of T , the Euclidean distance along the terrain qd between q and the left-most point of T , then the Euclidean +distance along the terrain between vertices q, r of T such that x(q) < x(r) is rd − qd. +9 + +Proof. Let pi and x be a viewpoint and a point of T achieving the maximum in the right hand expression. If r∗ < d(pi, x), +x would not be visible from pi in Vis(T , Pr∗). Since x belongs to the boundary of WT (pi, P∞), all other viewpoints +seeing x have a distance to x that is greater than or equal to d(pi, x); thus, x would also not be visible from any of them +in Vis(T , Pr∗). Since x is visible in Vis(T , P∞)7, we reach a contradiction. Therefore, r∗ ≥ d(pi, x). +On the other hand, to keep Vis(T , P∞) unchanged, it is enough to maintain the closure of WT (pi, P∞) visible for +all i, since Vis(T , P∞) is equal to the union of the closures of the regions WT (pi, P∞). If we set a visibility range of +sup +x∈WT (pi,P∞) +d(pi, x), the closure of WT (pi, P∞) indeed remains visible. Consequently, r∗ ≤ max +i=1,...,m{ +sup +x∈WT (pi,P∞) +d(pi, x)}. +Using this characterization of r∗, we can prove the following: +Theorem 5. The problem of computing the minimum value r∗ such that Vis(T , Pr∗) = Vis(T , P∞) can be solved in +O(n + (m2 + kc) log n) time. +Proof. By Lemma 3, it suffices to consider the distances between the vertices of VorVis(T , P∞) (that is, the points on +the boundary of the Voronoi viewsheds) and their associated viewpoints. Consequently, the problem can be trivially +solved in linear time if VorVis(T , P∞) is known. +6. Final remark +As indicated in [12], in the running time of the algorithm to compute ColVis(T , P), the term m2 log n disappears if +we assume that no two viewpoints change from invisible to visible at the same point of T . This can always be achieved +by infinitesimally perturbing the terrain. However, such a perturbation does not make the same term disappear from +the running time of the presented algorithm to compute VorVis(T , P). Given that one of the bounds in Theorem 1 +guarantees that kv = O(kc + m), it remains as an open problem to design an algorithm for VorVis(T , P) that is equally +faster than that for ColVis(T , P) for all possible instances. +References +[1] Afshani, P., de Berg, M., Casanova, H., Karsin, B., Lambrechts, C., Sitchinava, N., Tsirogiannis, C., 2018. An efficient algorithm for the 1D +total visibility-index problem and its parallelization. Journal of Experimental Algorithmics 23, 1–23. +[2] Ben-Moshe, B., Hall-Holt, O., Katz, M.J., Mitchell, J.S., 2004. Computing the visibility graph of points within a polygon, in: Proceedings of +the 20th Annual Symposium on Computational Geometry, pp. 27–35. +[3] Ben-Moshe, B., Katz, M.J., Mitchell, J.S., 2007. A constant-factor approximation algorithm for optimal 1.5 d terrain guarding. SIAM Journal +on Computing 36, 1631–1647. +[4] Catry, F.X., Rego, F.C., Santos, T., Almeida, J., Relvas, P., 2007. Forest fires prevention in Portugal - using GIS to help improving early fire +detection effectiveness, in: Proceedings of the 4th International Wildland Fire Conference. +[5] Chazelle, B., Edelsbrunner, H., Grigni, M., Guibas, L.J., Hershberger, J., Sharir, M., Snoeyink, J., 1994. Ray shooting in polygons using +geodesic triangulations. Algorithmica 12, 54–68. +[6] Daescu, O., Friedrichs, S., Malik, H., Polishchuk, V., Schmidt, C., 2019. Altitude terrain guarding and guarding uni-monotone polygons. +Computational Geometry: Theory and Applications 84, 22–35. +[7] Eisenbrand, F., Funke, S., Karrenbauer, A., Matijevic, D., 2008. Energy-aware stage illumination. International Journal of Computational +Geometry & Applications 18, 107–129. +[8] Ernestus, M., Friedrichs, S., Hemmer, M., Kokem¨uller, J., Kr¨oller, A., Moeini, M., Schmidt, C., 2017. Algorithms for art gallery illumination. +Journal of Global Optimization 68, 23–45. +[9] Friedrichs, S., Hemmer, M., King, J., Schmidt, C., 2016. The continuous 1.5D terrain guarding problem: Discretization, optimal solutions, and +PTAS. Journal of Computational Geometry 7, 256–284. +[10] Friedrichs, S., Hemmer, M., Schmidt, C., 2014. A PTAS for the continuous 1.5D terrain guarding problem, in: Proceedings of the 26th +Canadian Conference on Computational Geometry. +[11] Gibson, M., Kanade, G., Krohn, E., Varadarajan, K., 2009. An approximation scheme for terrain guarding, in: Approximation, Randomization, +and Combinatorial Optimization. Algorithms and Techniques. Springer, pp. 140–148. +[12] Hurtado, F., L¨offler, M., Matos, I., Sacrist´an, V., Saumell, M., Silveira, R.I., Staals, F., 2014. Terrain visibility with multiple viewpoints. +International Journal of Computational Geometry & Applications 24, 275–306. +7It follows from our definition of visibility that the maximal visible portions of T are closed and, hence, the points on the boundary of the Voronoi +viewsheds are visible. +10 + +[13] Joe, B., Simpson, R.B., 1987. Corrections to Lee’s visibility polygon algorithm. BIT Numerical Mathematics 27, 458–473. +[14] King, J., Krohn, E., 2011. Terrain guarding is NP-hard. SIAM Journal on Computing 40, 1316–1339. +[15] M¨oller, B., 2006. Changing wind-power landscapes: regional assessment of visual impact on land use and population in Northern Jutland, +Denmark. Applied Energy 83, 477–494. doi:10.1016/j.apenergy.2005.04.004. +[16] O’Rourke, J., 2006. Open problems from CCCG 2005., in: Proceedings of the 18th Canadian Conference on Computational Geometry. +[17] Yick, J., Mukherjee, B., Ghosal, D., 2008. Wireless sensor network survey. Computer Networks 52, 2292–2330. doi:10.1016/j.comnet. +2008.04.002. +11 + diff --git a/ydE4T4oBgHgl3EQfYgwr/content/tmp_files/load_file.txt b/ydE4T4oBgHgl3EQfYgwr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..12925e6b872b275a4a72988be7795f44c5191419 --- /dev/null +++ b/ydE4T4oBgHgl3EQfYgwr/content/tmp_files/load_file.txt @@ -0,0 +1,525 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf,len=524 +page_content='On Voronoi visibility maps of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D terrains with multiple viewpoints⋆ Vahideh Keikhaa,b, Maria Saumella,c,∗ aThe Czech Academy of Sciences, Institute of Computer Science, Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' bFaculty of Mathematics and Physics, Department of Applied Mathematics, Charles University, Prague, Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' cDepartment of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Abstract Given an n-vertex 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D terrain T and a set P of m < n viewpoints, the Voronoi visibility map VorVis(T , P) is a partitioning of T into regions such that each region is assigned to the closest (in Euclidean distance) visible viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The colored visibility map ColVis(T , P) is a partitioning of T into regions that have the same set of visible viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' In this paper, we propose an algorithm to compute VorVis(T , P) that runs in O(n + (m2 + kc) log n) time, where kc and kv denote the total complexity of ColVis(T , P) and VorVis(T , P), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' This improves upon a previous algorithm for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We also generalize our algorithm to higher order Voronoi visibility maps, and to Voronoi visibility maps with respect to other distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Finally, we prove bounds relating kv to kc, and we show an application of our algorithm to a problem on limited range of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Keywords: Visibility, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D terrains, Voronoi diagrams, multiple viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Introduction A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D terrain T is an x-monotone polygonal chain of n vertices in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Two points on T are visible if the segment connecting them does not contain any point strictly below T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Visibility problems in terrains are fundamental in geographical information science and have many applications, such as placing fireguard or telecommunication towers [4], identifying areas that are not visible from sensitive sites [15], or solving problems related to sensor networks [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Although 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D terrains are more interesting for modelling and forecasting, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D terrains are easier to visualize and to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' They give insights into the difficulties of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D terrains in terrain analysis, and their proper understanding is seen as an essential step towards the ultimate goal of settling the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' For this reason, visibility problems in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D terrains have been intensively studied by the computational geometry community during the last 15 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' In this paper, we focus on the variant where a set P of m < n viewpoints are located on vertices of T (we refer to the end of this section for a discussion on the assumption m < n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' For each viewpoint p ∈ P, the viewshed of p is the set of points of T that are visible from p (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 1 for an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Our goal is to efficiently extract information about the visibility of T with respect to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We continue the work initiated in [12], where the following structures are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The visibility map Vis(T , P) is a partitioning of T into a visible region (containing all portions of T that are visible by at least one element in P) and an invisible region (containing the portions that are not visible by any element in P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 2a for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The visible region of the visibility map is equal to the union of the viewsheds of all viewpoints in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' ⋆Supported by the Czech Science Foundation, grant number GJ19-06792Y, and with institutional support RVO:67985807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='K was also partially supported by Charles University project UNCE/SCI/004 and by the Czech Academy of Sciences (Praemium Academiae awarded to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Paluˇs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' ∗Corresponding author Email addresses: keikha@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='cas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='cz (Vahideh Keikha), maria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='saumell@fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='cvut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='cz (Maria Saumell) Preprint submitted to Elsevier January 13, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='05049v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='CG] 12 Jan 2023 p Figure 1: The viewshed of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' pk pj pi (a) pk (b) pj pi pk pj pi (c) Figure 2: (a) Vis(T , P) (the visible region is shown in gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' (b) ColVis(T , P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' (c) VorVis(T , P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The colored visibility map ColVis(T , P) is a partitioning of T into regions that have the same set of visible viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 2b for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Finally, the Voronoi visibility map VorVis(T , P) is a partitioning of T into regions that have the same closest visible viewpoint, where the distance used is the Euclidean distance (not the distance along the terrain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 2c for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Algorithms to compute these structures for both 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D terrains are proposed in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The algorithm to obtain VorVis(T , P) of a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D terrain runs in O(n + (m2 + kc) log n + kv(m + log n log m)) time, where kc and kv denote the total complexity of ColVis(T , P) and VorVis(T , P), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Both kc and kv have size O(mn), and this bound is asymptotically tight [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The algorithm first computes ColVis(T , P), and then it spends Θ(m) time to find each single region of VorVis(T , P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' In this paper, we show that VorVis(T , P) can be extracted from ColVis(T , P) in a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D terrain much more efficiently, resulting in an O(n + (m2 + kc) log n)-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We use an observation related to intersections of the terrain with bisectors of pairs of viewpoints that also allows us to prove a relationship between kc and kv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Let us point out that, apart from the mentioned output-sensitive algorithm for VorVis(T , P) of a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D terrain, the authors of [12] also propose a divide-and-conquer algorithm running in O(mn log m) time, which is worst-case nearly optimal (recall that the maximum complexity of VorVis(T , P) is Θ(mn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Therefore, our new algorithm does not represent an improvement in the worst-case instances, but in instances where the original output-sensitive algorithm is faster than the divide-and-conquer one, and kvm is the dominant term in the running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' An example of such an instance is m = Θ( √n), kc = Θ(n3/4) and kv = Θ(n3/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' In this paper, we also provide generalizations of our algorithm to compute Voronoi visibility maps of higher order (that is, containing the information about the k closest visible viewpoints, for some k > 1), and Voronoi visibility maps with respect to two other distances: the Euclidean distance along the terrain and the link distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' All of these generalizations have the same running time as the original algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Finally, the new algorithm for VorVis(T , P) also allows us to solve efficiently a problem related to limited range of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' These problems are motivated by the fact that, even though many visibility problems assume an infinite range of visibility, the intensity of light, signals and other phenomena modelled with viewpoints decreases over distance in realistic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' In this spirit, the problem of illuminating a polygonal area with the minimum total energy was introduced by O’Rourke [16], and studied in [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We consider a related problem on terrains, namely, computing the minimum value r∗ such that, if the viewpoints can only see objects within distance r∗, the obtained visibility map is the same as Vis(T , P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We show that this problem can also be solved in O(n + (m2 + kc) log n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Related Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' When there is only one viewpoint, computing the visibility map of a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D terrain can be done in O(n) time by converting the terrain into a simple polygon and applying the algorithm from [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' One of the first results on the variant with more than one viewpoint is an O((n + m) log m) time algorithm to detect if there are any visible pairs 2 pj pi q bi,j pj q bi,j (a) (b) r r pi Figure 3: Illustration of Lemma 1: (a) x(pi) > x(p j);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' (b) x(pi) < x(pj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' of viewpoints above a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D terrain [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Later, a systematic study of Vis(T , P), VorVis(T , P) and ColVis(T , P) was carried out in [12] for both 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D terrains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' A problem that is very related to the construction of Vis(T , P) is that of computing the total visibility index of the terrain, that is, the number of viewpoints that are visible from each of the viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' This problem can be solved in O(n log2 n) time [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The situation where the locations of the viewpoints are unknown has been thoroughly studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' It is well-known that computing the minimum number of viewpoints to keep a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D terrain illuminated is NP-complete [9, 14], but the problem admits a PTAS [9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' If the viewpoints are restricted to lie on a line, the same problem can be solved in linear time [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' As in [12], we assume that no three vertices of T are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' For the sake of simplicity, we also assume that no edge of T is contained in the bisector of two viewpoints in P, and that no point on T is at the same distance from three or more viewpoints in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' As mentioned earlier, we restrict to the case where the viewpoints lie on terrain vertices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' the same assumption is made in [12], and it has the implication that m ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Notice that no generality is lost because, if viewpoints are located in the interior of terrain edges, we can simply add vertices to the terrain and apply our algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Furthermore, placing a superlinear number of viewpoints on the terrain does not seem to make much sense: If more than two viewpoints lie on the same edge, it is easy to see that the union of the viewsheds of the leftmost and rightmost viewpoints contains the viewshed of any other viewpoint on the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Therefore, all the intermediate viewpoints are somewhat irrelevant for visibility purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Finally let us mention that, in [12], kv and kc do not only include the number of points of T that are on the boundary of two distinct regions of the respective diagrams, but also the total number of vertices of T , that is, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' For the sake of consistency, we follow the same convention in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Complexity of the Voronoi visibility map In [12], it is stated that the complexity of VorVis(T , P) can be higher than, lower than, or equal to that of ColVis(T , P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' In this section, we refine this statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Recall that in both cases the complexity is O(mn), and this bound is asymptotically tight [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Let us introduce some terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The Voronoi viewshed WT (p, P) of p is the set of points in the viewshed of p that are closer to p than to any other viewpoint that is visible from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Since we have assumed that no edge of T is contained in the bisector of two viewpoints, the shared boundary between two consecutive regions of VorVis(T , P) is always a single point of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We call such points event points of VorVis(T , P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Event points of ColVis(T , P) are defined analogously, that is, as points on the boundary of two consecutive regions of the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We denote by bi,j the perpendicular bisector of two viewpoints pi, p j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Additionally, we denote by qi,j an event point of VorVis(T , P) such that a point infinitesimally to the left and right of qi, j belongs to WT (pi, P) and WT (pj, P), respectively (notice that an event qi, j is different from an event qj,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' There are three (not mutually exclusive) possibilities: (i) pi becomes invisible at qi, j1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' (ii) pj becomes visible at qi,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' (iii) pi and p j are visible at qi, j, and qi, j is an intersection 1When we write that pi becomes invisible at qi,j, we mean that it is visible immediately to the left of qi,j and invisible immediately to its right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We use the same rational when we write that p j becomes visible at qi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 3 pj pi q bi,j (a) (b) t Figure 4: (a) Illustration of the charging scheme of events of VorVis(T , Pr) at the intersection of a bisector and T (proof of Theorem 1): The event q is charged to pi because no point t to the right of q belongs to WT (pi, Pr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' (b) An instance where kv = kc + 2m − 2: ColVis(T , P) consists of three portions (a portion visible by all viewpoints surrounded by two portions not visible by any viewpoint), while in VorVis(T , P) (illustrated in the figure) the visible portion is subdivided into 2m − 1 parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' point between bi, j and T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' In the following lemma, we prove the key observation of this paper: Even though a bisector bi, j might intersect the terrain Θ(n) times, only two such intersections are relevant and might produce events of type (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Let pi ∈ P be lower2 than pj ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Let q be an intersection point between bi, j and T to the left (respectively, right) of pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Then any point to the left (respectively, right) of q that is visible from pi is closer to p j than to pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Hence, there is no event qi,j or q j,i of type (iii) that lies to the left (respectively, right) of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Since pi is assumed to have a smaller y-coordinate than that of p j, the region of the plane closer to pi than to p j is the one below bi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' But any point r that is on bi, j or below it and to the left (respectively, right) of q is not visible from pi because the line segment pir contains a point (specifically, the point vertically aligned with q) that lies strictly below the terrain surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 3 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The second part of the statement follows because visibility from pi is one of the conditions of events of type (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' To prove our bounds, we also use this well-known property of visibility in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D terrains, known as order claim: Lemma 2 (Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='1 in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Let a, b, c, and d be four points on T such that x(a) < x(b) < x(c) < x(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' If a sees c and b sees d, then a sees d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We denote by VorVis(T , Pℓ) and ColVis(T , Pℓ) the Voronoi and colored visibility maps of T assuming that viewpoints can only see themselves and to their left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Further, we denote by WT (p, Pℓ) the Voronoi viewshed of p under the same assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' VorVis(T , Pr), ColVis(T , Pr) and WT (p, Pr) are defined analogously using visibility to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We can now prove the following: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Given a terrain T with n vertices and a set P of m viewpoints placed on vertices of T , the following bound holds: kv ≤ min{kc + m2, 2kc + 8m − 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Since the vertices of T are counted in both kv and kc, we exclude them from our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We start by proving that kv ≤ kc + m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Notice that events of type (i) and (ii) are also events of ColVis(T , P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Let us prove that there are at most m2 events of type (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Let pi, pj be a pair of viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' If pi and p j are at the same height, bi, j is vertical and only intersects T once, so there is at most one event of VorVis(T , P) on bi, j ∩ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Otherwise, we assume without loss of generality that pi is lower than pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' By Lemma 1 the only candidates for events qi,j or qj,i of type (iii) are the left-most intersection point of type 2We say that p is lower (respectively, higher) than q when it has a smaller (respectively, greater) y-coordinate than that of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 4 bi, j ∩ T among all such points to the right of pi and the right-most one among all points to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Thus, every pair of viewpoints creates at most two events of type (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We next prove the second upper bound for kv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We denote by kℓ v, kℓ c, kr v and kr c the total complexity of all the regions of VorVis(T , Pℓ), ColVis(T , Pℓ), VorVis(T , Pr) and ColVis(T , Pr), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Each event of ColVis(T , P) can be uniquely assigned to an event of either ColVis(T , Pℓ) or ColVis(T , Pr): If the event concerns viewpoint pi (becoming visible or invisible) and it is to the left of pi, the same event appears in ColVis(T , Pℓ) and it is assigned to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' If the event is to the right of pi, it is assigned to the same event in ColVis(T , Pr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' If the event is on pi, it is easy to see that pi is either the left-most point of T , in which case we assign it to the same event in ColVis(T , Pr), or the right-most point of T , in which case we assign it to the same event in ColVis(T , Pℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Each event of ColVis(T , Pℓ) or ColVis(T , Pr) that did not get any event of ColVis(T , P) assigned to it lies at the same position of some viewpoint that is not the left-most or the right-most point of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Indeed, if pi ∈ P is such a viewpoint, then, at the position where it lies, pi becomes visible in ColVis(T , Pr) and invisible in ColVis(T , Pℓ)3, but there are no such events in ColVis(T , P) (where there is a portion of T visible from pi containing pi not on its boundary but in its interior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' This proves that kℓ c + kr c ≤ kc + 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Next, we show a relationship between kr v and kr c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Suppose that we traverse VorVis(T , Pr) from left to right, and we stop at every event that is not an event of ColVis(T , Pr), that is, the event is at the intersection of a bisector bi, j and T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Since in VorVis(T , Pr) viewpoints can only see themselves and to their right, pi and p j are to the left of the event q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Without loss of generality, suppose that p j is to the left of pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' As in the proof of Lemma 1, if pi was higher than p j, no point on bi, j to the right of pi would be visible from pj, contradicting the existence of the event at the intersection of bi,j and T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Further, pi and pj are not at the same height because both are to the left of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Hence, pi is lower than p j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Let t be a point to the right of q that is visible from pi (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' By the Lemma 1 with a = pj, b = pi, c = q and d = t, t is also visible from p j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' By Lemma 1, t is closer to pj than to pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Therefore, t � WT (pi, Pr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' This implies that there is no portion of WT (pi, Pr) to the right of q and, in particular no more event caused by the intersection of a bisector of pi and another viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We charge the event q to pi, and obtain that there are at most m − 1 events of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Hence, kr v ≤ kr c + m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Finally, we derive a bound for kv based on kr v and kℓ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Let us take a continuous portion T ′ of T that belongs to the Voronoi viewshed of some viewpoint pi in VorVis(T , Pr), and to the Voronoi viewshed of some viewpoint pj in VorVis(T , Pℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Let T ′ be maximal with this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Notice that pi is to the left of T ′, while p j is to its right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Furthermore, in VorVis(T , P), every point of T ′ belongs to the Voronoi viewshed of pi or p j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We next show that bi, j intersects T ′ at most once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The claim is clear when y(pi) = y(pj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Otherwise, we assume without loss of generality that pi is lower than p j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Let q be the left-most intersection point (if any) between bi,j and T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We have that q is to the right of pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Additionally, all points of T ′ to the right of q are visible from pi because they belong to the Voronoi viewshed of pi in VorVis(T , Pr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' By Lemma 1, all points of T ′ to the right of q are closer to p j than to pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' In consequence, there is no intersection point between bi, j and T ′ to the right of q, and bi, j intersects T ′ at most once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' This implies that T ′ gets split into at most two portions of the final diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The situation where in at least one of VorVis(T , Pr) or VorVis(T , Pℓ) a portion does not have any visible viewpoint is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Consequently, kv ≤ 2(kr v + kℓ v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Putting everything together, kv ≤ 2(kr v + kℓ v) ≤ 2(kr c + kℓ c + 2m − 2) ≤ 2(kc + 4m − 2) = 2kc + 8m − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Regarding lower bounds, we show the following: Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' There exists a terrain with n vertices and a set of m viewpoints placed on vertices of the terrain such that kv = kc + 2m − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The construction is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 3Strictly speaking, in ColVis(T , Pℓ) pi becomes invisible immediately to its right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Computation of the Voronoi visibility map The algorithm we propose is simple: We sweep the terrain from left to right, and maintain the set of visible points in a balanced binary search tree, where the key of every viewpoint is the Euclidean distance to the point of the terrain currently swept by the sweep line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' the relevant viewpoint is always the closest visible one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The algorithm is based on the observation that maintaining the whole set of viewpoints sorted by distance to T might be expensive (since a bisector of two viewpoints might intersect the terrain Θ(n) times), while, by Lemma 1, maintaining the set of the visible ones is not (since, out of the potential Θ(n) intersections, the two viewpoints are visible in at most two).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Thanks to this observation, new events of VorVis(T , P) are found in O(log m) time rather than O(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We next present the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The algorithm sweeps the terrain from left to right and stops at points that are candidates for event points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The candidates for events of type (i) and (ii) are the events of ColVis(T , P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We explain in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='2 which are the candidates for events of type (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Events of ColVis(T , P) We compute ColVis(T , P) using the version of the algorithm from [12] that returns a doubly-linked list with the vertices of ColVis(T , P) sorted from left to right, together with the visibility information provided as follows: The visible viewpoints are specified for the first component of ColVis(T , P) and, for the other components, the algorithm outputs the changes in the set of visible viewpoints with respect to the component immediately to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Candidates for events of type (iii) We next describe the candidates for events of type (iii) associated with a pair of viewpoints pi, pj ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' If pi and p j are at the same height, the only intersection point of bi,j with T lies between both viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' If such point is visible from both pi and pj, we add it as a candidate for event of type (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Otherwise, we may assume, without loss of generality, that pi is lower than pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' By Lemma 1 the only candidates for events of type (iii) involving pi and pj are the left-most intersection point of type bi, j ∩ T among all such points to the right of pi and the right-most one among all points to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' For the sake of simplicity, we first assume that bi, j is not tangent to T at any of these intersection points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Then each of these intersection points is added to the list of candidates for events swept by the line if and only if it is visible from both pi and pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Finally, let q be one of the two candidates for events of type (iii) involving pi and pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Suppose that q is to the right of pi (the other case is symmetric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' If bi, j is tangent to T at q, points of T infinitesimally to the left or right of q are closer to pi than to pj (while q is equidistant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Additionally, pi becomes invisible right after q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' In consequence, it is not needed to add q to the list of candidates for events of type (iii): Right before q, the algorithm knows that pi is closer to the terrain than p j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' At q, the algorithm processes that pi becomes invisible, and pj (if it is visible) automatically gets higher priority than pi in the list of candidates for the “owner” of the current Voronoi visibility region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The key argument (there are no more candidates for events of type (iii) to the right of q) also holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Data structures The algorithm uses the following data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We maintain a balanced binary search tree H that contains the viewpoints that are visible at the current point of the sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' These viewpoints are sorted in the tree according to their corresponding key, which is the distance from the viewpoint to the current intersection point between the sweep line and the terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The keys are not stored in the tree because they change as the sweep line moves, but each of them can be computed when needed in constant time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The algorithm always chooses as the “owner” of the current Voronoi visibility region the viewpoint of H with the minimum key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' In H, we perform insertions and deletions when viewpoints become visible and invisible, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' During these operations, when at some node of the tree we need to decide whether we move to its left or right subtree, we simply compute the key associated to the viewpoint in that node, and compare it with the key of the viewpoint that we want to insert or delete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Therefore, insertions and deletions can be performed in the standard way in O(log m) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' When the sweep line encounters a candidate for an event of type (iii) (let us call it q), the relative order of two visible viewpoints with respect to their current distance to the terrain changes (formally speaking, it changes right after q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' A possible way to reflect this in H is to delete from the tree one of the two viewpoints associated with q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=" and then 6 ' & $ % Input: T ," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' E Output: VorVis(T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' P) 1: H := ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' tℓ := left-most point of T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' and p∗ := ⊥ 2: while E � ∅ do 3: extract the next element q of E 4: if q is the last element of E then 5: output ((tℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' p∗) 6: break 7: else if some viewpoint v becomes visible at q then 8: insert v in H 9: else if some viewpoint v becomes invisible at q then 10: delete v from H 11: else if q is an intersection point between T and bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' j then 12: update the positions of pi and p j in H 13: end if 14: update pmin 15: if pmin � p∗ then 16: output ((tℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' p∗) 17: tℓ := q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' p∗ := pmin 18: end if 19: end while Figure 5: Computation of VorVis(T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' E is the list of potential events, H is the tree containing the viewpoints that are currently visible, tℓ is the left endpoint of the current portion of T , p∗ is the closest visible viewpoint in that portion, and pmin is the viewpoint in H with the minimum key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' insert it again using as keys the distances from the viewpoints to a point of T infinitesimally to the right of q (and still to the left of the next event in the list).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Thus, candidates for events of type (iii) can be processed in H in O(log m) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Additionally, we use a data structure that allows us to answer ray-shooting queries in T in O(log n) time [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Such queries are used to decide whether a given pair of points are mutually visible, and to find the relevant intersections between T and the bisector of a pair of viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Description of the algorithm Given q, r on T with x(q) < x(r), we denote by T (q, r) and T [q, r] the open and closed portion of the terrain between q and r, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Our algorithm, outlined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 5, takes as input T , P and a list E of potential events sorted from left to right containing all events of ColVis(T , P) together with the O(m2) candidates for events of type (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The list E also contains an event at the right-most point of the terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The algorithm outputs VorVis(T , P) as a list of pairs ((q, r), pi) such that pi is the closest visible viewpoint in T (q, r) (if T (q, r) is not visible from any viewpoint, we output ((q, r), ⊥)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The variables tℓ and p∗ in the algorithm refer to the left endpoint of the portion of T currently analyzed by the algorithm and the closest visible viewpoint in that portion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The variable pmin refers to the viewpoint in H with the minimum key (if H is empty, pmin = ⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Initially, H := ∅, tℓ := left-most point of T , and p∗ := ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We repeat the following procedure until E is empty: We extract the next element q from E, and proceed according to four cases, corresponding to lines 4, 7, 9, and 11 of the pseudocode in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' For the sake of simplicity, in the description in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 5 we deliberately ignore the situation where several events of distinct type occur at the same point of T , which we tackle in the next paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The cases in lines 4, 7 and 9 are clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Regarding the case starting at line 11, in line 12 we update the positions of pi and pj in H as explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='3 (see the paragraph where we discuss the case where the sweep line encounters a candidate for an event of type (iii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We also point out that, if q is an intersection point between T and more than one bisector of type bi,j, the bisectors can be processed in any order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='4 4By our general position assumptions, q is not equidistant from three or more viewpoints, so at most one of the bisectors through q might involve 7 It remains to explain how to deal with the situation where several events of distinct type occur at the same point of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' In this case, we first perform the modifications in H triggered by all the events at that point (insertions of viewpoints becoming visible, deletions of viewpoints becoming invisible and updates of the positions of pairs of viewpoints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' After updating H in this way, we update pmin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' if pmin � p∗, we output ((tℓ, q), p∗), set tℓ := q, and set p∗ := pmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Correctness and running time We first show that the algorithm for VorVis(T , P) always selects the closest visible viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Changes in the visibility status of the viewpoints correspond to events of ColVis(T , P), which are added to E, so the set of visible viewpoints contained in H is correct at any time of the sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Regarding the distances from the viewpoints to the terrain, every time that a viewpoint is swept or becomes visible, it is inserted in H correctly (according to its current distance to the terrain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Changes in the order of the visible viewpoints with respect to their distances to T coincide with intersections of T with the bisectors among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' As argued in the proof of Theorem 1, for every pair of viewpoints it happens at most twice that both viewpoints are visible at an intersection point between T and their bisector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Such an event is precomputed and stored in E, and later processed by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We next analyze the complexity of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The map ColVis(T , P) can be computed in O(n + (m2 + kc) log n) time using the algorithm in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' This map has at most kc regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' however, due to the fact that several viewpoints might become visible or invisible at the same time, when sweeping ColVis(T , P) from left to right, the number of times that a viewpoint becomes visible or invisible, added over all viewpoints, can be higher;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' an upper bound of kc + m2 is given in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Each time that a viewpoint changes its visibility status, we perform an insertion or a deletion in H, which takes O(log m) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The algorithm processes at most m2 intersections between the terrain and bisectors of endpoints in O(log m) time each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Consequently, VorVis(T , P) can be extracted from ColVis(T , P) in O((m2 + kc) log m) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The space complexity of the algorithm is the space required to store the terrain, the events and the data structures, that is, O(n + m2 + kc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We conclude with the following: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The Voronoi visibility map of a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D terrain can be constructed in O(n + (m2 + kc) log n) time and O(n + m2 + kc) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Extensions In this section, we present adaptations of the previous algorithm to compute related maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Higher order Voronoi visibility maps We define the kth-order Voronoi visibility map VorVisk(T , P) as a partitioning of T into regions that have the same set of ℓ closest visible viewpoints, where ℓ is the minimum of k and the number of visible viewpoints in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Observe that the mth-order Voronoi visibility map is equal to ColVis(T , P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We can easily compute VorVisk(T , P) by adapting the algorithm from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' In this case, we need to maintain two additional variables: the total number b of viewpoints that are visible at the point currently swept by the line, and, from the current set of ℓ closest visible viewpoints, the furthest one, denoted pmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Analogously to the algorithm for ColVis(T , P), for space reasons our algorithm for VorVisk(T , P) returns a doubly-linked list with the vertices of VorVisk(T , P) sorted from left to right, together with the following information: The set of ℓ closest visible viewpoints is specified for the first component of VorVisk(T , P) and, for the other components, the algorithm outputs the changes in the set of ℓ closest visible viewpoints with respect to the component immediately to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Let q be the next element from the list of events E, computed as in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We explain in detail the case where one or more viewpoints become visible at q, and leave the remaining cases to the interested reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Let P′ denote the set of viewpoints becoming visible at q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We update b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' If, after this update, b ≤ k, we report vertex q together with the set P′ (containing the new viewpoints in the set of ℓ closest visible viewpoints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We also insert the viewpoints of P′ in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Otherwise, let b′ and b be the number of visible viewpoints right before q and at q, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' If b′ < k, we remove from P′ the set of k − b′ closest viewpoints to q (obtained after sorting the viewpoints of P′ according to p∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 8 their distance to q), we add these viewpoints to a set P′ in, and we insert them in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' After possibly performing this operation in P′, we proceed as follows: We extract the closest viewpoint to q of P′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' if it is closer to q than pmax, we add this viewpoint to P′ in, we insert it in H, we add viewpoint pmax to Pout, and we update pmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Notice that pmax can be updated by finding the predecessor in H of the “old” pmax, that is, in O(log m) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We repeat this process until P′ is empty or the next element in P′ is farther to q than pmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Then we insert the remaining viewpoints of P′ (if any) in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Finally, we report vertex q together with the set P′ in (containing the new viewpoints in the set of ℓ closest visible viewpoints) and the set Pout (containing the viewpoints that stop belonging to the set of ℓ closest visible viewpoints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Clearly, every change in the visibility status of a viewpoint and every intersection of T with the bisector of two visible viewpoints can be processed in O(log m + log n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Hence, we obtain: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The kth-order Voronoi visibility map of a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D terrain can be constructed in O(n + (m2 + kc) log n) time and O(n + m2 + kc) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Other distances Given q, r on T with x(q) < x(r), two other natural distances between q and r are the Euclidean length of the portion T [q, r], which we will call Euclidean distance along the terrain, and the number of vertices in the portion T (q, r), which we will call link distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5 We may define the Voronoi visibility map of T based on these distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The relevant difference with respect to the standard case is the shape of the bisectors between two viewpoints pi and pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' In the case of the Euclidean distance along the terrain, there is exactly one point of T that is equidistant to pi and pj, and this point can be computed in O(log n) time after preprocessing T so that the Euclidean distance along the terrain between any pair of vertices of T can be computed in O(1) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='6 Regarding the link distance, if there is an odd number of vertices between pi and pj, there is exactly one vertex of T that is equidistant to pi and pj, and this vertex can be computed in O(1) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' However, if there is an even number of vertices between pi and p j, there is an open edge of T such that all of its points are at the same link distance from pi and pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' In this case, we must either allow the border between two consecutive Voronoi regions to be 1-dimensional, or, if simplicity is more desirable, we might (artificially) select an interior point of this edge as the intersection point between T and the bisector of pi and p j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' After adding the corresponding candidates for events of type (iii) based on the explanations in the previous paragraph, the rest of the algorithm is equal to the one for the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The running time remains the same because, given a pair of points on T , in both cases the distance between them can be computed in O(1) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Therefore, we conclude: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The Voronoi visibility map of a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='5D terrain with respect to the Euclidean distance along the terrain or to the link distance can be constructed in O(n + (m2 + kc) log n) time and O(n + m2 + kc) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Computation of r∗ We recall that r∗ is the minimum value of r such that, if the viewpoints can only see objects that are within distance r, the visibility map of T does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Let Pr denote the set of viewpoints P with the restriction that the visibility range of the viewpoints is r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' We then may define Vis(T , Pr), VorVis(T , Pr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='in the natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Notice that, for P∞, we obtain the same objects as in the standard case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Let d(x, y) denote the Euclidean distance between two points x, y ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' r∗ = max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=',m{ sup x∈WT (pi,P∞) d(pi, x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 5For the link distance, we take the open portion of the terrain T (q, r) so that any two points on the same edge (including the endpoints) are at (link) distance zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 6If we store, for every vertex q of T , the Euclidean distance along the terrain qd between q and the left-most point of T , then the Euclidean distance along the terrain between vertices q, r of T such that x(q) < x(r) is rd − qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Let pi and x be a viewpoint and a point of T achieving the maximum in the right hand expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' If r∗ < d(pi, x), x would not be visible from pi in Vis(T , Pr∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Since x belongs to the boundary of WT (pi, P∞), all other viewpoints seeing x have a distance to x that is greater than or equal to d(pi, x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' thus, x would also not be visible from any of them in Vis(T , Pr∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Since x is visible in Vis(T , P∞)7, we reach a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Therefore, r∗ ≥ d(pi, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' On the other hand, to keep Vis(T , P∞) unchanged, it is enough to maintain the closure of WT (pi, P∞) visible for all i, since Vis(T , P∞) is equal to the union of the closures of the regions WT (pi, P∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' If we set a visibility range of sup x∈WT (pi,P∞) d(pi, x), the closure of WT (pi, P∞) indeed remains visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Consequently, r∗ ≤ max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=',m{ sup x∈WT (pi,P∞) d(pi, x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Using this characterization of r∗, we can prove the following: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' The problem of computing the minimum value r∗ such that Vis(T , Pr∗) = Vis(T , P∞) can be solved in O(n + (m2 + kc) log n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' By Lemma 3, it suffices to consider the distances between the vertices of VorVis(T , P∞) (that is, the points on the boundary of the Voronoi viewsheds) and their associated viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Consequently, the problem can be trivially solved in linear time if VorVis(T , P∞) is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Final remark As indicated in [12], in the running time of the algorithm to compute ColVis(T , P), the term m2 log n disappears if we assume that no two viewpoints change from invisible to visible at the same point of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' This can always be achieved by infinitesimally perturbing the terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' However, such a perturbation does not make the same term disappear from the running time of the presented algorithm to compute VorVis(T , P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfYgwr/content/2301.05049v1.pdf'} +page_content=' Given that one of the bounds in Theorem 1 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0000000000000000000000000000000000000000..7feac43de48432cf587c8338738c0b54cc3642a3 --- /dev/null +++ b/ytE0T4oBgHgl3EQf-wLX/content/tmp_files/2301.02819v1.pdf.txt @@ -0,0 +1,1598 @@ +EXCELFORMER: A Neural Network Surpassing GBDTs on Tabular Data +Jintai Chen * 1 Jiahuan Yan * 1 Danny Z. Chen 2 Jian Wu 3 +Abstract +Though neural networks have achieved enormous +breakthroughs on various fields (e.g., computer +vision) in supervised learning, they still trailed +the performances of GBDTs on tabular data thus +far. Delving into this issue, we identify that a +proper handling of feature interactions and feature +embedding is crucial to the success of neural net- +works on tabular data. We develop a novel neural +network called EXCELFORMER, which alternates +in turn two attention modules that respectively ma- +nipulate careful feature interactions and feature +embedding updates. A bespoke training method- +ology is jointly introduced to facilitate the model +performances. By initializing parameters with +minuscule values, these attention modules are at- +tenuated when the training begins, and the effects +of feature interactions and embedding updates +progressively grow up to optimum levels under +the guidance of the proposed specific regulariza- +tion approaches SWAP-MIX and HIDDEN-MIX as +the training proceeds. Experiments on 25 public +tabular datasets show that our EXCELFORMER is +superior to extremely-tuned GBDTs, which is an +unprecedented achievement of neural networks in +supervised tabular learning. +1. Introduction +Neural networks have been firmly established as state of the +art approaches in various fields such as computer vision (Sri- +vastava et al., 2015; Khan et al., 2022), natural language pro- +cessing (Hochreiter & Schmidhuber, 1997; Vaswani et al., +2017), and automatic speech recognition (Dong et al., 2018). +However, on tabular data, one of the most ubiquitous data +formats, neural networks cannot yet achieve comparable per- +*Equal contribution 1College of Computer Science and Tech- +nology, Zhejiang University, Hangzhou, China 2Department of +Computer Science and Engineering, University of Notre Dame, +Notre Dame, IN 46556, USA 3The First Affiliated Hospital, and +Department of Public Health, Zhejiang University School of +Medicine, Hangzhou, China. Correspondence to: Jian Wu . +formances against classical gradient boosting decision trees +(GBDTs) (Chen & Guestrin, 2016; Prokhorenkova et al., +2018; Duan et al., 2020) in supervised learning despite nu- +merous efforts (Borisov et al., 2021), which hinders the +widespread adoption of neural networks and the progression +towards general artificial intelligence. +The investigation in (Grinsztajn et al., 2022) pointed out +three inherent characteristics of tabular data that impeded +known neural networks from top-tier performances, in- +cluding irregular patterns of the target function, the +negative effects of uninformative features, and the non- +rotationally-invariant features. Based on this, we further- +more identify two points that highly promote the capabil- +ities of neural networks on tabular data. (i) An appropri- +ate feature embedding approach. Though it was demon- +strated (Rahaman et al., 2019; Grinsztajn et al., 2022) that +neural networks are likely to predict overly smooth solutions +on tabular data, a deep learning model was also observed +to be capable of memorizing random labels (Zhang et al., +2021). Since the target function patterns are irregular and +spurious correlations between the targets and features exist, +an appropriate feature embedding network should well fit +the irregular patterns while maintaining generalizability. (ii) +A careful feature interaction approach. Since features of +tabular data are non-rotationally-variant and a considerable +portion of them are uninformative, it harms the generaliza- +tion when a model incorporates needless feature interactions. +However, theoretical analysis (Ng, 2004) suggested current +neural networks are naturally ineffective in dealing with +data that have very limited relevant features, taking a cost +of high worst-case sample complexity. +Some previous approaches either designed feature embed- +ding approaches (Gorishniy et al., 2022) to alleviate overly +smooth solutions inspired by (Tancik et al., 2020) or em- +ployed regularization (Katzir et al., 2020) and shallow mod- +els (Cheng et al., 2016) to promote the model generalization, +while some neural networks were equipped with sophisti- +cated feature interaction approaches (Yan et al., 2023; Chen +et al., 2022; Gorishniy et al., 2021) for better selectively +feature interactions. Although these tailored designs gained +the performances on supervised tabular data tasks, which +still underperform GBDT approaches (e.g., XGboost) on a +diverse array of datasets (Borisov et al., 2021). +arXiv:2301.02819v1 [cs.LG] 7 Jan 2023 + +EXCELFORMER: A Neural Network Surpassing GBDTs on Tabular Data +This paper pushes the envelop forward: we develop a neu- +ral network that, for the first time, surpasses the GBDTs +on a wide range of public tabular datasets. This result is +achieved based on the cooperative effects of a new tabular +data tailored architecture EXCELFORMER and a bespoke +training methodology, which jointly learn appropriate fea- +ture embedding and careful feature interactions. For better +feature embedding, we propose an attention module, called +attentive intra-feature update module (AiuM), that is more +powerful than previous non-attentive embedding update ap- +proaches (e.g., linear or non-linear projection networks). +For feature interactions, we present a conservative approach +conducted by a novel module called directed inter-feature +attention module (DiaM), which avoids compromising the +semantics of critical features by only allowing features of +lower importance to fetch the information from those of +higher importance. Our EXCELFORMER is mainly built by +stacking these two kinds of modules in turns. +Since the major ingredients AiuM and DiaM are both flexi- +ble attention based modules, our training methodology aims +to an adversarial objective that prevents our model from +converging to an overcomplicated embedding functions that +overfit to irregular target functions and from introducing +useless feature interactions that hurt the generalization. At +the start of training, a novel initialization approach assigns +minuscule values to the weights of DiaM and AiuM, so as +to attenuate the intra-feature embedding updates and inter- +feature interactions. During training, the effects of DiaM +and AiuM then progressively grow to optimum levels under +the guidance of proposed regularization approaches SWAP- +MIX and HIDDEN-MIX. The newly proposed HIDDEN- +MIX and SWAP-MIX are two variants of Mixup (Zhang +et al., 2018) specific to tabular data, which fix the disadvan- +tages of the original Mixup approach (as we will discuss in +Sec. 4) and respectively prioritize to promote the learning +of feature embedding and feature interactions. +The main contributions are summarized as follows. +• We present the first neural network that performs su- +perior than GBDTs on 25 public tabular datasets, with +the assistance of a novel training methodology. +• We summarize two crucially-required capabilities of +neural networks to succeed in dealing with tabular data, +which will inspire the future researches. +• We propose two tabular-data-specific Mixup variants, +HIDDEN-MIX and SWAP-MIX, which outperform the +original Mixup approach on tabular data. +2. Related Work +Supervised Tabular Learning. +Since neural networks +were demonstrated efficient on various data types such as +images (Khan et al., 2022), plentiful efforts have been made +to harness the powers of neural networks for tabular data. +However, GBDT approaches (e.g., XGboost) remained the +go-to choice (Katzir et al., 2020) in various supervised tab- +ular tasks (Borisov et al., 2021; Grinsztajn et al., 2022) +to date, due to its superior performances on diverse tab- +ular datasets. To achieve GBDT-level results, recent re- +searches focused on presenting sophisticated neural mod- +ules for heterogeneous feature interactions (Gorishniy et al., +2021; Chen et al., 2022; Yan et al., 2023), mimicking tree- +like approaches (Katzir et al., 2020; Popov et al., 2019; +Arik & Pfister, 2021) to seek decision paths, or resorting +to conventional approaches (Cheng et al., 2016; Guo et al., +2017). Apart from model designs, various data represen- +tation approaches, such as feature embedding (Gorishniy +et al., 2022), discretization of continuous features (Guo et al., +2021; Wang et al., 2020), and rule search approaches (Wang +et al., 2021), were proposed against the irregular target pat- +terns (Tancik et al., 2020; Grinsztajn et al., 2022). These +attempts showcased the potentials of neural networks, but +still attained inferior performances in comparison to GB- +DTs on a wide range of tabular datasets. Investigations +in (Grinsztajn et al., 2022) summarized several challenges +for neural networks on tabular data, however, no solution +was given and these challenges were still open. Besides, +there were some attempts (Wang & Sun, 2022; Arik & Pfis- +ter, 2021; Yoon et al., 2020) to apply self-supervision on +tabular datasets. However, these approaches are dataset- or +domain- specific that is hardly to be widely adopted, due to +the heterogeneity of tabular datasets. +Mixup and its Variants. +The original Mixup (Zhang +et al., 2018) generated a new data by convex interpola- +tions of two given data, which was proved beneficial on +various image datasets (Tajbakhsh et al., 2020; Touvron +et al., 2021a) and some tabular datasets. However, we found +that the original Mixup is conflict with the irregular target +patterns (as we will discuss in Sec. 4) and hardly cooper- +ates with the cutting-edge models (Gorishniy et al., 2021; +Somepalli et al., 2021). ManifoldMix (Verma et al., 2019) +and Flow-Mixup (Chen et al., 2020) applied the convex +interpolations on the hidden states, which did not funda- +mentally alter the way to synthesize new data and exhib- +ited similar characteristics as the original Mixup. Then, +the follow-up variants CutMix (Yun et al., 2019), Atten- +tiveMix (Walawalkar et al., 2020), SaliencyMix (Uddin +et al., 2020), ResizeMix (Qin et al., 2020), PuzzleMix (Kim +et al., 2020) spliced two images spatially, which defended +the local patterns of images but are not directly available to +tabular data. Darabi et al (Darabi et al., 2021) and Gowthami +et al (Somepalli et al., 2021) used Mixup and CutMix-like +approach in tabular data pre-training, however, we did not at- +tain performance gains when we trained these models under +the guidance of these Mixup approaches in the supervised +learning fashion. + +EXCELFORMER: A Neural Network Surpassing GBDTs on Tabular Data +𝒙𝟏 +𝒙𝟐 +𝒙𝟑 +𝒙𝟒 +0.3 +1.54 +Yes +0.43 +Importance +Input table with sorted features +×L +DiaM +Linear +Linear +Linear +Q +K +V +Element-wise product +Element-wise add +Matrix product +Tanh +𝒙 ∈ ℝ𝒇 +𝒛 ∈ ℝ𝒇×𝒅 +𝒛′ ∈ ℝ𝒇×𝒅 +AiuM +Linear +Linear +Norm +Norm +Embedding Layer +Prediction Head +Figure 1. The demonstration of the proposed EXCELFORMER. The AiuM and DiaM are abbreviations for attentive intra-feature update +module and directed inter-feature attention module, respectively. “Norm” indicates the LayerNorm layer (Ba et al., 2016). Before being +fed into the model, the features are sorted according to a feature importance metric (e.g., mutual information). +3. EXCELFORMER +3.1. The Overall Architecture +Fig. 1 illustrates our proposed EXCELFORMER. Our EX- +CELFORMER is mainly built based on two simple ingredi- +ents — the attentive intra-feature update module (AiuM) +and the directed inter-feature attention module (DiaM) — +which respectively conduct the feature embedding update +and feature interactions. In processing, f features of an +input data x ∈ Rf are first tokenized by a neural embedding +layer into representations of size d, denoted as z(0) ∈ Rf×d. +It is then successively processed by L DiaMs and L AiuMs +alternately. Both of these two modules were with a Layer- +Norm ahead, and are accompanied with additive shortcut +connections as illustrated in Fig. 1. Finally, the probability +vector of C categories p ∈ RC for classification or a scale +value p ∈ R1 for regression is yielded by a prediction head. +3.2. Attentive Intra-feature Update Module +The investigations (Grinsztajn et al., 2022) highlighted the +seeming contradiction between the irregularity of target +functions and the over-smooth solutions obtained by neural +networks. In previous Transformer-like models (Yan et al., +2023; Gorishniy et al., 2021), the commonly-used position- +wise feed-forward network (FFN) (Vaswani et al., 2017) +is employed for feature embedding update. However, we +empirically discovered that the FFN containing two linear +projections and a ReLU activation is not flexible enough to +fit the irregular target functions, and we design an attention +approach to handle the intra-feature embedding updates, by: +z′ = tanh (zW (l) +1 ++ b(l) +1 ) ⊙ (zW (l) +2 ++ b(l) +2 ) +(1) +in which W (l) +1 +∈ Rd×d, W (l) +2 +∈ Rd×d, b(l) +1 +∈ Rd, and +b(l) +2 +∈ Rd are all learnable parameters for the l-th layer, and +⊙ means element-wise product. The z and z′ denote the +input and output representations. Our experiment showed +that Eq.(1) is more powerful than FFN at the same com- +putational costs. Notably, the operations in Eq.(1) do not +conduct any feature interactions. +3.3. Directed Inter-feature Attention Module +It was pointed out (Ng, 2004) that neural networks are in- +herently inefficient to organize the feature interactions, but +previous works empirically demonstrated the benefits of +feature interactions (Chen et al., 2022; Cheng et al., 2016). +Thus, we present a conservative approach for feature inter- +actions that only allows the lower target-relevant feature to +get access to the information of the higher target-relevant +features. Before feeding features into EXCELFORMER, we +sort them in a descending order according to the feature +importance (mutual information) w.r.t. the targets on the +training set. For feature interactions, our approach is con- +ducted by computing self-attention with a mask M, which +sets the upper triangle part of the attention map to be zeros. +Formally, the attention is computed by: +z′ = σ((zWq)(zWk)T ⊙ M/ +√ +d)(zWv) +(2) +where z and z′ are respectively the input and output repre- +sentations, Wq, Wk, Wv ∈ Rd×d are all learnable matrices, +and σ is the “softmax” operating along the last dimension. +The mask M ∈ Rf×f is not optimizable, whose elements in +the upper triangle part are set negative infinity (using −105 +as default) and rest elements (including those on the main +diagonal) are equal to 1. In practice, Eq.(2) is extended into +a multi-head attention version, with 32 heads by default. +Remarks. By our DiaM, a feature is updated by features +of higher importance, but not vice versa. It remains the +interactions of any two features while protecting important +features to a large extent if some interactions performed +by the model are inappropriate. Our DiaM looks similar +to some self-attention mechanisms (Radford et al., 2018), +however, a distinguishing feature of our approach is that it +is only plausible when the features have been sorted in the +descending order according to the feature importance (e.g., + +EXCELFORMER: A Neural Network Surpassing GBDTs on Tabular Data +: sample 𝑥! +: sample 𝑥" +Figure 2. Decision boundaries of k-Nearest Neighbor (kNN, k = +8) for the 2 most important features (by mutual information) of a +zoomed-in part of Higgs dataset. The convex combinations (points +on the black line) of two samples x1 and x2 of two different cate- +gories is likely to be in conflict with the irregular target function. +mutual information used in this paper). +3.4. Embedding Layer +Our embedding layer is also an attention based module that +is similar to the AiuM. In Eq.(1), the parameters W1, W2, b1 +and b2 are shared among features, while in the embedding +layer, the parameters are not shared among features, as: +z(0) = tanh (x ⊙ W (0) +1 ++ b(0) +1 ) ⊙ (x ⊙ W (0) +2 ++ b(0) +2 ) (3) +in which the input features x ∈ Rf, the learnable parameters +W (0) +1 +, W (0) +2 +∈ Rf×d and b(0) +1 , b(0) +2 +∈ Rf×d, and ⊙ means +element-wise product. Before being fed into the embedding +layer, the numeric features are normalized and the categor- +ical features are transformed into numeric features by the +Sklearn python package 1. +3.5. Prediction Head +In our EXCELFORMER, we do not use a class token for +target function prediction, since collecting information into +a class token depends on the feature interaction capability of +the neural network. Our prediction head contains two linear +projection layers to separately compress the information +along feature dimension and embedding dimension, by: +p = φ(Wd(PReLU((z(L))T Wf + bf))T + bd) +(4) +where z(L) denotes the output of the top-most AiuM, Wf ∈ +Rf×C and bf ∈ RC (C indicates the target category count in +classification (C > 2) and C = 1 for regression and binary +classification) compress the features, while Wd ∈ Rd×1 and +bd ∈ R1 jointly compress the embedding size d into 1. φ is +sigmoid when C = 1, while φ denotes softmax for C > 2. +features +emb. +sample 𝑥! +synthesized 𝑥! +by SWAP-MIX +synthesized 𝑥! +by HIDDEN-MIX +sample 𝑥" +Figure 3. An illustration of HIDDEN-MIX and SWAP-MIX opera- +tions to input data. “emb.” means “embedding”. +4. Training Methodology +The proposed AiuM and DiaM satisfy (i) and (ii) mentioned +in Sec. 1 respectively, while we argue that their effectiveness +would be improved by using tailored training methodology +since vanilla neural network training strategy was consid- +ered to be inefficient on tabular data (Ng, 2004; Rahaman +et al., 2019). Mixup (Zhang et al., 2018) is one of the most +effective regularization approaches for neural networks, but +our tests showed that it cannot well cooperate with some +the cutting-edge approaches like (Gorishniy et al., 2021; +Somepalli et al., 2021). Besides, such element-wise convex +interpolation operation is intuitively in conflict with the ir- +regular target function of tabular datasets. Fig. 2 showcases +an example of the irregular target function of tabular data, +and it is obvious that the data synthesized by original Mixup +(e.g., convex combination) conflicts with the target function. +To fix this, this paper introduces two Mixup approaches, +HIDDEN-MIX and SWAP-MIX, for tabular data, which can +enhance the model performances and avoid the conflicts +showed in Fig. 2. Besides, we also propose an attenuated +initialization approach for these two modules. For better +understanding, we introduce HIDDEN-MIX and SWAP-MIX +first, and then the attenuated initialization approach. +4.1. HIDDEN-MIX +The proposed HIDDEN-MIX is applied on the representa- +tions after the embedding layer and the labels. It exchanges +some embedding elements of two samples (see Fig. 3), by: +� +z(0) +m += S ⊙ z(0) +1 ++ (1 − S) ⊙ z(0) +2 , +ym = λy1 + (1 − λ)y2 +(5) +where z(0) +1 , z(0) +2 , z(0) +m +∈ Rf×d denote the feature embed- +ding of the two samples and the synthesized sample, while +y1, y2, ym are the labels of the two samples and synthesized +sample. The coefficient matrix S and the all-one matrix +1 are of size f × d. S = [s1, s2, . . . , sf]T , where all the +vector sk ∈ Rd (k = 1, 2, . . . , d) are identical and with +1https://scikit-learn.org/stable/modules/ +classes.html#module-sklearn.preprocessing + +EXCELFORMER: A Neural Network Surpassing GBDTs on Tabular Data +⌊λ×d⌋ randomly selected elements equalling to 1 while the +rest equalling to 0. Similar to previous work (Zhang et al., +2018), the scalar coefficient λ ∼ Beta(α, α). +Interpretation. +The proposed HIDDEN-MIX encourages +to learn linear feature embedding solutions. Consider a sim- +ple situation that there are two data x1 and x2 with f = 1, +d = 2 and λ = 0.5. Since ym = 1 +2(y1 + y2), we can in- +fer the constraint to the optimizable neural network g that +g(x(0,0) +1 +, x(0,1) +2 +) = g(x(0,0) +2 +, x(0,1) +1 +) = 1 +2g(x(0,0) +1 +, x(0,1) +1 +) + +1 +2g(x(0,0) +2 +, x(0,1) +2 +), in which the subscript (i, j) indicates the +j-th element of the i-th feature embedding. For a simple +neural network g(x(0,0), x(0,1)) = w1x(0,0) + w2x(0,1) + +w3x(0,0)x(0,1), it is obvious that HIDDEN-MIX requires +w3x(0,0) +1 +≡ w3x(0,0) +2 +for any x1 and x2, and thus w3 = 0. +In our EXCELFORMER, our AiuM and embedding layer are +implemented with flexible attention operations for fitting +the irregular target functions, while our HIDDEN-MIX pri- +oritizes to learn a linear embedding approach for a feature +and avoids over-fitting. +4.2. SWAP-MIX +See Fig. 3, unlike HIDDEN-MIX acting on the embedding +dimension, SWAP-MIX swaps parts of features between two +samples (x1, y1) and (x2, y2), by: +� +xm = ⊙x1 + (1 − S) ⊙ x2, +ym = ΛSy1 + (1 − ΛS)y2 +(6) +where S = [s1, s2, . . . , sd] and the all-one matrix 1 are of +size f × d, and all sk (k = 1, 2, . . . , d) are identical and +contain ⌊λ×f⌋ elements equalling to 1 which are randomly +located (λ ∼ Beta(α, α)). The rest elements in sk equal to +0. The label mixing coefficient ΛS is the normalized sum +of the mutual information of the features that are selected +by S, which is computed by: +ΛS = +� +s(i) +k =1 MI(i) +�f +i=1 MI(i) +(7) +where s(i) +k indicates the i-th element of the vector sk (denot- +ing whether the i-th feature is used for swap), and MI(i) is +the mutual information of the i-th feature. +Interpretation. +Consider two tabular data denoted by x1 +and x2 (with f = 2, d = 1, λ = 0.5, and MI(1) = MI(2)) +are processed by SWAP-MIX. One can easily infer that +g(x(0,0) +1 +, x(1,0) +2 +) = g(x(0,0) +2 +, x(1,0) +1 +) = 1 +2g(x(0,0) +1 +, x(1,0) +1 +) + +1 +2g(x(0,0) +2 +, x(1,0) +2 +). For a neural network g(x(0,0), x(1,0)) = +w1x(0,0) + w2x(1,0) + w3x(0,0)x(1,0), it is suggested that +w3 is likely to be 0 and SWAP-MIX is disposed to make g +learn a non-feature-interaction function. It encourages the +DiaM to solely include necessary interactions, dispensing +with useless ones. +4.3. An Attenuated Initialization +The function of the attenuated initialization approach is to +reduce the effects of DiaM and AiuM when the model train- +ing begins. Our attenuated initialization approach is built +upon the commonly used He’s initialization approach (He +et al., 2015) and the Xavier initialization approach (Glorot +& Bengio, 2010), by rescaling the weight variance with γ +(< 1) while keeping the expectation at 0: +Var(w) = γVarprev.(w) +(8) +where Varprev.(w) denotes the weight variance used in previ- +ous work (He et al., 2015; Glorot & Bengio, 2010). In this +paper, we set γ = 10−4. To reduce the impacts of AiuM and +DiaM, we can either apply Eq.(8) on all the parameters in +these modules or in part of them, and we empirically witness +these options all perform similarly. We apply Eq.(8) on all +the parameters in AiuM and in DiaM as default, and thus +these two modules have almost no effects before training. +Interpretation. +As demonstrated in the interpretations +for HIDDEN-MIX and SWAP-MIX, these Mixup approaches +encourage a neural network to learn linear feature embed- +ding functions and non-feature-interaction solutions by re- +quiring the coefficient interaction terms w3 to be 0. To +cooperate with these two approaches, our initialization ap- +proach suppresses the intra-feature embedding updates and +inter-feature interactions when the training begins, and the +effects of necessary non-linear (i.e., attentive) feature em- +bedding and crucial feature interactions can be progressively +added under the driving force of data. +On the other hand, for a module with additive identity short- +cut like y = F(x) + x, our initialization approach atten- +uates the sub-network F(x) and satisfies the property of +dynamical isometry (Saxe et al., 2014) for better trainability. +Some previous literature (Bachlechner et al., 2021; Tou- +vron et al., 2021b) suggested to rescale the F(x) path by +y = ηF(x) + x, in which η is a learnable scalar initial- +ized by 0 or a learnable diagonal matrix whose elements +equal to very small values. Different from these works, our +attenuated initialization approach directly gives minuscule +values to the model weights in initialization, which is more +flexible and allows every feature to learn adaptive degrees +of interactions and embedding updates. +4.4. Model Training and Loss Functions +Our EXCELFORMER can handle both of the classification +and regression tasks on tabular datasets. In training, the +two proposed Mixup approaches can be applied succes- +sively to the data by HIDDEN-MIX(SWAP-MIX(x, y)) or + +EXCELFORMER: A Neural Network Surpassing GBDTs on Tabular Data +SWAP-MIX(HIDDEN-MIX(x, y)). However, our test sug- +gests that the performance of EXCELFORMER on a certain +dataset is fixedly better with only SWAP-MIX or HIDDEN- +MIX. Thus, we only use one Mixup approach in dealing +with a certain tabular dataset. The cross-entropy loss func- +tion is used for classification and the mean square error loss +function is used for regression tasks. +5. Experiments +In this section, we evaluate the effects of the proposed EX- +CELFORMER on 25 public tabular datasets and compare +to extremely tuned GBDT approaches, XGboost and Cat- +boost. We report the performances of our proposed EX- +CELFORMER under several configurations: (a) all the hyper- +parameters fixed, (b) the settings for HIDDEN-MIX and +SWAP-MIX are tuned, and (c) all the hyper-parameters are +tuned. Besides, we conduct ablation studies on a part of +datasets to inspect the effects of proposed architecture de- +signs (including the proposed DiaM, AiuM) and training +methodology (including HIDDEN-MIX, SWAP-MIX, and +the attenuated initialization approach). +5.1. Experimental Setup +Datasets. For fair and comprehensive comparisons, we +involve 25 public tabular datasets in our experiments, in- +cluding large-, middle-, or small- scale ones with numerical +or categorical features, and for regression, binary classifica- +tion, or multi-class classification tasks. The detailed dataset +descriptions are listed in Appendix A. +Implementation Details. The codes of EXCELFORMER +and the training methodology are implemented by us- +ing PyTorch on Python 3.8. All the experiments of EX- +CELFORMER were run on NVIDIA GTX 3090. We set +the numbers of DiaM and AiuM layers L = 3, the fea- +ture embedding size d = 256, and the dropout rate for +attention map is set 0.3. The optimizer for our approach is +AdamW (Loshchilov & Hutter, 2018) with default settings. +We use He’s initialization approach (He et al., 2015) and +our attenuated initialization approach. In hyper-parameter +tuning, the Optuna library (Akiba et al., 2019) is used for +all the approaches. Following (Gorishniy et al., 2021), we +randomly select 80% data as the training samples and take +the rest as the test samples. In training, we utilize 20% of +all the training samples for validation. For EXCELFORMER +with fixed hyper-parameters, the learning rate is set 10−4 +without weight decay, and α for Beta distributions is set 0.5. +As for the tuned versions (Mixup tuned or fully tuned), the +hyper-parameter tuning settings are listed in Appendix B. +Compared Models. We compare our EXCELFORMER with +the popular GBDT approaches XGboost (Chen & Guestrin, +2016) and Catboost (Prokhorenkova et al., 2018). The imple- +mentations of XGboost and Catboost mainly follow (Gor- +ishniy et al., 2021). More stringent than from previous +work (Gorishniy et al., 2021), for XGboost and Catboost, +we increase the number of estimators / iterations (i.e., de- +cision trees) from 2000 to 4096 and increase the tuning +iterations from 100 to 500 for better performances. +Evaluation. For each fixed or tuned configurations, we +run the codes by 5 times with different random seeds and +reported the average performance on the test set. For our +proposed EXCELFORMER, we do not use any ensemble +strategy. For binary classification (binclass) tasks, we com- +pute the area under the ROC Curve (AUC) for evaluation. +We use accuracy (ACC) for multi-class classification (multi- +class) tasks and while we use the inverse root mean square +error (iRMSE) for regression tasks. On all of these metrics, +the higher the obtained values, the better. +5.2. Performances of Untuned EXCELFORMER +Performances on all the 25 datasets are reported in Table 1. +Both of EXCELFORMER-SWAP-MIX and EXCELFORMER- +HIDDEN-MIX are trained with pre-given hyper-parameters, +dispensing with any hyper-parameter tuning. The aver- +age performance rank of EXCELFORMER-SWAP-MIX is +4.68, that is very close to Catboost 4.24. +The average +rank of EXCELFORMER-HIDDEN-MIX is 3.76, that falls +between that of Catboost (4.24) and XGboost (3.36). In pair- +wise comparison, EXCELFORMER-SWAP-MIX beats the +extremely tuned XGboost on 9 out of 25 datasets, while EX- +CELFORMER-HIDDEN-MIX beats XGboost on 12 out of 25 +datasets. In comparison with extremely tuned Catboost, EX- +CELFORMER-SWAP-MIX obtains better performances on +9 out of 25 datasets while EXCELFORMER-HIDDEN-MIX +obtains better performances on 13 out of 25 datasets. These +findings suggest that, by directly using EXCELFORMER +with the default hyper-parameters, one can easily obtain +GBDT-level performances on tabular datasets. Due to the +diversity of tabular datasets, a well-performed foolproof +approach without tuning is very user-friendly and has great +potential for practical applications, since most users are not +proficient in conducting hyper-parameter tuning. +5.3. Performances of Tuned EXCELFORMER +By tuning the configurations on Mixup approaches (Mixup +types and α of Beta distributions), the performance rank +of EXCELFORMER, 2.48, is significantly superior than XG- +boost (3.36) and Catboost (4.24). In direct comparison to +extremely tuned XGboost, EXCELFORMER with Mixup tun- +ing is superior on 16 out of 25 datasets, while it is superior +on 21 out of 25 datasets when comparing with extremely +tuned Catboost. Such results suggest that, a user can easily +obtain obviously better performances than extremely tuned +XGboost and Catboost by only tuning two hyper-parameters + +EXCELFORMER: A Neural Network Surpassing GBDTs on Tabular Data +Table 1. Performance Comparison with extremely tuned XGboost and Catboost. The performances of EXCELFORMERs outperform- +ing both of XGboost and Catboost are bold. The performances of XGboost and Catboost are bold if they are the best. +Datasets +AN +CP +VI +YP +GE +CH +SU +BA +BR +XGboost +-0.1076 +-2.1370 +-0.1140 +-0.0275 +68.75 +85.66 +-0.0177 +88.97 +-0.0769 +Catboost +-0.0929 +-2.5160 +-0.1181 +-0.0275 +66.54 +86.62 +-0.0220 +89.16 +-0.0931 +EXCELFORMER-SWAP-MIX +-0.0782 +-2.6590 +-1.6220 +-0.0276 +70.38 +85.89 +-0.0184 +89.00 +-0.1123 +EXCELFORMER-HIDDEN-MIX +-0.0786 +-2.2320 +-0.2440 +-0.0276 +70.72 +85.89 +-0.0174 +88.65 +-0.0696 +EXCELFORMER (Mixup Tuned) +-0.0876 +-2.2020 +-0.1070 +-0.0275 +68.36 +85.80 +-0.0173 +89.21 +-0.0627 +EXCELFORMER (Fully Tuned) +-0.0778 +-2.1980 +-0.0899 +-0.0275 +68.94 +85.89 +-0.0161 +89.16 +-0.0641 +Datasets (Continued) +EY +MA +AI +PO +BP +CR +CA +HS +HO +XGboost +72.88 +93.69 +-0.0001605 +-4.331 +99.96 +85.11 +-0.4359 +-0.1707 +-3.139 +Catboost +72.41 +93.66 +-0.0001616 +-4.622 +99.95 +85.12 +-0.4359 +-0.1746 +-3.279 +EXCELFORMER-SWAP-MIX +71.44 +93.38 +-0.0001689 +-5.694 +99.94 +85.23 +-0.4331 +-0.1835 +-3.305 +EXCELFORMER-HIDDEN-MIX +72.09 +93.66 +-0.0001627 +-2.862 +99.95 +85.22 +-0.4587 +-0.1773 +-3.147 +EXCELFORMER (Mixup Tuned) +74.14 +94.04 +-0.0001615 +-2.629 +99.93 +85.26 +-0.4316 +-0.1726 +-3.159 +EXCELFORMER (Fully Tuned) +78.94 +94.11 +-0.0001612 +-2.636 +99.96 +85.36 +-0.4336 +-0.1727 +-3.214 +Datasets (Continued) +DI +HE +JA +HI +RO +ME +SG +rank +XGboost +-0.2353 +37.39 +72.45 +80.28 +90.48 +-0.0820 +-0.01635 +3.36 +Catboost +-0.2362 +37.81 +71.97 +80.22 +89.55 +-0.0829 +-0.02038 +4.24 +EXCELFORMER-SWAP-MIX +-0.2368 +37.22 +72.51 +80.60 +88.65 +-0.0821 +-0.01587 +4.68 +EXCELFORMER-HIDDEN-MIX +-0.2387 +38.20 +72.79 +80.75 +88.15 +-0.0808 +-0.01531 +3.76 +EXCELFORMER (Mixup Tuned) +-0.2359 +38.65 +73.15 +80.88 +89.33 +-0.0809 +-0.01465 +2.48 +EXCELFORMER (Fully Tuned) +-0.2358 +38.61 +73.55 +81.22 +89.27 +-0.0808 +-0.01454 +1.84 +of Mixup configurations, dispensing with tuning the config- +urations about the model architecture. +Furthermore, one can obtain better performances by tuning +all the configurations listed in Table 3. In this way, EX- +CELFORMER can obtain a better performance rank at 1.84, +outperforming the extremely tuned XGboost and Catboost +on 17 and 20 datasets out of 25 datasets, respectively. +Observe on the performance ranks over 25 datasets, one can +learn that the fully tuned EXCELFORMER performs better +than the Mixup tuned model, which is much better than EX- +CELFORMER with fixed hyper-parameters. From the aspect +of practice, EXCELFORMER with fixed hyper-parameters is +sufficient to get results on par with extremely tuned XGboost +and Catboost, while the tuned versions can obtain remark- +ably better results. In practice, we suggest that one can +use our EXCELFORMER in steps: (1) try EXCELFORMER +with fixed hyper-parameters first and it can meet the needs +in most situations, (2) try to tune the hyper-parameters of +Mixup if the fixed hyper-parameter version is not satisfac- +tory, and (3) finally tune the other hyper-parameters if better +performances are desired. +5.4. Ablation Study +In this section, we inspect the effects of the proposed in- +gredients empirically on 6 tabular datasets, and we find the +conclusions on the other datasets are similar. We take the +best performed model in EXCELFORMER-SWAP-MIX and +EXCELFORMER-HIDDEN-MIX (without hyper-parameter +tuning) as the baseline, and either remove or replace one +ingredient each time for comparison. We report the per- +formances of EXCELFORMER that (1) He’s initialization +is used to replace the proposed attenuated initialization ap- +proach for AiuM and DiaM, (2) a vanilla self-attention mod- +ule (vanilla SA) is used to replace DiaM for heterogeneous +feature interactions, (3) the linear feed forward network +(FFN) is used to replace AiuM for feature embedding up- +dates, (4) both of SWAP-MIX and HIDDEN-MIX are not +used, (5) the input Mixup (Zhang et al., 2018) (α = 0.5) is +used to replace our proposed Mixup approach. See Fig. 4, +the performances often decrease when an ingredient is re- +moved or replaced, suggesting that all of the ingredients +are beneficial in general. But it is also witnessed that the +compared models perform better than the baselines on 1 +or 2 datasets out of 6 datasets, showing that an ingredient + +EXCELFORMER: A Neural Network Surpassing GBDTs on Tabular Data +CP +GE +CA +HS +HO +HE +Figure 4. Ablation studies on the proposed ingredients. “-” indicates removal and “+” indicates addition. The bars colored by +“purple” indicates worse performances compared to the baseline, while bars in “orange” indicates better performances. On the +metric “Accuracy”, the higher values a model achieve the better. On the metric “RMSE”, the lower values a model achieve the +better. +may have negative impacts on some datasets. In the model +development, we remain all of these designs since they show +positive impacts on most of the datasets. Notably, it is dif- +ficult to find a design that is always effective, since tabular +data are of high diversity and our goal is to present a neural +network that can accommodate as many cases as possible. +Compared the baselines with the models with no Mixup and +with the input Mixup, it is evident that our proposed Mixup +approaches are more suitable to tabular data and outperform +them on 5 out of 6 datasets, respectively. Comparing the no- +Mixup models and the models with input Mixup, the model +with the input Mixup performed better on 3 out of 6 datasets +and no-Mixup model is better on the rest 3 datasets. This +result indicates that the use of input Mixup is not consis- +tently effective across varied tabular datasets, even it beats +our proposed Mixup approaches on the GE dataset. +6. Conclusions +This paper presented a neural network EXCELFORMER +for supervised tabular data tasks (e.g., classification and +regression), and achieved performances beyond the level +of GBDTs without whistle and bell. The proposed EX- +CELFORMER can achieve competitive performances com- +pared to the extremely tuned XGboost and Catboost without +hyper-parameter tuning, while the hyper-parameter tuning +can promote EXCELFORMER’s performances further. Such +superiority is supported by comprehensive experiments on +25 public tabular datasets. Such good performances were +achieved by the cooperation of a simple but efficient model +architecture and an accompanied training methodology. We +expect that our EXCELFORMER with the training method- +ology will serve as a strong baseline in supervised tabular +tasks and inspire future work to develop better approaches +in dealing with tabular data. +Acknowledgements +This research was partially supported by the National Key +R&D Program of China under grant No. 2018AAA0102102 +and National Natural Science Foundation of China under +grants No. 62132017. +References +Akiba, T., Sano, S., Yanase, T., Ohta, T., and Koyama, M. +Optuna: A next-generation hyperparameter optimization +framework. In The ACM SIGKDD International Confer- +ence on Knowledge Discovery & Data Mining, 2019. +Arik, S. ¨O. and Pfister, T. TabNet: Attentive interpretable +tabular learning. In The AAAI Conference on Artificial +Intelligence, 2021. +Ba, J. L., Kiros, J. R., and Hinton, G. E. Layer normalization. +arXiv preprint arXiv:1607.06450, 2016. +Bachlechner, T., Majumder, B. P., Mao, H., Cottrell, G., and +McAuley, J. 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Details of the involved datasets. “# Num” and “# Cat” indicate the numbers of the numeric and categorical features. “# Sample” +means the size of datasets. +Dataset +Abbr. Task Type Metric # Feature # Num # Cat # Sample +Link +analcatdata supreme +AN +regression iRMSE +7 +2 +5 +4,052 +https://www.openml.org/d/44055 +cpu act +CP +regression iRMSE +21 +21 +0 +8,192 +https://www.openml.org/d/44132 +visualizing soil +VI +regression iRMSE +4 +3 +1 +8,641 +https://www.openml.org/d/44056 +yprop 4 1 +YP +regression iRMSE +62 +42 +20 +8,885 +https://www.openml.org/d/44054 +gesture +GE +multiclass +ACC +32 +32 +0 +9,873 +https://www.openml.org/d/4538 +churn +CH +binclass +AUC +11 +10 +1 +10,000 +https://www.kaggle.com/ +shrutimechlearn/churn-modelling +sulfur +SU +regression iRMSE +6 +6 +0 +10,081 +https://www.openml.org/d/44145 +bank-marketing +BA +binclass +AUC +7 +7 +0 +10,578 +https://www.openml.org/d/44126 +Brazilian houses +BR +regression iRMSE +8 +8 +0 +10,692 +https://www.openml.org/d/44141 +eye +EY +multiclass +ACC +26 +26 +0 +10,936 +http://www.cis.hut.fi/ +eyechallenge2005 +MagicTelescope +MA +binclass +AUC +10 +10 +0 +13,376 +https://www.openml.org/d/44125 +Ailerons +AI +regression iRMSE +33 +33 +0 +13,750 +https://www.openml.org/d/44137 +pol +PO +regression iRMSE +26 +26 +0 +15,000 +https://www.openml.org/d/722 +binarized-pol +BP +binclass +AUC +48 +48 +0 +15,000 +https://www.openml.org/d/722 +credit +CR +binclass +AUC +10 +10 +0 +16,714 +https://www.openml.org/d/44089 +california +CA +regression iRMSE +8 +8 +0 +20,640 +https://www.dcc.fc.up.pt/˜ltorgo/ +Regression/cal_housing.html +house sales +HS +regression iRMSE +15 +15 +0 +21,613 +https://www.openml.org/d/44144 +house +HO +regression iRMSE +16 +16 +0 +22,784 +https://www.openml.org/d/574 +diamonds +DI +regression iRMSE +6 +6 +0 +53,940 +https://www.openml.org/d/44140 +helena +HE +multiclass +ACC +27 +27 +0 +65,196 +https://www.openml.org/d/41169 +jannis +JA +multiclass +ACC +54 +54 +0 +83,733 +https://www.openml.org/d/41168 +higgs-small +HI +binclass +AUC +28 +28 +0 +98,049 +https://www.openml.org/d/23512 +road-safety +RO +binclass +AUC +32 +29 +3 +111,762 +https://www.openml.org/d/44161 +medicalcharges +ME +regression iRMSE +3 +3 +0 +163,065 +https://www.openml.org/d/44146 +SGEMM GPU kernel performance +SG +regression iRMSE +9 +3 +6 +241,600 +https://www.openml.org/d/44069 +A. The Descriptions of the Involved Datasets. +The details of the used tabular datasets are summarized in Table 2. We use the same train-test-valid split for all the approaches +and data pre-processing as in (Gorishniy et al., 2021). +B. The Hyper-Parameter Tuning +For XGboost and Catboost, we follow the implementations of (Gorishniy et al., 2021) while increase the number of the +estimators / iterations (i.e., decision trees) and the tuning iterations, so as to obtain the best-performed models. For our +EXCELFORMER, we use the optuna based tuning (Akiba et al., 2019). The hyper-parameter search spaces of EXCELFORMER, +XGboost and Catboost are reported in Table 3, 4, and 5, respectively. For EXCELFORMER, we only tune 50 iterations +on Mixup configurations (Mixup tuning), while for full tuning, we tune further 50 iterations by using the acquired hyper- +parameters from Mixup tuning as initialization. + +EXCELFORMER: A Neural Network Surpassing GBDTs on Tabular Data +Table 3. Hyper-parameter tuning spaces for EXCELFORMER. The items marked with “*” are used in the Mixup tuning, while all the items +are used in the full tuning. +Hyper-parameter +Distribution +# Layers L +UniformInt[2,5] +Embedding size d +{64,128,256} +# Heads +{4, 8, 16, 32} +Residual dropout rate +Uniform[0, 0.5] +Learning rate +LogUniform[3 × 10−5, 10−3] +Weight decay +{0.0, LogUniform[10−6,10−3]} +(*) Mixup Type +{SWAP-MIX, HIDDEN-MIX } +(*) α of Beta distribution +Uniform[0.1, 3.0] +Table 4. Hyper-parameter tuning spaces for XGboost. +Hyper-parameter +Distribution +Booster +“gbtree” +N-estimators +Const(4096) +Early-stopping-rounds +Const(50) +Max depth +UniformInt[3, 10] +Min child weight +LogUniform[10−8, 105] +Subsample +Uniform[0.5, 1.0] +Learning rate +LogUniform[10−5, 1] +Col sample by level +Uniform[0.5, 1] +Col sample by tree +Uniform[0.5, 1] +Gamma +{0, LogUniform[10−8, 102]} +Lambda +{0, LogUniform[10−8, 102]} +Alpha +{0, LogUniform[10−8, 102]} +# Tuning Iterations +500 +Table 5. Hyper-parameter tuning spaces for Catboost. +Hyper-parameter +Distribution +Iterations (number of trees) +Const(4096) +Od pval +Const(0.001) +Early-stopping-rounds +Const(50) +Max depth +UniformInt[3, 10] +Learning rate +LogUniform[10−5, 1] +Bagging temperature +Uniform[0, 1] +L2 leaf reg +LogUniform[1, 10] +Leaf estimation iterations +UniformInt[1, 10] +# Tuning Iterations +500 + diff --git a/ytE0T4oBgHgl3EQf-wLX/content/tmp_files/load_file.txt b/ytE0T4oBgHgl3EQf-wLX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b9876e11b4ca3a7509e35c47c6a20388552e092f --- /dev/null +++ b/ytE0T4oBgHgl3EQf-wLX/content/tmp_files/load_file.txt @@ -0,0 +1,1085 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE0T4oBgHgl3EQf-wLX/content/2301.02819v1.pdf,len=1084 +page_content='EXCELFORMER: A Neural Network Surpassing GBDTs on Tabular Data Jintai Chen * 1 Jiahuan Yan * 1 Danny Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE0T4oBgHgl3EQf-wLX/content/2301.02819v1.pdf'} +page_content=' Chen 2 Jian Wu 3 Abstract Though neural networks have achieved enormous breakthroughs on various fields (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE0T4oBgHgl3EQf-wLX/content/2301.02819v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE0T4oBgHgl3EQf-wLX/content/2301.02819v1.pdf'} +page_content=', computer vision) in supervised learning, they still trailed the performances of GBDTs on tabular data thus far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE0T4oBgHgl3EQf-wLX/content/2301.02819v1.pdf'} +page_content=' Delving into this issue, we identify that a proper handling of feature interactions and feature embedding is crucial to the success of neural net- works on tabular data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE0T4oBgHgl3EQf-wLX/content/2301.02819v1.pdf'} +page_content=' We develop a novel neural network called EXCELFORMER, which alternates in turn two attention modules that respectively ma- nipulate careful feature interactions and feature embedding updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE0T4oBgHgl3EQf-wLX/content/2301.02819v1.pdf'} +page_content=' A bespoke training method- ology is jointly introduced to facilitate the model performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE0T4oBgHgl3EQf-wLX/content/2301.02819v1.pdf'} +page_content=' By initializing parameters with minuscule values, these attention modules are at- tenuated when the training begins, and the effects of feature interactions and embedding updates progressively grow up to optimum levels under the guidance of the proposed specific regulariza- tion approaches SWAP-MIX and HIDDEN-MIX as the training proceeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE0T4oBgHgl3EQf-wLX/content/2301.02819v1.pdf'} +page_content=' Experiments on 25 public tabular datasets show that our EXCELFORMER is superior to extremely-tuned GBDTs, which is an unprecedented achievement of neural networks in supervised tabular learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE0T4oBgHgl3EQf-wLX/content/2301.02819v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE0T4oBgHgl3EQf-wLX/content/2301.02819v1.pdf'} +page_content=' Introduction Neural networks have been firmly established as state of the art approaches in various fields such as computer vision (Sri- vastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE0T4oBgHgl3EQf-wLX/content/2301.02819v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE0T4oBgHgl3EQf-wLX/content/2301.02819v1.pdf'} +page_content=' Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE0T4oBgHgl3EQf-wLX/content/2301.02819v1.pdf'} +page_content=', 2022), natural language pro- cessing (Hochreiter & Schmidhuber, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE0T4oBgHgl3EQf-wLX/content/2301.02819v1.pdf'} +page_content=' Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE0T4oBgHgl3EQf-wLX/content/2301.02819v1.pdf'} +page_content=', 2017), and automatic speech recognition (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE0T4oBgHgl3EQf-wLX/content/2301.02819v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE0T4oBgHgl3EQf-wLX/content/2301.02819v1.pdf'} +page_content=' However, on tabular data, one of the most ubiquitous data formats, neural networks cannot yet achieve comparable per- Equal contribution 1College of Computer Science and Tech- nology, Zhejiang University, Hangzhou, China 2Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN 46556, USA 3The First Affiliated Hospital, and Department of Public Health, Zhejiang University School of Medicine, Hangzhou, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE0T4oBgHgl3EQf-wLX/content/2301.02819v1.pdf'} +page_content=' Correspondence to: Jian Wu 0 such that +−C1ε−θ ≤ log P +� +∥BH∥ ≤ ε +� +≤ −C2ε−θ, +ε ∈ (0, 1]. +For the uniform norm, it is known that this holds with θ = 1/H, +− C1ε−1/H ≤ log P +� +sup +0≤t≤1 +|BH(t)| ≤ ε +� +≤ −C2ε−1/H. +(2.5) +Assumption 2.1 also holds for the γ-H¨older norm, where 0 < γ < H, and for +the L2-norm. See [4, 13, 14] for the corresponding values of θ, and for much +more information on small ball probabilities for fBm and other Gaussian +processes. +The examples we just mentioned are norms in the classical sense, and +so we stick to this terminology in our statements. From our proofs, it is +clear that it would suffice throughout to assume that ∥ · ∥ is a measurable +non-negative homogeneous functional. +3 + +Proposition 2.2. Let 0 < α < 2, 0 < β < 1. If the norm ∥ · ∥ satisfies +Assumption 2.1, then the cdf of ∥Bα,β∥ is an analytic function on R+. In +particular, this holds for the cdf of ∥Bα,β∥∞ = sup0≤t≤1 |Bα,β(t)|. +Proof. Recall that FH denotes the cdf of ∥BH∥. From (1.1) we find +P +� +∥Bα,β∥ ≤ ε +� += +� ∞ +0 +FH(εx−1/2)Mβ(x)dx += 2ε2 +� ∞ +0 +FH(y)Mβ(ε2/y2)y−3dy. +(2.6) +As Mβ extends to an entire function (see above), the last integrand clearly +is an entire function of ε for any fixed y > 0. The function Mβ is bounded +on R+, as follows, e.g., from (2.1) and (2.4). Thus, the integrand in (2.6) +can be bounded by an integrable function of y, independently of ε. Thus, +the conditions of a standard criterion for complex differentiation under the +integral sign [8, Theorem IV.5.8] are satisfied, which yields the assertion. +Note that fBm, i.e. β = 1, is not covered by Proposition 2.2. In [12], it is +shown by Malliavin calculus that sup0≤t≤1 BH (without the absolute value) +has a C∞ density. +We now show that, for β < 1, the small ball ball probability of ggBm is +of order ε2 as ε ↓ 0. For 2/θ + β < 1 (α + β < 1 for the uniform norm), we +express it as a power series, which yields a full asymptotic expansion. We +write +ηk(H) := E +� +∥BH∥−k� +, +k ∈ N, +for the negative moments of the norm of fBm, omitting the dependence on +the norm ∥ · ∥ in the notation ηk(H). By integration by parts, it is easy to +see that ηk(H) is finite under Assumption 2.1. +Theorem 2.3. Let 0 < α < 2, 0 < β < 1, and define H = α/2. Under +Assumption 2.1, the small ball probability of ggBm satisfies +P +� +∥Bα,β∥ ≤ ε +� +∼ η2(H)ε2 +Γ(1 − β), +ε ↓ 0. +(2.7) +If, additionally, 2/θ + β < 1, then it has the convergent series representation +P +� +∥Bα,β∥ ≤ ε +� += 2 +∞ +� +n=0 +(−1)nη2n+2(H) +(2n + 2)n!Γ(1 − β − βn)ε2n+2, +ε ≥ 0. +(2.8) +In particular, if ∥ · ∥ = ∥ · ∥∞, then (2.8) holds for α + β < 1. +4 + +Proof. By integration by parts, we have +� ∞ +0 +FH(y) +y2n+3 dy = +1 +2n + 2 +� ∞ +0 +y−2n−2FH(dy) = η2n+2(H) +2n + 2 . +(2.9) +The assertion (2.7) follows from (2.6), (2.1) for x = 0, (2.9) for n = 0, and +dominated convergence, because Mβ is a bounded function. For the next +statement, define +GN(ε, y) := +∞ +� +n=N+1 +(−1)nε2n−2N+1 +y2nn!Γ(1 − β − βn), +y > 0, ε ∈ [0, 1], +so that (2.6) yields, for N ∈ N, +P +� +∥Bα,β∥ ≤ ε +� += 2ε2 +� ∞ +0 +FH(y) +y3 +N +� +n=0 +(−ε2/y2)n +n!Γ(1 − β − βn)dy + 2ε2N+1 +� ∞ +0 +FH(y) +y3 +GN(ε, y)dy. +(2.10) +For the finite sum, we can use (2.9) to rewrite the summands as in (2.8). We +now provide an integrable bound for the last integrand in (2.10) that does +not depend on ε ∈ [0, 1]. It is clear that +|GN(ε, y)| ≤ +∞ +� +n=N+1 +1 +y2nn!Γ(1 − β − βn), +y > 0, ε ∈ [0, 1]. +(2.11) +By (2.2) and Stirling’s formula, +1 +n!|Γ(1 − β − βn)| ≤ n−(1−β)n+o(n) ≤ Cn−(1− ˆβ)n, +n ∈ N, +(2.12) +for any ˆβ > β; we will fix ˆβ later. From (2.11), (2.12), and Stirling’s formula, +we conclude +|GN(ε, y)| ≤ C +∞ +� +n=N+1 +1 +y2nΓ((1 − ˆβ)n) += y−2E1− ˆβ,1− ˆβ(y−2) − +N +� +n=1 +1 +y2nΓ((1 − ˆβ)n) +, +(2.13) +where +Eu,v(z) = +∞ +� +n=0 +zn +Γ(un + v), +u, v > 0, z ∈ C, +5 + +denotes the two-parameter Mittag-Leffler function. We now use the uniform +bound (2.13) in (2.10). +Integrability at ∞ is obvious, and we now show +integrability at zero. By [10, Theorem 4.3], +E1− ˆβ,1− ˆβ(y−2) = exp +� +y +− +2 +1− ˆβ � +1 + o(1) +�� +, +y ↓ 0. +We see, using Assumption 2.1 for FH, that the last integrand in (2.10) satisfies +FH(y) +y3 +GN(ε, y) ≤ exp +� +−C1y−θ + y +− +2 +1− ˆβ + o +� +y +− +� +θ∧ +2 +1− ˆβ +��� +, +y ↓ 0, +uniformly w.r.t. ε ∈ [0, 1]. This is integrable if θ > 2/(1− ˆβ), i.e., 2/θ+ ˆβ < 1. +Clearly, our assumption that 2/θ + β < 1 allows us to chose such a ˆβ > β. +By the following lemma, η2n+2(H) = 22n/θ+o(n). Using (2.12), we can thus +take the limit N ↑ ∞ in (2.10) for fixed ε ∈ [0, 1], which proves (2.8) for +these ε. The extension to any ε ≥ 0 follows by analytic continuation, using +Proposition 2.2. +In the preceding proof, we applied the following estimate for negative mo- +ments of the supremum of fBm. Note that moments with positive exponent +are estimated in [21]; see also [20]. +Lemma 2.4. Under Assumption 2.1, for k ↑ ∞, we have ηk(H) = kk/θ+o(k). +Proof. We show only the upper estimate, as the lower one can be proven +analogously. By (2.5), there is ε0 > 0 such that +FH(y) ≤ 2 exp(−C2y−θ), +0 < y ≤ ε0. +Define ˜K := 2 ∨ exp(C2ε−θ +0 ). Then, +FH(y) ≤ ˜K exp(−C2y−θ), +y > 0; +note that the right hand side is ≥ 1 for y ≥ ε0. This implies +ηk(H) = +� ∞ +0 +y−kFH(dy) = k +� ∞ +0 +y−k−1FH(y)dy +≤ k ˜K +� ∞ +0 +exp(−C2y−θ)y−k−1dy += eO(k) +� ∞ +0 +e−wwk/θ−1dw += eO(k)Γ(k/θ − 1) = kk/θ+o(k), +by Stirling’s formula (2.3) for the gamma function. +6 + +If 2/θ + β < 1, then the series in (2.8) diverges for any ε > 0. Indeed, +there is an increasing sequence (nj) in N such that the lower bound +dist(1 − β − βnj, Z) ≥ C > 0, +j ∈ N, +holds. For rational β ∈ (0, 1), this is clear by periodicity. For irrational β, it +follows from the classical fact that the sequence of fractional parts {nβ} is +dense in [0, 1] (Kronecker’s approximation theorem). Hence, again by (2.2) +and Stirling’s formula, +1 +|Γ(1 − β − βnj)| ≥ n +βnj+o(nj) +j +, +which, together with Lemma 2.4, shows divergence. We leave it as an open +problem if (2.8) still holds in the sense of an asymptotic expansion of the +small ball probability, if 2/θ + β ≤ 1. +3 +Large deviations +For fractional Brownian motion, it is well known that +P +� +sup +0≤t≤1 +|BH(t)| ≥ y +� += exp +� +−1 +2y2 + o(y2) +� +, +y ↑ ∞. +(3.1) +Indeed, the upper estimate follows from +P +� +sup +0≤t≤1 +|BH(t)| ≥ y +� +≤ 2 P +� +sup +0≤t≤1 +BH(t) ≥ y +� +and the Borell-TIS inequality [19, Theorem 4.2], and the lower one is clear +from sup0≤t≤1 |BH(t)| ≥ BH(1). The following result gives a large deviation +estimate for ggBm. For β = 1, the distribution has a Gaussian upper tail, +of course. For 0 < β < 1, the decay is between exponential and Gaussian, +which is sometimes called compressed exponential. +Theorem 3.1. Let 0 < α < 2 and 0 < β ≤ 1, and assume that ∥ · ∥ is a +norm on the H¨older space Cγ +0 [0, 1], where 0 < γ < H = α/2, such that +P +� +∥BH∥ ≥ y +� += exp +� +−κy2 + o(y2) +� +, +y ↑ ∞, +(3.2) +for some κ > 0. Then there are constants K1, K2 > 0 such that +exp +� +−K1y +2 +2−β � +1 + o(1) +�� +≤ P +� +∥Bα,β∥ ≥ y +� +(3.3) +≤ exp +� +−K2y +2 +2−β � +1 + o(1) +�� +, +y ↑ ∞. +(3.4) +7 + +Proof. We may assume β < 1, because for β = 1 we have Bα,1 = BH and the +assumption (3.2) makes the statement trivial. With ¯FH = 1 − FH the tail +distribution function of ∥BH∥, we have, from (1.1), +P +� +∥Bα,β∥ ≥ y +� += +� ∞ +0 +¯FH(yx−1/2)Mβ(x)dx. +If κ = 1 +2, then ¯FH satisfies +¯FH(y) = exp +� +−1 +2y2 + o(y2) +� +, +y ↑ ∞, +(3.5) +by (3.2). +We assume κ = +1 +2 for rest of the proof, as κ > 0 is a trivial +extension. Let 0 < ˆκ < 1 +2 be arbitrary. Since Mβ is bounded, we obtain +� 1 +0 +¯FH(yx−1/2)Mβ(x)dx ≤ C +� 1 +0 +e−ˆκy2/xMβ(x)dx +≤ C +� 1 +0 +e−ˆκy2/xdx += C +� +e−ˆκy2 − ˆκy2Γ(0, ˆκy2) +� +, +where Γ(a, z) = +� ∞ +z ta−1e−tdt is the incomplete gamma function. Using a +well-known expansion of that function [7, §8.11], we conclude +� 1 +0 +¯FH(yx−1/2)Mβ(x)dx ≤ exp +� +−ˆκy2 + o(y2) +� +, +y ↑ ∞. +(3.6) +As β < 1, this is negligible compared to the decay rate claimed in (3.3) +and (3.4). Now define h(y) := y2/(log y). Since ¯FH ≤ 1, and using (2.4), we +have +� ∞ +h(y) +¯FH(yx−1/2)Mβ(x)dx ≤ +� ∞ +h(y) +Mβ(x)dx +≤ +� ∞ +h(y) +exp +� +−Cx +1 +1−β � +dx +≤ exp +� +−Ch(y) +1 +1−β � +. +Since 2/(1 − β) > 2/(2 − β), this is of faster decay than (3.4). +It remains to show that the integral +� h(y) +1 +¯FH(yx−1/2)Mβ(x)dx has the +claimed growth order (3.4). By dividing the exponent in (2.4) by 2, which +makes the decay slower, we obtain +Mβ(x) ≤ C exp +� +−1 − β +2β (βx) +1 +1−β +� +, +x ≥ 1. +8 + +Similarly, (3.5) implies +¯FH(y) ≤ Ce−y2/3, +y ≥ 1. +Analogously, we can increase the constants in the exponents to find lower +estimates, for which the following reasoning is analogous, and yields (3.3). +Therefore, we only discuss the upper estimate for +� h(y) +1 +. This is a straight- +forward application of the Laplace method [5, Chapter 4] to the integral +� h(y) +1 +exp +� +− y2 +3x − 1 − β +2β (βx) +1 +1−β +� +dx, +which results from the two preceding estimates. The integrand is a strictly +concave function with a maximum at +x0(y) = cy +2(1−β) +2−β +∈ (1, h(y)) +for some constant c > 0. As we are not concerned with lower order terms, it +suffices to evaluate the integrand at x0(y) to conclude +� h(y) +1 +¯F(yx−1/2)Mβ(x)dx ≤ exp +� +−Cy +2 +2−β (1 + o(1)) +� +. +This completes the proof. +We now comment on applying Theorem 3.1 to other norms than the sup +norm, which requires verifying (3.2). As mentioned above, for the sup norm, +this follows from the Borell-TIS inequality. For an arbitrary norm ∥ · ∥ on +H¨older space, we have +P +� +∥BH∥ ≥ y +� += P +� +y−1BH ∈ {∥f∥ ≥ 1} +� +. +In principle, this is in the scope of the general LDP (large deviation principle) +for Gaussian measures [6, Theorem 3.4.12], but it may not be trivial to verify +the assumptions. For H = +1 +2 and the H¨older norm, this was done in [3], +extending Schilder’s theorem. Note that choosing a stronger topology than +the uniform one enlarges the dual space of path space, making it harder to +verify the defining property of a Gaussian measure. For the H¨older topology, +we are on safe grounds, though, by another approach: Using the double sum +method for Gaussian fields, Fatalov has shown that (3.1) holds for the γ- +H¨older norm [9, Theorem 1.3], and so Theorem 3.1 is applicable to this norm +(with 0 < γ < H, of course). +9 + +References +[1] F. Aurzada, M. Lifshits, and W. Linde, Small deviations of stable +processes and entropy of the associated random operators, Bernoulli, 15 +(2009), pp. 1305–1334. +[2] F. Aurzada and T. Simon, Small ball probabilities for stable convo- +lutions, ESAIM Probab. Stat., 11 (2007), pp. 327–343. +[3] P. Baldi, G. Ben Arous, and G. Kerkyacharian, Large devi- +ations and the Strassen theorem in H¨older norm, Stochastic Process. +Appl., 42 (1992), pp. 171–180. +[4] J. C. Bronski, Small ball constants and tight eigenvalue asymptotics +for fractional Brownian motions, J. Theoret. Probab., 16 (2003), pp. 87– +100. +[5] N. 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Schneider, Grey noise, in Stochastic processes, physics and +geometry (Ascona and Locarno, 1988), World Sci. Publ., Teaneck, NJ, +1990, pp. 676–681. +[23] E. M. Wright, The generalized Bessel function of order greater than +one, Quart. J. Math. Oxford Ser., 11 (1940), pp. 36–48. +11 + diff --git a/z9E4T4oBgHgl3EQfZQyl/content/tmp_files/load_file.txt b/z9E4T4oBgHgl3EQfZQyl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..27de12025e0409200c9596d313c67240a3c629c3 --- /dev/null +++ b/z9E4T4oBgHgl3EQfZQyl/content/tmp_files/load_file.txt @@ -0,0 +1,438 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf,len=437 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='05055v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='PR] 12 Jan 2023 Small ball probabilities and large deviations for grey Brownian motion Stefan Gerhold TU Wien sgerhold@fam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='tuwien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='at January 13, 2023 Abstract We show that the uniform norm of generalized grey Brownian mo- tion over the unit interval has an analytic density, excluding the special case of fractional Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Our main result is an asymptotic expansion for the small ball probability of generalized grey Brownian motion, which extends to other norms on path space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' The decay rate is not exponential but polynomial, of degree two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' For the uniform norm and the H¨older norm, we also prove a large deviations estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' MSC2020: 60G22, 60F99 Keywords: grey Brownian motion, fractional Brownian motion, small ball probabilities, small deviations, large deviations, Wright M-function 1 Introduction Generalized grey Brownian motion (ggBm) is a two-parameter stochastic process Bα,β, which is in general not Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Introduced in [16, 17], ggBm has been considered in the physics literature to model anomalous diffusions with non-Gaussian marginals, including both slow (variance grows slower than linearly) and fast diffusive behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' The process Bα,β has stationary increments and is self-similar with parameter H = α/2 [16, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' The marginal density of ggBm satisfies a fractional partial integro-differential equation [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Special cases of ggBm include fractional Brownian motion (fBm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' β = 1), grey Brownian motion ([22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' α = β), and Brownian motion (α = β = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Our focus is mainly on the case β < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' In [16], a generalized grey noise space is defined, motivated by white noise space, but with the 1 Gaussian characteristic function replaced by the Mittag-Leffler function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' The ggBm is then defined by evaluating generalized grey noise at the test function 1[0,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' We do not go into details, because for our purposes, the representation Bα,β(t) = � LβBα/2(t), 0 < α < 2, 0 < β < 1, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1) which was proved in [17], is more convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Here, Bα/2 is a fBm with Hurst parameter H = α/2, and Lβ is an independent positive random variable whose density is the M-Wright function (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' The representation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1) makes sense also in the limiting case β = 1, but we will not require this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' The problem of small ball probabilities, also called small deviations, con- sists of estimating P � sup 0≤t≤1 |Bα,β(t)| ≤ ε � , ε ↓ 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='2) asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' More generally, we can consider P � ∥Bα,β∥ ≤ ε � , ε ↓ 0, where ∥ · ∥ is a norm on Cγ 0 [0, 1], the space of γ-H¨older continuous functions, with 0 < γ < H = α/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' For ggBm with β < 1, our main result (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='3) shows that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='2) is of order ε2, and that this also holds for some other norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' For Gaussian processes, such as fBm (β = 1), the small ball problem has been studied extensively [13], and exponential decay is typical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' But there are also many works studying small ball probabilities for non-Gaussian processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=', [1, 2] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' We refer to [11, 18] for other examples of processes with the small ball rate ε2 of ggBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' In Section 2, we will show that the known exponential small ball estimates for fBm can be used to deduce our quadratic small ball estimate for ggBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' As a byproduct, we show that the uniform norm (sup norm) of ggBm has a smooth, even analytic, pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' In Section 3, we provide a large deviations estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' The decay rate is exponential, but slower than Gaussian, depending on the parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Notation: When we write Bα,β, we always mean the process on the time interval [0, 1], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Bα,β = (Bα,β(t))0≤t≤1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' We write FH for the cdf of ∥BH∥, assuming that the choice of the norm ∥·∥ is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' As usual, R+ = (0, ∞) denotes the positive reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' The letter C denotes various positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' 2 2 Analyticity of the cdf and small ball prob- ability The M-Wright function, which is the pdf of Lβ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1), is defined by Mβ(x) = ∞ � n=0 (−x)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='Γ(1 − β − βn), x ≥ 0, 0 < β < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1) It is not obvious that Mβ is a pdf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' for this, and more information on Mβ and its generalizations, we refer to [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' For later use, we note that it follows from Euler’s reflection formula that 1 Γ(1 − β − βn) = sin � π(β + βn) � π Γ(β + βn), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='2) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' [23, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' 41] and [15, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='8)]), which shows, by Stirling’s formula for the gamma function, that the series in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1) defines an entire function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' For this, the crude version Γ(x) = xx+o(x), x ↑ ∞, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='3) of Stirling’s formula suffices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' We will also need the asymptotic behavior of Mβ at infinity [15, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='5)], Mβ(x) = exp � −1 − β β (βx) 1 1−β + O(log x) � , x ↑ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='4) Our main assumption is that fBm satisfies an exponential small ball es- timate w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' to the chosen norm ∥ · ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' For 0 < H < 1, there are θ, C1, C2 > 0 such that −C1ε−θ ≤ log P � ∥BH∥ ≤ ε � ≤ −C2ε−θ, ε ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' For the uniform norm, it is known that this holds with θ = 1/H, − C1ε−1/H ≤ log P � sup 0≤t≤1 |BH(t)| ≤ ε � ≤ −C2ε−1/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='5) Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1 also holds for the γ-H¨older norm, where 0 < γ < H, and for the L2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' See [4, 13, 14] for the corresponding values of θ, and for much more information on small ball probabilities for fBm and other Gaussian processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' The examples we just mentioned are norms in the classical sense, and so we stick to this terminology in our statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' From our proofs, it is clear that it would suffice throughout to assume that ∥ · ∥ is a measurable non-negative homogeneous functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' 3 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Let 0 < α < 2, 0 < β < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' If the norm ∥ · ∥ satisfies Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1, then the cdf of ∥Bα,β∥ is an analytic function on R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' In particular, this holds for the cdf of ∥Bα,β∥∞ = sup0≤t≤1 |Bα,β(t)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Recall that FH denotes the cdf of ∥BH∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' From (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1) we find P � ∥Bα,β∥ ≤ ε � = � ∞ 0 FH(εx−1/2)Mβ(x)dx = 2ε2 � ∞ 0 FH(y)Mβ(ε2/y2)y−3dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='6) As Mβ extends to an entire function (see above), the last integrand clearly is an entire function of ε for any fixed y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' The function Mβ is bounded on R+, as follows, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=', from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Thus, the integrand in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='6) can be bounded by an integrable function of y, independently of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Thus, the conditions of a standard criterion for complex differentiation under the integral sign [8, Theorem IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='8] are satisfied, which yields the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Note that fBm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' β = 1, is not covered by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' In [12], it is shown by Malliavin calculus that sup0≤t≤1 BH (without the absolute value) has a C∞ density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' We now show that, for β < 1, the small ball ball probability of ggBm is of order ε2 as ε ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' For 2/θ + β < 1 (α + β < 1 for the uniform norm), we express it as a power series, which yields a full asymptotic expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' We write ηk(H) := E � ∥BH∥−k� , k ∈ N, for the negative moments of the norm of fBm, omitting the dependence on the norm ∥ · ∥ in the notation ηk(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' By integration by parts, it is easy to see that ηk(H) is finite under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Let 0 < α < 2, 0 < β < 1, and define H = α/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1, the small ball probability of ggBm satisfies P � ∥Bα,β∥ ≤ ε � ∼ η2(H)ε2 Γ(1 − β), ε ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='7) If, additionally, 2/θ + β < 1, then it has the convergent series representation P � ∥Bα,β∥ ≤ ε � = 2 ∞ � n=0 (−1)nη2n+2(H) (2n + 2)n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='Γ(1 − β − βn)ε2n+2, ε ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='8) In particular, if ∥ · ∥ = ∥ · ∥∞, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='8) holds for α + β < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' 4 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' By integration by parts, we have � ∞ 0 FH(y) y2n+3 dy = 1 2n + 2 � ∞ 0 y−2n−2FH(dy) = η2n+2(H) 2n + 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='9) The assertion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='7) follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1) for x = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='9) for n = 0, and dominated convergence, because Mβ is a bounded function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' For the next statement, define GN(ε, y) := ∞ � n=N+1 (−1)nε2n−2N+1 y2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='Γ(1 − β − βn), y > 0, ε ∈ [0, 1], so that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='6) yields, for N ∈ N, P � ∥Bα,β∥ ≤ ε � = 2ε2 � ∞ 0 FH(y) y3 N � n=0 (−ε2/y2)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='Γ(1 − β − βn)dy + 2ε2N+1 � ∞ 0 FH(y) y3 GN(ε, y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='10) For the finite sum, we can use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='9) to rewrite the summands as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' We now provide an integrable bound for the last integrand in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='10) that does not depend on ε ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' It is clear that |GN(ε, y)| ≤ ∞ � n=N+1 1 y2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='Γ(1 − β − βn), y > 0, ε ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='11) By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='2) and Stirling’s formula, 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='|Γ(1 − β − βn)| ≤ n−(1−β)n+o(n) ≤ Cn−(1− ˆβ)n, n ∈ N, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='12) for any ˆβ > β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' we will fix ˆβ later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='11), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='12), and Stirling’s formula, we conclude |GN(ε, y)| ≤ C ∞ � n=N+1 1 y2nΓ((1 − ˆβ)n) = y−2E1− ˆβ,1− ˆβ(y−2) − N � n=1 1 y2nΓ((1 − ˆβ)n) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='13) where Eu,v(z) = ∞ � n=0 zn Γ(un + v), u, v > 0, z ∈ C, 5 denotes the two-parameter Mittag-Leffler function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' We now use the uniform bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='13) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Integrability at ∞ is obvious, and we now show integrability at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' By [10, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='3], E1− ˆβ,1− ˆβ(y−2) = exp � y − 2 1− ˆβ � 1 + o(1) �� , y ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' We see, using Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1 for FH, that the last integrand in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='10) satisfies FH(y) y3 GN(ε, y) ≤ exp � −C1y−θ + y − 2 1− ˆβ + o � y − � θ∧ 2 1− ˆβ ��� , y ↓ 0, uniformly w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' ε ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' This is integrable if θ > 2/(1− ˆβ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=', 2/θ+ ˆβ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Clearly, our assumption that 2/θ + β < 1 allows us to chose such a ˆβ > β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' By the following lemma, η2n+2(H) = 22n/θ+o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='12), we can thus take the limit N ↑ ∞ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='10) for fixed ε ∈ [0, 1], which proves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='8) for these ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' The extension to any ε ≥ 0 follows by analytic continuation, using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' In the preceding proof, we applied the following estimate for negative mo- ments of the supremum of fBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Note that moments with positive exponent are estimated in [21];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' see also [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1, for k ↑ ∞, we have ηk(H) = kk/θ+o(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' We show only the upper estimate, as the lower one can be proven analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='5), there is ε0 > 0 such that FH(y) ≤ 2 exp(−C2y−θ), 0 < y ≤ ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Define ˜K := 2 ∨ exp(C2ε−θ 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Then, FH(y) ≤ ˜K exp(−C2y−θ), y > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' note that the right hand side is ≥ 1 for y ≥ ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' This implies ηk(H) = � ∞ 0 y−kFH(dy) = k � ∞ 0 y−k−1FH(y)dy ≤ k ˜K � ∞ 0 exp(−C2y−θ)y−k−1dy = eO(k) � ∞ 0 e−wwk/θ−1dw = eO(k)Γ(k/θ − 1) = kk/θ+o(k), by Stirling’s formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='3) for the gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' 6 If 2/θ + β < 1, then the series in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='8) diverges for any ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Indeed, there is an increasing sequence (nj) in N such that the lower bound dist(1 − β − βnj, Z) ≥ C > 0, j ∈ N, holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' For rational β ∈ (0, 1), this is clear by periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' For irrational β, it follows from the classical fact that the sequence of fractional parts {nβ} is dense in [0, 1] (Kronecker’s approximation theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Hence, again by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='2) and Stirling’s formula, 1 |Γ(1 − β − βnj)| ≥ n βnj+o(nj) j , which, together with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='4, shows divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' We leave it as an open problem if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='8) still holds in the sense of an asymptotic expansion of the small ball probability, if 2/θ + β ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' 3 Large deviations For fractional Brownian motion, it is well known that P � sup 0≤t≤1 |BH(t)| ≥ y � = exp � −1 2y2 + o(y2) � , y ↑ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1) Indeed, the upper estimate follows from P � sup 0≤t≤1 |BH(t)| ≥ y � ≤ 2 P � sup 0≤t≤1 BH(t) ≥ y � and the Borell-TIS inequality [19, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='2], and the lower one is clear from sup0≤t≤1 |BH(t)| ≥ BH(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' The following result gives a large deviation estimate for ggBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' For β = 1, the distribution has a Gaussian upper tail, of course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' For 0 < β < 1, the decay is between exponential and Gaussian, which is sometimes called compressed exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Let 0 < α < 2 and 0 < β ≤ 1, and assume that ∥ · ∥ is a norm on the H¨older space Cγ 0 [0, 1], where 0 < γ < H = α/2, such that P � ∥BH∥ ≥ y � = exp � −κy2 + o(y2) � , y ↑ ∞, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='2) for some κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Then there are constants K1, K2 > 0 such that exp � −K1y 2 2−β � 1 + o(1) �� ≤ P � ∥Bα,β∥ ≥ y � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='3) ≤ exp � −K2y 2 2−β � 1 + o(1) �� , y ↑ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='4) 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' We may assume β < 1, because for β = 1 we have Bα,1 = BH and the assumption (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='2) makes the statement trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' With ¯FH = 1 − FH the tail distribution function of ∥BH∥, we have, from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1), P � ∥Bα,β∥ ≥ y � = � ∞ 0 ¯FH(yx−1/2)Mβ(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' If κ = 1 2, then ¯FH satisfies ¯FH(y) = exp � −1 2y2 + o(y2) � , y ↑ ∞, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='5) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' We assume κ = 1 2 for rest of the proof, as κ > 0 is a trivial extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Let 0 < ˆκ < 1 2 be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Since Mβ is bounded, we obtain � 1 0 ¯FH(yx−1/2)Mβ(x)dx ≤ C � 1 0 e−ˆκy2/xMβ(x)dx ≤ C � 1 0 e−ˆκy2/xdx = C � e−ˆκy2 − ˆκy2Γ(0, ˆκy2) � , where Γ(a, z) = � ∞ z ta−1e−tdt is the incomplete gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Using a well-known expansion of that function [7, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='11], we conclude � 1 0 ¯FH(yx−1/2)Mβ(x)dx ≤ exp � −ˆκy2 + o(y2) � , y ↑ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='6) As β < 1, this is negligible compared to the decay rate claimed in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Now define h(y) := y2/(log y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Since ¯FH ≤ 1, and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='4), we have � ∞ h(y) ¯FH(yx−1/2)Mβ(x)dx ≤ � ∞ h(y) Mβ(x)dx ≤ � ∞ h(y) exp � −Cx 1 1−β � dx ≤ exp � −Ch(y) 1 1−β � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Since 2/(1 − β) > 2/(2 − β), this is of faster decay than (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' It remains to show that the integral � h(y) 1 ¯FH(yx−1/2)Mβ(x)dx has the claimed growth order (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' By dividing the exponent in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='4) by 2, which makes the decay slower, we obtain Mβ(x) ≤ C exp � −1 − β 2β (βx) 1 1−β � , x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' 8 Similarly, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='5) implies ¯FH(y) ≤ Ce−y2/3, y ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Analogously, we can increase the constants in the exponents to find lower estimates, for which the following reasoning is analogous, and yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Therefore, we only discuss the upper estimate for � h(y) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' This is a straight- forward application of the Laplace method [5, Chapter 4] to the integral � h(y) 1 exp � − y2 3x − 1 − β 2β (βx) 1 1−β � dx, which results from the two preceding estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' The integrand is a strictly concave function with a maximum at x0(y) = cy 2(1−β) 2−β ∈ (1, h(y)) for some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' As we are not concerned with lower order terms, it suffices to evaluate the integrand at x0(y) to conclude � h(y) 1 ¯F(yx−1/2)Mβ(x)dx ≤ exp � −Cy 2 2−β (1 + o(1)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' We now comment on applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1 to other norms than the sup norm, which requires verifying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' As mentioned above, for the sup norm, this follows from the Borell-TIS inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' For an arbitrary norm ∥ · ∥ on H¨older space, we have P � ∥BH∥ ≥ y � = P � y−1BH ∈ {∥f∥ ≥ 1} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' In principle, this is in the scope of the general LDP (large deviation principle) for Gaussian measures [6, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='12], but it may not be trivial to verify the assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' For H = 1 2 and the H¨older norm, this was done in [3], extending Schilder’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Note that choosing a stronger topology than the uniform one enlarges the dual space of path space, making it harder to verify the defining property of a Gaussian measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' For the H¨older topology, we are on safe grounds, though, by another approach: Using the double sum method for Gaussian fields, Fatalov has shown that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1) holds for the γ- H¨older norm [9, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='3], and so Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content='1 is applicable to this norm (with 0 < γ < H, of course).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' 9 References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Aurzada, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Lifshits, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Linde, Small deviations of stable processes and entropy of the associated random operators, Bernoulli, 15 (2009), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' 1305–1334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' [2] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Aurzada and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Simon, Small ball probabilities for stable convo- lutions, ESAIM Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfZQyl/content/2301.05055v1.pdf'} 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100644 index 0000000000000000000000000000000000000000..f7d1762e82fef5531d147ff4b4537f8411b81a3c --- /dev/null +++ b/zNE1T4oBgHgl3EQfkQT7/content/tmp_files/2301.03273v1.pdf.txt @@ -0,0 +1,816 @@ +arXiv:2301.03273v1 [astro-ph.SR] 9 Jan 2023 +Astronomy & Astrophysics manuscript no. main +©ESO 2023 +January 10, 2023 +Impact of opacity effects on chromospheric oscillations +inferred from NLTE inversions +T. Felipe1, 2 and H. Socas-Navarro1, 2 +1 Instituto de Astrofísica de Canarias, 38205, C/ Vía Láctea, s/n, La Laguna, Tenerife, Spain +2 Departamento de Astrofísica, Universidad de La Laguna, 38205, La Laguna, Tenerife, Spain +January 10, 2023 +ABSTRACT +Context. Spectropolarimetric inversions are a fundamental tool to diagnose the solar atmosphere. Chromospheric infer- +ences rely on the interpretation of spectral lines that are formed under Non Local Thermodynamic Equilibrium (NLTE) +conditions. In the presence of oscillations, changes in the opacity impact the response height of the spectral lines and +hinder the determination of the real properties of the fluctuations. +Aims. We aim to explore the relationship between the chromospheric oscillations inferred by NLTE inversion codes and +the intrinsic fluctuations in velocity and temperature produced by the waves. +Methods. Numerical simulations of wave propagation in a sunspot umbra have been computed with the code MANCHA. +The NLTE synthesis and inversion code NICOLE has been used to compute spectropolarimetric Ca ii 8542 Å line profiles +for the atmospheric models obtained as the output from the simulations. The synthetic profiles have been inverted and +the inferences from the inversions have been compared with the known atmospheres from the simulations. +Results. NLTE inversions of the Ca ii 8542 Å line capture low frequency oscillations, including those in the main band +of chromospheric oscillations around 6 mHz. In contrast, waves with frequencies above 9 mHz are poorly characterized +by the inversion results. Velocity oscillations at those higher frequencies exhibit clear insights of opacity fluctuations +since the power of the signal at constant optical depth greatly depart from the power of the oscillations at constant +geometrical height. The main response of the line to velocity fluctuations comes from low chromospheric heights, whereas +the response to temperature shows sudden jumps between the high photosphere and the low chromosphere. This strong +variation in the heights where the line is sensitive to temperature is revealed as a strong oscillatory power in the inferred +fluctuations, much stronger than the actual power from the intrinsic temperature oscillations. +Conclusions. Our results validate the use of NLTE inversions to study chromospheric oscillations with frequencies below +∼9 mHz. However, the interpretation of higher frequency oscillations and the power of temperature oscillations must +be addressed with care since they exhibit signatures of opacity oscillations. +Key words. Methods: numerical – Sun: chromosphere – Sun: oscillations – sunspots – Techniques: polarimetric +1. Introduction +The study of the solar atmosphere heavily relies on the +observation and interpretation of the solar spectra, often +not only using spectroscopic data but also full polarimetry. +The Ca ii 8542 Å line is one of the most exploited spectral +lines for probing the solar chromosphere. The interpreta- +tion of its spectral profiles requires NLTE diagnostics. Sev- +eral NLTE inversion codes have been developed with this +aim, such as NICOLE (Socas-Navarro et al. 2015), STiC +(de la Cruz Rodríguez et al. 2019), SNAPI (Milić & van +Noort 2018), and DeSIRe (Ruiz Cobo et al. 2022). Due to +the large amount of computational resources required by +these inversions, pioneering studies using these tools were +mostly restricted to analyzing a few spectral profiles (e.g., +Socas-Navarro et al. 2000) or spatially coherent maps with +a reduced resolution for a few time steps (de la Cruz Ro- +dríguez et al. 2013). Until recently, chromospheric oscilla- +tions have been out of the scope of the works using NLTE +inversions since they require the analysis of long temporal +series with high temporal cadence and, thus, the inversion +of numerous spectral profiles. Several works have employed +alternative techniques to derive the chromospheric plasma +properties from the interpretation of the Ca ii 8542 Å line, +such as the lambdameter (e.g., Chae et al. 2018) or bi-sector +(e.g., Grant et al. 2022) methods to infer the velocity. How- +ever, in umbral regions, these methods are challenged by the +common display of emission near the core of the line as a +manifestation of umbral flashes. The interpretation of these +profiles requires sophisticated analysis tools, like inversion +codes. +Thanks to the improvement of the computational capa- +bilities, recent works have been able to perform more com- +prehensive studies of sunspot chromospheres using NLTE +inversions of the Ca ii 8542 Å line (Henriques et al. 2017; +Joshi & de la Cruz Rodríguez 2018; Henriques et al. 2020; +Houston et al. 2020). Also, new methods based on machine +learning techniques are being developed to diagnose the so- +lar chromosphere in a fast and computationally efficient way +(Vicente Arévalo et al. 2022). The availability of physical +information from larger maps and longer temporal series +will enable the study of their oscillations in the common +ground of Fourier and/or wavelet analyses. +In the dynamic solar atmosphere, the contribution of +different atmospheric heights to the formation of a spectral +Article number, page 1 of 9 + +A&A proofs: manuscript no. main +line changes with time (Uitenbroek 2003). These variations +are especially troubling for the study of solar oscillations. +Spurious oscillations produced by the change in the forma- +tion height of the spectral lines in atmospheres with vertical +gradients (known as opacity effects) can overlap the intrin- +sic fluctuations due to wave propagation. Disentangling the +intrinsic oscillations from those produced by the opacity ef- +fect is thus fundamental for a proper interpretation of wave +phenomena. +Many reported magnetic field fluctuations in sunspots +have been interpreted as a result of opacity effects (Bellot +Rubio et al. 2000; Rüedi & Cally 2003; Khomenko et al. +2003). Active regions are known to harbor vertical mag- +netic field gradients (see Solanki 2003; Borrero & Ichimoto +2011, for a review). In these atmospheres, the periodical +displacements of the formation region of spectral lines in- +troduce spurious oscillations in the inferred magnetic field. +In contrast, the opacity effect is barely discussed in the ex- +amination of oscillations in other quantities, such as tem- +perature and velocity. Indeed, the lower solar atmosphere +exhibits significant vertical gradients in temperature. Ve- +locity gradients are also expected since the amplitude of +the oscillations increases with height due to the drop of the +density (Centeno et al. 2006). Both vertical gradients will +certainly leave an imprint in the temperature/velocity fluc- +tuations measured from any spectral line formed at those +heights. +In this study, we focus on the umbral oscillations in- +ferred from the analysis of the Ca ii 8542 Å line. This spec- +tral line is sensitive to a broad range of heights, from the +photosphere at the wings to the chromosphere at the core of +the line (Cauzzi et al. 2008). Nowadays, it is one of the most +employed lines for the study of the solar chromosphere (e.g., +Socas-Navarro 2005; Kleint 2012; Rouppe van der Voort & +de la Cruz Rodríguez 2013; Kuridze et al. 2018; Murabito +et al. 2019). It is also commonly used for the inspection of +umbral oscillations and, more specifically, the development +of umbral flashes (Socas-Navarro et al. 2000; de la Cruz +Rodríguez et al. 2013; Henriques et al. 2017; Houston et al. +2018; Bose et al. 2019; Houston et al. 2020). +Variations in the geometrical heights where the Ca ii +8542 Å line is sensitive during umbral flashes have been +reported from the analysis of observations (Joshi & de la +Cruz Rodríguez 2018) and numerical modeling (Felipe et al. +2021a). Here, we address how how this opacity effect im- +pacts the chromospheric umbral oscillations inferred with +the Ca ii 8542 Å line. We have constructed synthetic Stokes +profiles from the atmospheres computed with numerical +simulations and then inverted those profiles. The numerical +methods are briefly described in Sect. 2 and the response +functions of the line are introduced in Sect. 3. In Sects. 4 +and 5 we present the results obtained for velocity and tem- +perature oscillations, respectively. Finally, conclusions are +discussed in Sect. 6. +2. Numerical methods +We have analyzed numerical simulations of wave propaga- +tion in a sunspot umbra computed with the code MANCHA +(Khomenko & Collados 2006; Felipe et al. 2010). These +are the same simulations previously studied in Felipe et al. +(2021a,b). In those works, we describe the numerical setup, +the calculation of the synthetic Ca ii 8542 Å spectropolari- +metric profiles, and the inversion of those synthetic pro- +Fig. 1. Stokes profiles and atmospheric models from an umbral +flash. Top panels: Stokes I (panel a) and Stokes V (panel b) pro- +files. Bottom panels: vertical stratification of the velocity (panel +c), temperature (d), and magnetic field (e) as a function of the +optical depth (bottom axis) and geometrical height (top axis). +In all panels, the red lines illustrated the quantities directly ob- +tained from the numerical simulation, whereas black and grey +lines are the results obtained from inversions with one or three +velocity nodes, respectively. +files (only in Felipe et al. 2021b). We refer the reader to +those publications for a detailed description of the numeri- +cal methods. For completeness, in this manuscript we briefly +describe them. +The numerical code computes the evolution of the per- +turbations to a background model. This model is a modified +Avrett (1981) umbral model, which was extended to the +solar interior and corona. Simulations were computed us- +ing the 2.5D approximation (two-dimensional domain but +keeping the three coordinates from vectors). The vertical +domain spans from z = −1.14 Mm to z = 3.50 Mm, with +a constant vertical step of 10 km. In the horizontal dimen- +sion, we set the same background stratification at all spa- +tial positions (96 points with a horizontal spatial step of 50 +km), and periodic boundary conditions were established. +The background model is permeated by a constant vertical +magnetic field with a strength of 2000 G. +Waves are excited by a driver that reproduces ac- +tual umbral oscillations, including typical photospheric and +chromospheric velocity amplitudes and power spectra. The +spatial and temporal evolution of the driver were retrieved +from photospheric umbral observations acquired with a slit +spectrograph in the Si i 10827 Å line (Felipe et al. 2018). +The driver was introduced as a vertical force directly imple- +Article number, page 2 of 9 + +T. Felipe and H. Socas-Navarro: Opacity effects in NLTE inversions +mented in the equations, following Felipe et al. (2011) and +Felipe & Sangeetha (2020). It changes along the horizontal +dimension of the computational domain, corresponding to +the direction along the slit of the spectrograph. This setup +introduces horizontal variations in a simulation that other- +wise would be purely one-dimensional (the magnetic field +is constant and the stratification of the background is the +same for all spatial positions). Instead, a two-dimensional +simulation was employed to improve the statistical signif- +icance of the results since it allows us to sample a larger +number of spectral profiles (including the synthesis and in- +version of numerous umbral flashes with various properties) +and to compute the spatial average of some of the quantities +of interest, such as power spectra. +Synthetic spectropolarimetric Ca ii 8542 Å profiles were +computed by feeding the NLTE code NICOLE (Socas- +Navarro et al. 2015) with the output from the simulation. +The spectral resolution has been degraded by convolving +the high-resolution profiles with Gaussians with an FWHM +of 100 mÅ, obtaining a spectral step of 55 mÅ (approx- +imately the Nyquist frequency of the CRISP instrument +at 8542 Å; Scharmer et al. 2008). Random noise has been +added to produce profiles with a signal-to-noise of 1 × 10−3 +in units of continuum intensity, comparable to that ob- +tained in actual umbral flash observations (e.g., de la Cruz +Rodríguez et al. 2013). +The central part of the umbra (46 points spanning 2.3 +Mm in the horizontal direction) have been inverted for +the whole temporal series (55 min of simulations/syntheses +with a cadence of 5 s). A total of 30,360 profiles have been +inverted. The NICOLE code was also employed to carry out +these inversions. Two independent inversions of the whole +set were performed. All of them employed a single cycle, +with 6 nodes in temperature and 3 nodes in vertical mag- +netic field. Since the simulation has a purely vertical mag- +netic field and Stokes Q and U exhibit very weak signals, +we did not invert the transversal magnetic field. Three ve- +locity nodes were selected in one of the inversion sets, while +in the other a single velocity node was imposed. +An example of the simulated atmospheric models, their +corresponding synthetic profiles, and the outcome from the +inversions during an umbral flash is illustrated in Figure +1. Both inversion setups (with a different number of veloc- +ity nodes) provide a good fit of the Stokes profiles and a +fair characterization of the atmospheric stratification. The +inversion with one velocity node (black lines) captures the +actual velocity around log τ = −5, where the sensitivity of +the Ca ii 8542 Å line to velocity is maximum during umbral +flashes (see Section 4.2). The inferred chromospheric mag- +netic field exhibits a discrepancy of almost 200 G between +both inversions. See Felipe et al. (2021b) for a discussion of +the limitations of the Ca ii 8542 Å line to measure magnetic +fields. +3. Temporal evolution of Ca ii 8542 Å response +functions +Umbral chromospheres experience remarkable changes dur- +ing the passage of waves. Fluctuations in velocity, temper- +ature, and density modify the atmospheric stratification, +with strong implications for radiative transfer. The imprint +of these fluctuations is clearly visible in the Stokes profiles, +which develop into umbral flashes. They also produce vari- +Fig. 2. Response functions of the Ca ii 8542 Å intensity to line- +of-sight velocity (left panels) and temperature (right panels) at a +randomly chosen time step and spatial position. Top panels show +the dependence of the normalized response function with wave- +length and optical depth. Bottom panels illustrates the response +functions integrated in the core of the line. The spectral region +employed for the integral is delimited by the horizontal dashed +lines in the top panels. Vertical red lines with different thickness +indicate the optical depth (bottom axis) and geometrical height +(top axis) where the normalized integrated response function +is maximum (thicker line), above 0.7 (middle thick lines), and +above 0.5 (thinner lines). +ations in the optical depth, source function, and opacity, +which must be taken into account to understand the out- +put radiation. In this work, we characterize the contribution +of the different atmospheric layers to the measured spectra +by computing the response functions of the intensity. +Figure 2 illustrates the response functions of Ca ii 8542 +Å intensity to velocity and temperature for a case when +the core of the line is in absorption. A positive (negative) +value of the response function indicates that an increase in +the velocity/temperature at that optical depth will produce +an increase (reduction) of the intensity at the corresponding +wavelength of the line profile. In the illustrated example, the +main contribution of the velocity to the line profile is con- +centrated at chromospheric heights. The two lobes with op- +posite signs indicate the intensity changes associated with +the Doppler shift. A positive (downward) velocity shifts the +line core to higher wavelengths and, thus, the intensity will +increase at one side of the line and will decrease and the +other side. In this work, we focus on the study of the chro- +mospheric oscillations that can be measured with the Ca ii +8542 Å line. In the following, we will restrict the discus- +sion of the response functions to those computed for the +core of the spectral line. With this aim, we have integrated +the absolute value of the response functions in the central +part, in the range ±17.5 mÅ (for velocity) and ±7.5 mÅ +(for temperature) from the line center. Bottom panels from +Figure 2 show the integrated response functions. +Figure 3 illustrates an umbral flash event, including the +Stokes I and V profiles at certain time steps and their cor- +responding intensity response functions to temperature and +velocity. The response functions are plotted as a function +of geometrical height. They are computed from the simu- +lated models, where we have access to an accurate stratifi- +cation in geometrical scale. This situation differs from the +analysis of observations, where the inversions return the +stratification in optical depth and the geometrical height is +Article number, page 3 of 9 + +A&A proofs: manuscript no. main +Fig. 3. Temporal evolution of a synthetic umbral flash in Ca ii +8542 Å. Panel a: Vertical velocity (black solid line, left axis), +temperature perturbation (black dashed line, right axis), and +density perturbation (red dashed line, red right axis) as a func- +tion of time. The blue-shaded area denotes the times when the +core of the line is in emission. Vertical dotted lines indicate the +time steps plotted in panels b-s. Bottom panels: Stokes I (left +column), Stokes V (middle column), and response functions of +the intensity to velocity (solid line) and temperature (dashed +line) as a function of geometrical height (right column). Each +row correspond to the time shown in the left column and indi- +cated by vertical dotted lines in panel a. +calculated assuming hydrostatic equilibrium. The validity +of this assumption in the case of umbral flashes is limited +since they take place in a magnetized atmosphere and are +associated with strong plasma flows. Our approach is not +affected by this limitation. +The top panel of Figure 3 shows the chromospheric tem- +poral evolution at a randomly chosen position during one +period (around three minutes). During this time, an um- +bral flash is developed, as seen in the appearance of the +intensity emission core (left column) and the reversal of +Stokes V (middle column). The core-integrated response +functions (right column) exhibit remarkable variations dur- +ing the different phases of the umbral flash. Initially, when +the atmosphere is approximately at rest (t1 = 1115 s), the +response function to velocity (panel d) shows a high contri- +bution from a broad range of heights (around 1 Mm wide), +with the maximum at z ∼ 0.9 Mm. In contrast, during +the umbral flash (e.g., t5 = 1210 s), the response function +to velocity is narrower and the peak is shifted to z ∼ 1.2 +Mm. The core-integrated response function of the intensity +to temperature exhibits even more striking variations than +the response functions to velocity. Whereas during the um- +bral flash (last three rows) they are similar, in the initial +stages the response function to temperature shows a peak +at z ∼ 0.5 Mm, indicating that at those time steps the in- +formation from Stokes I contains a remarkable contribution +from the high-photosphere. +The results illustrated in Figure 3 clearly show that the +radiation measured in the core of the Ca ii 8542 Å line not +only includes the imprint from a wide range of heights but +also that those heights significantly change during the evo- +lution of umbral flashes. +4. Velocity oscillations +4.1. Comparison of velocity signals at constant geometrical +height and constant optical depth +The velocity measured from the core of the Ca ii 8542 Å line +samples atmospheric heights in the range z ∼ [0.6, 1.5] Mm +(Figure 3). Figure 4 illustrates a comparison between veloc- +ity oscillations at fixed geometrical heights in that range, +from z = 0.59 Mm (green) to z = 1.39 Mm (violet), with +velocity fluctuations at constant optical depth. For the lat- +ter, three different measurements are plotted: velocities ob- +tained from the inversions of the synthetic profiles with one +(black lines) or three (grey lines) velocity nodes, and the +actual velocity directly obtained from the simulation (red +lines). The illustrated signals at constant optical depth cor- +respond to the average velocity between log τ = −4.8 and +log τ = −5.4 since this range provides the main contribu- +tion to the core of the Ca ii 8542 Å, as given by the exam- +ination of the response functions. +The velocity inferred from the inversions captures fairly +well the actual oscillations at constant optical depth (top +panel from Figure 4). Both inversions (which differ in the +number of velocity nodes) show similar results, although +the agreement with the actual simulated velocity of the in- +version with three velocity nodes is slightly better (see the +amplitude of the upflows (negative velocity) at t ∼ 600 s +and t ∼ 700). However, there is a systematic shift in the +maximum positive velocity (downflows) when the higher +chromospheric layers (above z ∼ 1.1 Mm) exhibit a high +amplitude (wavefronts between t = 800 s and t = 1900 +s). In those cases, the maximum positive velocity obtained +from the inversions (black and grey lines) takes place in +phase with the velocity at z ∼ 1.1 Mm and lags the max- +imum velocity at constant optical depth from the simula- +tion (red line). The velocity fluctuations at constant optical +depth (both from inversions and actual simulated values) +can hardly be associated with a single geometrical height. +Instead, they exhibit contributions from different layers, +which depend on the phase of the oscillation. +Article number, page 4 of 9 + +T. Felipe and H. Socas-Navarro: Opacity effects in NLTE inversions +Fig. 4. Comparison between velocity oscillations in geometri- +cal scale, optical depth, and those inferred from the inversion of +synthetic Ca ii 8542 Å profiles. Top panel: temporal evolution of +the vertical velocity at a randomly chosen spatial position. Bot- +tom panel: Power spectra of the vertical velocity averaged for +all the spatial position from the numerical simulation. In both +panels, the black lines correspond to the velocity inferred from +inversions of the synthetic profiles with one node in velocity, the +grey lines indicate the velocity inferred from inversions with 3 +nodes in velocity averaged in log τ = [−4.8, −5.4], and the red +lines correspond to the actual velocity from the simulations av- +eraged in the same range of optical depths. Color lines with a +gradient green-blue-violet represent the velocity (top panel) and +power spectra (bottom panel) at constant geometrical height. +They span from z = 590 km to z = 1390 km, which is approxi- +mately the range of geometrical heights where the Ca ii 8542 Å +line is sensitive to the velocity (Figure 3), with a step of 50 km +between consecutive lines. The legend in the bottom panel indi- +cates the heights corresponding to some colors, as a reference. +The power spectra (bottom panel from Figure 4) show +that Ca ii 8542 Å inversions provide an excellent charac- +terization of the main frequency (period) of chromospheric +oscillations. The power of that main peak is comparable +to that from oscillations at z ∼ 0.75 Mm. In contrast, dis- +crepancies are found at higher frequencies. Oscillations at +constant optical depth exhibit a secondary power peak at +around 13 mHz. This peak is absent in the power peak of +oscillations at constant geometrical height (only the power +spectra at z ∼ 1.2 Mm shows some hints of a power en- +hancement at that frequency). This power excess is not an +artifact from the solution of the inversion problem, but it +is already present in the oscillations at a constant opti- +cal depth directly extracted from the simulation (red line). +This fact points to the change in the layers where the spec- +tral line is sensitive as a key feature to interpret oscillations +in the 10-15 mHz band. In the case of even higher frequen- +cies, the inversions greatly depart from the actual simulated +power. The small amplitude of these fluctuations is beyond +the expected precision of the inversion results. +Fig. 5. Oscillations in a randomly chosen location of the sim- +ulated umbra in the high chromosphere and low chromosphere +(top two panels) and the regions where the Ca ii 8542 Å inten- +sity is sensitive to the velocity (bottom two panels). Top panels +show the temporal evolution of the vertical velocity (panel a) +and temperature (middle panel) at constant geometrical heights. +The colors have the same meaning as in Figure 4. Bottom pan- +els illustrate the range of optical depths (panel c) or geometrical +heights (panel d) where the Ca ii 8542 Å intensity is sensitive +to the velocity, as given by the examination of the response +functions. The thickest lines indicate the layer where the re- +sponse function is maximum. Thinner lines delimit successively +the range of heights where the normalized response function is +above 0.7, and 0.5. Blue-shaded regions indicate the times when +the core of the line is in emission. +4.2. Fluctuations in the formation height of the Ca ii 8542 Å +line core +The response functions of intensity to velocity have been +computed for all the atmospheres in the central part of the +simulation, individually for each of the 660 time steps. Fig- +ure 5 illustrates the variations in the optical depths (panel +c) and geometrical heights (panel d) where the line core is +sensitive to velocity. For each atmospheric model, we have +integrated the response function in the central wavelengths +and determined the height of the maximum response (thick +red line) and the heights where the normalized response +function exhibit some selected values (red lines with smaller +thickness as lower response values are considered). The def- +inition of these lines is illustrated in the bottom panels from +Figure 2. +The heights with higher sensitivity to the velocity ex- +hibit periodic behavior, both in optical depth and geomet- +rical height. During umbral flashes (blue-shaded areas, tak- +ing place at the times when the chromospheric temperature +is maximum) the core of the Ca ii 8542 Å line is sensi- +tive to a deeper optical depth (with the maximum response +at around log τ = −5.0). Interestingly, this deeper optical +Article number, page 5 of 9 + +A&A proofs: manuscript no. main +depth is associated with high geometrical heights (around +z = 1.2 Mm). Generally, the geometrical height where the +response of the core is maximum is maintained approxi- +mately constant during the entire umbral flash event. In +contrast, during the quiescent phase of the oscillations, the +core of the line is in absorption and the peak of its sensi- +tivity is shifted to optical depths around log τ = −5.7 and +geometrical heights as low as z = 0.7 Mm. +Fig. 6. Power spectra of the formation height of the Ca ii 8542 +Å line averaged for all locations inside the umbra. The lines rep- +resent the maximum of the response function (black line), the +minimum height where the response function of the intensity to +velocity is above a selected threshold (red lines), and the maxi- +mum height where the response function is above the threshold +(blue lines). The thickness of the color lines indicates the thresh- +old concerning the normalized response function, corresponding +to 0.7, 0.5, and 0.3 from thicker to thinner. +The periodicity of the fluctuations in the formation +height of the line core has been evaluated by computing the +average power spectra of the geometrical heights plotted in +Figure 5 that characterize the regions where the line is sen- +sitive to velocity. Figure 6 shows that the main frequency of +the oscillations is 6 mHz, in agreement with the oscillations +in the 3-minute band widely reported from chromospheric +umbral oscillations. Some secondary peaks appear at 10- +16 mHz, especially for the peak of the response function +(black line) and the lower height of the selected contours +(red lines). These peaks coincide with the power enhance- +ment found in the velocity inferred from the inversions (bot- +tom panel from Figure 4) and point to fluctuations in the +sensitivity of the line core to height as the origin of the +measured secondary power peak in velocity at 12-13 mHz +(Figure 4). +4.3. Velocity gradients and opacity effects +The effective formation height of the Ca ii 8542 Å line core +significantly changes with the atmospheric variations pro- +duced by umbral oscillations. The presence of vertical gra- +dients in the stratification of the atmosphere can thus lead +to spurious oscillations in the quantities inferred from the +interpretation of the spectral line. In addition to the in- +trinsic oscillations associated with wave travel, a second +component produced by changes in the opacity can leave +an imprint in the measurements. +Figure 7 illustrates the stratification of the vertical ve- +locity between the high photosphere and the chromosphere +during ∼18 min of simulation. In the top panel, the veloc- +Fig. 7. Top panel: Vertical velocity as a function of height (verti- +cal axis) and time (horizontal axis) at a randomly chosen spatial +position. Red lines have the same meaning as those in the bot- +tom panel from Figure 5. Bottom panel: Vertical stratification +of the velocity at some selected time steps. The color of the line +indicates the corresponding time step, as given by the dashed +vertical lines in the top panel. +ity (grey scale) is saturated outside the ±1 km s−1 range to +better visualize its variations in the layers where the core +of the line is formed (indicated by the red lines). The bot- +tom panel shows the vertical velocity at some selected time +steps. All of them correspond to times when the response +function of the Ca ii 8542 Å line core is undergoing striking +variations, with its maximum shifting from lower to higher +layers (blueish lines) or vice versa (greenish lines). During +those time steps, the atmospheric velocity exhibits strong +vertical gradients. The combination of vertical gradients in +the velocity with the variations in the layers where the line +core is sensitive to velocity produces the aforementioned +additional component in the measured velocity as a result +of opacity changes. +5. Temperature oscillations +Figure 3 shows that the response function of the intensity +to temperature also experiences striking variations during +umbral oscillations. The range of heights where the tem- +perature is probed by the core of the Ca ii 8542 Å line is +even broader than that for the velocity, with the peak of +the response function changing from below z ∼ 0.5 Mm +(t2 = 1140 s) up to z ∼ 1.2 Mm (t4 = 1190 s). The top +panel from Figure 8 shows the periodicity of these fluctu- +ations. The region where the line core is sensitive to tem- +perature exhibits sudden variations from the photosphere +(when the temperature perturbation is low) to the chromo- +sphere (at those times when the chromospheric temperature +increases). In the umbral atmospheres computed during the +simulation, the chromosphere is generally ∼2000 K hotter +than the photosphere (bottom panel from 8). This way, the +intrinsic temperature enhancements produced by the waves +will be accompanied by an additional temperature increase +caused by the displacement in the region where the line is +sensitive to temperature. +Article number, page 6 of 9 + +T. Felipe and H. Socas-Navarro: Opacity effects in NLTE inversions +Fig. 8. Top panel: Temperature as a function of height (vertical +axis) and time (horizontal axis) at a randomly chosen spatial po- +sition. Red lines indicate the height of the peak of the response +function of intensity to temperature (thickest line) and the range +of heights where the normalized response function is above 0.7 +(medium-thick line), and 0.5 (thinner line). Bottom panel: Verti- +cal stratification of the temperature at some selected time steps. +The color of the line indicates the corresponding time step, as +given by the dashed vertical lines in the top panel. +The comparison between the temperature oscillations +at constant geometrical height and constant optical depth +illustrated in Figure 9 shows clear indications of height vari- +ations in the origin of the temperature signals. During the +first 500 s of simulation, when the fluctuations driven under +the photosphere have not yet reached upper atmospheric +layers and the chromosphere is approximately at rest, the +temperature at constant optical depth (both from inver- +sions and directly extracted from the simulation) is con- +sistent with the temperature at z ∼ 0.89 Mm (light-blue +line). However, later times exhibit oscillations with ampli- +tude significantly higher than that found for that geomet- +rical height. When the chromospheric temperature is maxi- +mum (during umbral flashes), the temperature at constant +optical depth is comparable to the temperature at z ∼ 1.2 +Mm (dark-blue/violet line). In contrast, the lowest values +of the temperature at constant optical depth are similar to +the temperature measured at z ∼ 0.5 Mm (light green line). +The temperature oscillations inferred from the inversions of +the Ca ii 8542 Å line show a fair agreement with the real +temperature oscillations directly extracted from the output +of the simulation, although some discrepancies are found at +the times when the temperature is maximum. +The oscillations at constant optical depth have peak- +to-peak amplitude around 2000 K, although the strongest +wavefronts exhibit a change between the minimum and +maximum temperature up to 4000 K. This amplitude is +comparable to that from oscillations at z ∼ 1.3 Mm (vio- +let line in Figure 9). This is also illustrated by the power +spectra (bottom panel from Figure 9). The frequency of +the main power peak at 6 mHz, present in oscillations at +constant geometrical height and optical depth, is perfectly +reproduced by the inversion results. Its power is similar to +the power from oscillations at z ∼ 1.3 Mm, which is the +Fig. 9. Comparison between temperature oscillations in geomet- +rical scale, optical depth, and those inferred from the inversion +of synthetic Ca ii 8542 Å profiles. Top panel: temporal evolution +of the temperature at a randomly chosen spatial position. Bot- +tom panel: Power spectra of the temperature averaged for all +the spatial positions from the numerical simulation. The color +lines have the same meaning as in Figure 4. +upper limit of the region sampled by the Ca ii 8542 Å line +core. In contrast, the power of the oscillations at ∼ 0.89 Mm +(whose temperature is probed by the core of the line when +the atmosphere is approximately at rest, first 500 s of simu- +lation) is almost two orders of magnitude lower. This result +indicates that opacity oscillations are a significant compo- +nent of the total temperature variations measured during +umbral flashes. Both intrinsic and opacity oscillations take +place in phase, with the core of the line shifting to upper +(hotter) layers when the temperature of the intrinsic oscil- +lations increases. +Similarly to the case of the velocity (Figure 4), the power +of temperature oscillations at constant optical depth also +exhibits a secondary peak at around 12-13 mHz (red line in +the bottom panel from Figure 9). However, the strength of +this power peak is lower than the fluctuations that can be +captured by the inversions. +6. Discussion and conclusions +In this paper, we have characterized the relationship be- +tween the quantities inferred from inversions of the Ca ii +8542 Å line and the intrinsic oscillations at constant geo- +metrical height. The former is affected by the opacity effect, +which changes the response height of the spectral line and +severely affects the interpretation of the results. This goal +has been addressed through numerical simulations of wave +Article number, page 7 of 9 + +A&A proofs: manuscript no. main +propagation in an umbral model. The stratification of the +simulated velocity and temperature fluctuations has been +compared with the fluctuations of the same quantities at +constant optical depth. In addition, synthetic Ca ii 8542 Å +profiles have been computed from the output of the simu- +lations, and the inferences from their inversions have also +been critically evaluated. +Our analysis focused on the examination of the chro- +mospheric velocity and temperature fluctuations, obtained +by averaging the atmosphere (both the actual simulated +model and those inferred from the inversion of the syn- +thetic profiles) in a range of optical depths where the Ca ii +8542 Å line is generally sensitive to changes in the atmo- +sphere. Our results show that inversions of the Ca ii 8542 +Å line provide a good characterization of the velocity os- +cillations in the low chromosphere. They capture the am- +plitude (and power) of the oscillations and the frequency +of the main power peak at around 6 mHz, indicating that +they provide a reliable estimation of the main oscillatory +period. However, some discrepancies are also obvious. The +time steps of the velocity maximum (downflow) inferred +from the inversions lag those found in the simulations at +the low chromosphere (around 60 s delay). Instead, they +go in phase with the maximum velocity and higher mid- +chromospheric layers (top panel from Fig. 4). This points +to a change in the optical depths where the Ca ii 8542 Å line +is sensitive to the velocity during different phases of the os- +cillation. An examination of the evolution of the response +function shows that during the downflowing phase in the +mid-chromosphere (z ∈ [1000, 1400]), the response of the +line is shifted upward to the range log τ ∈ [−5.0, −6.1], with +the maximum of the response reaching up to log τ = −5.9 +(Fig. 2). +The velocity power spectrum inferred from the inver- +sions with three velocity nodes closely matches the sim- +ulated spectrum (averaged in log τ) for frequencies lower +than 8 mHz. For higher frequencies, the inferred power is +significantly stronger than the actual simulated power. In- +terestingly, the simulated power averaged in log τ exhibits a +remarkable dip at around 9.5 mHz (bottom panel from Fig. +4). This dip is absent in the power computed at constant +geometrical heights. The dip and subsequent peak are par- +tially captured by the inversions, which show a power excess +between 12 and 14 mHz. The examination of the response +functions shows that a similar power peak is found in the +height where the line is sensitive to the velocity (Fig. 6) +and that changes in the response height take place when +strong velocity gradients are present (Fig. 8). The presence +of power peaks (or dips) at the sunspot chromosphere has +been suggested to be caused by the existence of a chro- +mospheric resonant cavity (Jess et al. 2020; Felipe et al. +2020) or by harmonics resulting from the non-linearity of +the three-minute oscillations (Chae et al. 2018; Felipe 2021; +Chai et al. 2022). Our results suggest caution with the in- +terpretation of power peaks with frequencies higher than 9 +mHz in Ca ii 8542 Å observations. +The opacity effect also has a remarkable impact on the +measured temperature fluctuations. The region where the +core of the Ca ii 8542 Å line exhibits a significant response +to temperature oscillates between z ∼ 500 km and z ∼ 1200 +km, that is, a region with strong vertical gradients in tem- +perature (Fig. 8). Temperature fluctuations produced by +the opacity effect are in phase with the intrinsic tempera- +ture oscillations associated with wave propagation. When +the temperature (at constant geometrical height) increases, +the response of the line is shifted to upper (hotter) lay- +ers. This leads to stronger inferred temperature fluctua- +tions, which exhibit a high power in the three-minute band +(higher than the power in most of the geometrical heights +where the core of the line forms). The frequency of this +power peak is well captured by the inversions. +Our goals are in line with the recent work by Keys +et al. (2021). They evaluated the inversion results of os- +cillations in the photospheric 6301 Å and 6302 Å lines us- +ing a similar approach of comparing the output from the +inversions with known simulated atmospheres. Their anal- +ysis focused on short-period (24 s) waves, finding a good +match between simulations and inversions except for some +discrepancies during the passage of waves. These deviations +are due to the small height range perturbed by the waves +in comparison with the range where the lines are sensitive. +Our results for chromospheric fluctuations inferred from a +line formed under NLTE conditions indicate that oscilla- +tions with frequencies higher than 9 mHz (periods shorter +than ∼ 110 s) are hardly well characterized by the out- +put of the inversions. Low-frequency acoustic waves (or, +more precisely, slow magnetoacoustic waves) have a longer +wavelength, which is comparable to the range of heights +where the Ca ii 8542 Å line is formed. However, at chromo- +spheric heights the temperature and velocity change at spa- +tial scales much shorter than the photon free path, which +makes it impossible to precisely determine their vertical +stratification. The inversion can only be interpreted as the +average properties in a region large enough to affect the +emerging radiation. This way, our analysis has focused on +the average chromospheric fluctuations. A comparison be- +tween the models inferred from the inversions and the ac- +tual vertical stratification is illustrated in Fig. 1. This limi- +tation could be lessened by performing multi-line inversions +with several spectral lines that probe similar atmospheric +layers. +Acknowledgements. Financial +support +from +grants +PGC2018- +097611-A-I00 and PID2021-127487NB-I00, funded by MCIN/AEI/ +10.13039/501100011033 and by “ERDF A way of making Europe” is +gratefully acknowledged. TF acknowledges grant RYC2020-030307-I +funded +by +MCIN/AEI/ +10.13039/501100011033 +and +by +“ESF +Investing in your future”. We acknowledge the contribution of Teide +High-Performance Computing facilities to the results of this research. +TeideHPC facilities are provided by the Instituto Tecnológico y de +Energías Renovables (ITER, SA). URL: http://teidehpc.iter.es. +References +Avrett, E. H. 1981, in The Physics of Sunspots, ed. L. E. Cram & +J. H. Thomas, 235–255 +Bellot Rubio, L. R., Collados, M., Ruiz Cobo, B., & Rodríguez Hi- +dalgo, I. 2000, ApJ, 534, 989 +Borrero, J. M. & Ichimoto, K. 2011, Liv. Rev. Solar Phys., 8 +Bose, S., Henriques, V. M. J., Rouppe van der Voort, L., & Pereira, +T. M. D. 2019, A&A, 627, A46 +Cauzzi, G., Reardon, K. P., Uitenbroek, H., et al. 2008, A&A, 480, +515 +Centeno, R., Collados, M., & Trujillo Bueno, J. 2006, ApJ, 640, 1153 +Chae, J., Cho, K., Song, D., & Litvinenko, Y. E. 2018, ApJ, 854, 127 +Chai, Y., Gary, D. E., Reardon, K. 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K. 2003, A&A Rev., 11, 153 +Uitenbroek, H. 2003, ApJ, 592, 1225 +Vicente Arévalo, A., Asensio Ramos, A., & Esteban Pozuelo, S. 2022, +ApJ, 928, 101 +Article number, page 9 of 9 + +This figure "Orcid-ID.png" is available in "png"� format from: +http://arxiv.org/ps/2301.03273v1 + diff --git a/zNE1T4oBgHgl3EQfkQT7/content/tmp_files/load_file.txt b/zNE1T4oBgHgl3EQfkQT7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..be1cd57a9b3de17625c884899dbbbc28f5fa669b --- /dev/null +++ b/zNE1T4oBgHgl3EQfkQT7/content/tmp_files/load_file.txt @@ -0,0 +1,629 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf,len=628 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='03273v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='SR] 9 Jan 2023 Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' main ©ESO 2023 January 10, 2023 Impact of opacity effects on chromospheric oscillations inferred from NLTE inversions T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Felipe1, 2 and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Socas-Navarro1, 2 1 Instituto de Astrofísica de Canarias, 38205, C/ Vía Láctea, s/n, La Laguna, Tenerife, Spain 2 Departamento de Astrofísica, Universidad de La Laguna, 38205, La Laguna, Tenerife, Spain January 10, 2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Spectropolarimetric inversions are a fundamental tool to diagnose the solar atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Chromospheric infer- ences rely on the interpretation of spectral lines that are formed under Non Local Thermodynamic Equilibrium (NLTE) conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In the presence of oscillations, changes in the opacity impact the response height of the spectral lines and hinder the determination of the real properties of the fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' We aim to explore the relationship between the chromospheric oscillations inferred by NLTE inversion codes and the intrinsic fluctuations in velocity and temperature produced by the waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Numerical simulations of wave propagation in a sunspot umbra have been computed with the code MANCHA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The NLTE synthesis and inversion code NICOLE has been used to compute spectropolarimetric Ca ii 8542 Å line profiles for the atmospheric models obtained as the output from the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The synthetic profiles have been inverted and the inferences from the inversions have been compared with the known atmospheres from the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' NLTE inversions of the Ca ii 8542 Å line capture low frequency oscillations, including those in the main band of chromospheric oscillations around 6 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In contrast, waves with frequencies above 9 mHz are poorly characterized by the inversion results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Velocity oscillations at those higher frequencies exhibit clear insights of opacity fluctuations since the power of the signal at constant optical depth greatly depart from the power of the oscillations at constant geometrical height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The main response of the line to velocity fluctuations comes from low chromospheric heights, whereas the response to temperature shows sudden jumps between the high photosphere and the low chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' This strong variation in the heights where the line is sensitive to temperature is revealed as a strong oscillatory power in the inferred fluctuations, much stronger than the actual power from the intrinsic temperature oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Our results validate the use of NLTE inversions to study chromospheric oscillations with frequencies below ∼9 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' However, the interpretation of higher frequency oscillations and the power of temperature oscillations must be addressed with care since they exhibit signatures of opacity oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Methods: numerical – Sun: chromosphere – Sun: oscillations – sunspots – Techniques: polarimetric 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Introduction The study of the solar atmosphere heavily relies on the observation and interpretation of the solar spectra, often not only using spectroscopic data but also full polarimetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The Ca ii 8542 Å line is one of the most exploited spectral lines for probing the solar chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The interpreta- tion of its spectral profiles requires NLTE diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Sev- eral NLTE inversion codes have been developed with this aim, such as NICOLE (Socas-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2015), STiC (de la Cruz Rodríguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2019), SNAPI (Milić & van Noort 2018), and DeSIRe (Ruiz Cobo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Due to the large amount of computational resources required by these inversions, pioneering studies using these tools were mostly restricted to analyzing a few spectral profiles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=', Socas-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2000) or spatially coherent maps with a reduced resolution for a few time steps (de la Cruz Ro- dríguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Until recently, chromospheric oscilla- tions have been out of the scope of the works using NLTE inversions since they require the analysis of long temporal series with high temporal cadence and, thus, the inversion of numerous spectral profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Several works have employed alternative techniques to derive the chromospheric plasma properties from the interpretation of the Ca ii 8542 Å line, such as the lambdameter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=', Chae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2018) or bi-sector (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=', Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2022) methods to infer the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' How- ever, in umbral regions, these methods are challenged by the common display of emission near the core of the line as a manifestation of umbral flashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The interpretation of these profiles requires sophisticated analysis tools, like inversion codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Thanks to the improvement of the computational capa- bilities, recent works have been able to perform more com- prehensive studies of sunspot chromospheres using NLTE inversions of the Ca ii 8542 Å line (Henriques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Joshi & de la Cruz Rodríguez 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Henriques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Houston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Also, new methods based on machine learning techniques are being developed to diagnose the so- lar chromosphere in a fast and computationally efficient way (Vicente Arévalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The availability of physical information from larger maps and longer temporal series will enable the study of their oscillations in the common ground of Fourier and/or wavelet analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In the dynamic solar atmosphere, the contribution of different atmospheric heights to the formation of a spectral Article number, page 1 of 9 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' main line changes with time (Uitenbroek 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' These variations are especially troubling for the study of solar oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Spurious oscillations produced by the change in the forma- tion height of the spectral lines in atmospheres with vertical gradients (known as opacity effects) can overlap the intrin- sic fluctuations due to wave propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Disentangling the intrinsic oscillations from those produced by the opacity ef- fect is thus fundamental for a proper interpretation of wave phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Many reported magnetic field fluctuations in sunspots have been interpreted as a result of opacity effects (Bellot Rubio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Rüedi & Cally 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Khomenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Active regions are known to harbor vertical mag- netic field gradients (see Solanki 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Borrero & Ichimoto 2011, for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In these atmospheres, the periodical displacements of the formation region of spectral lines in- troduce spurious oscillations in the inferred magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In contrast, the opacity effect is barely discussed in the ex- amination of oscillations in other quantities, such as tem- perature and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Indeed, the lower solar atmosphere exhibits significant vertical gradients in temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Ve- locity gradients are also expected since the amplitude of the oscillations increases with height due to the drop of the density (Centeno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Both vertical gradients will certainly leave an imprint in the temperature/velocity fluc- tuations measured from any spectral line formed at those heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In this study, we focus on the umbral oscillations in- ferred from the analysis of the Ca ii 8542 Å line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' This spec- tral line is sensitive to a broad range of heights, from the photosphere at the wings to the chromosphere at the core of the line (Cauzzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Nowadays, it is one of the most employed lines for the study of the solar chromosphere (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=', Socas-Navarro 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Kleint 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Rouppe van der Voort & de la Cruz Rodríguez 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Kuridze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Murabito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' It is also commonly used for the inspection of umbral oscillations and, more specifically, the development of umbral flashes (Socas-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' de la Cruz Rodríguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Henriques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Houston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Houston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Variations in the geometrical heights where the Ca ii 8542 Å line is sensitive during umbral flashes have been reported from the analysis of observations (Joshi & de la Cruz Rodríguez 2018) and numerical modeling (Felipe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Here, we address how how this opacity effect im- pacts the chromospheric umbral oscillations inferred with the Ca ii 8542 Å line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' We have constructed synthetic Stokes profiles from the atmospheres computed with numerical simulations and then inverted those profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The numerical methods are briefly described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2 and the response functions of the line are introduced in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 4 and 5 we present the results obtained for velocity and tem- perature oscillations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Finally, conclusions are discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Numerical methods We have analyzed numerical simulations of wave propaga- tion in a sunspot umbra computed with the code MANCHA (Khomenko & Collados 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Felipe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' These are the same simulations previously studied in Felipe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' (2021a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In those works, we describe the numerical setup, the calculation of the synthetic Ca ii 8542 Å spectropolari- metric profiles, and the inversion of those synthetic pro- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Stokes profiles and atmospheric models from an umbral flash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Top panels: Stokes I (panel a) and Stokes V (panel b) pro- files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Bottom panels: vertical stratification of the velocity (panel c), temperature (d), and magnetic field (e) as a function of the optical depth (bottom axis) and geometrical height (top axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In all panels, the red lines illustrated the quantities directly ob- tained from the numerical simulation, whereas black and grey lines are the results obtained from inversions with one or three velocity nodes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' files (only in Felipe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' We refer the reader to those publications for a detailed description of the numeri- cal methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' For completeness, in this manuscript we briefly describe them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The numerical code computes the evolution of the per- turbations to a background model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' This model is a modified Avrett (1981) umbral model, which was extended to the solar interior and corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Simulations were computed us- ing the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='5D approximation (two-dimensional domain but keeping the three coordinates from vectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The vertical domain spans from z = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='14 Mm to z = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='50 Mm, with a constant vertical step of 10 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In the horizontal dimen- sion, we set the same background stratification at all spa- tial positions (96 points with a horizontal spatial step of 50 km), and periodic boundary conditions were established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The background model is permeated by a constant vertical magnetic field with a strength of 2000 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Waves are excited by a driver that reproduces ac- tual umbral oscillations, including typical photospheric and chromospheric velocity amplitudes and power spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The spatial and temporal evolution of the driver were retrieved from photospheric umbral observations acquired with a slit spectrograph in the Si i 10827 Å line (Felipe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The driver was introduced as a vertical force directly imple- Article number, page 2 of 9 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Felipe and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Socas-Navarro: Opacity effects in NLTE inversions mented in the equations, following Felipe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' (2011) and Felipe & Sangeetha (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' It changes along the horizontal dimension of the computational domain, corresponding to the direction along the slit of the spectrograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' This setup introduces horizontal variations in a simulation that other- wise would be purely one-dimensional (the magnetic field is constant and the stratification of the background is the same for all spatial positions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Instead, a two-dimensional simulation was employed to improve the statistical signif- icance of the results since it allows us to sample a larger number of spectral profiles (including the synthesis and in- version of numerous umbral flashes with various properties) and to compute the spatial average of some of the quantities of interest, such as power spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Synthetic spectropolarimetric Ca ii 8542 Å profiles were computed by feeding the NLTE code NICOLE (Socas- Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2015) with the output from the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The spectral resolution has been degraded by convolving the high-resolution profiles with Gaussians with an FWHM of 100 mÅ, obtaining a spectral step of 55 mÅ (approx- imately the Nyquist frequency of the CRISP instrument at 8542 Å;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Scharmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Random noise has been added to produce profiles with a signal-to-noise of 1 × 10−3 in units of continuum intensity, comparable to that ob- tained in actual umbral flash observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=', de la Cruz Rodríguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The central part of the umbra (46 points spanning 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='3 Mm in the horizontal direction) have been inverted for the whole temporal series (55 min of simulations/syntheses with a cadence of 5 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' A total of 30,360 profiles have been inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The NICOLE code was also employed to carry out these inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Two independent inversions of the whole set were performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' All of them employed a single cycle, with 6 nodes in temperature and 3 nodes in vertical mag- netic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Since the simulation has a purely vertical mag- netic field and Stokes Q and U exhibit very weak signals, we did not invert the transversal magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Three ve- locity nodes were selected in one of the inversion sets, while in the other a single velocity node was imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' An example of the simulated atmospheric models, their corresponding synthetic profiles, and the outcome from the inversions during an umbral flash is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Both inversion setups (with a different number of veloc- ity nodes) provide a good fit of the Stokes profiles and a fair characterization of the atmospheric stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The inversion with one velocity node (black lines) captures the actual velocity around log τ = −5, where the sensitivity of the Ca ii 8542 Å line to velocity is maximum during umbral flashes (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The inferred chromospheric mag- netic field exhibits a discrepancy of almost 200 G between both inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' See Felipe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' (2021b) for a discussion of the limitations of the Ca ii 8542 Å line to measure magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Temporal evolution of Ca ii 8542 Å response functions Umbral chromospheres experience remarkable changes dur- ing the passage of waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Fluctuations in velocity, temper- ature, and density modify the atmospheric stratification, with strong implications for radiative transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The imprint of these fluctuations is clearly visible in the Stokes profiles, which develop into umbral flashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' They also produce vari- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Response functions of the Ca ii 8542 Å intensity to line- of-sight velocity (left panels) and temperature (right panels) at a randomly chosen time step and spatial position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Top panels show the dependence of the normalized response function with wave- length and optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Bottom panels illustrates the response functions integrated in the core of the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The spectral region employed for the integral is delimited by the horizontal dashed lines in the top panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Vertical red lines with different thickness indicate the optical depth (bottom axis) and geometrical height (top axis) where the normalized integrated response function is maximum (thicker line), above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='7 (middle thick lines), and above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='5 (thinner lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' ations in the optical depth, source function, and opacity, which must be taken into account to understand the out- put radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In this work, we characterize the contribution of the different atmospheric layers to the measured spectra by computing the response functions of the intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Figure 2 illustrates the response functions of Ca ii 8542 Å intensity to velocity and temperature for a case when the core of the line is in absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' A positive (negative) value of the response function indicates that an increase in the velocity/temperature at that optical depth will produce an increase (reduction) of the intensity at the corresponding wavelength of the line profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In the illustrated example, the main contribution of the velocity to the line profile is con- centrated at chromospheric heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The two lobes with op- posite signs indicate the intensity changes associated with the Doppler shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' A positive (downward) velocity shifts the line core to higher wavelengths and, thus, the intensity will increase at one side of the line and will decrease and the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In this work, we focus on the study of the chro- mospheric oscillations that can be measured with the Ca ii 8542 Å line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In the following, we will restrict the discus- sion of the response functions to those computed for the core of the spectral line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' With this aim, we have integrated the absolute value of the response functions in the central part, in the range ±17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='5 mÅ (for velocity) and ±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='5 mÅ (for temperature) from the line center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Bottom panels from Figure 2 show the integrated response functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Figure 3 illustrates an umbral flash event, including the Stokes I and V profiles at certain time steps and their cor- responding intensity response functions to temperature and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The response functions are plotted as a function of geometrical height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' They are computed from the simu- lated models, where we have access to an accurate stratifi- cation in geometrical scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' This situation differs from the analysis of observations, where the inversions return the stratification in optical depth and the geometrical height is Article number, page 3 of 9 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Temporal evolution of a synthetic umbral flash in Ca ii 8542 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Panel a: Vertical velocity (black solid line, left axis), temperature perturbation (black dashed line, right axis), and density perturbation (red dashed line, red right axis) as a func- tion of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The blue-shaded area denotes the times when the core of the line is in emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Vertical dotted lines indicate the time steps plotted in panels b-s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Bottom panels: Stokes I (left column), Stokes V (middle column), and response functions of the intensity to velocity (solid line) and temperature (dashed line) as a function of geometrical height (right column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Each row correspond to the time shown in the left column and indi- cated by vertical dotted lines in panel a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' calculated assuming hydrostatic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The validity of this assumption in the case of umbral flashes is limited since they take place in a magnetized atmosphere and are associated with strong plasma flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Our approach is not affected by this limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The top panel of Figure 3 shows the chromospheric tem- poral evolution at a randomly chosen position during one period (around three minutes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' During this time, an um- bral flash is developed, as seen in the appearance of the intensity emission core (left column) and the reversal of Stokes V (middle column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The core-integrated response functions (right column) exhibit remarkable variations dur- ing the different phases of the umbral flash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Initially, when the atmosphere is approximately at rest (t1 = 1115 s), the response function to velocity (panel d) shows a high contri- bution from a broad range of heights (around 1 Mm wide), with the maximum at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='9 Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In contrast, during the umbral flash (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=', t5 = 1210 s), the response function to velocity is narrower and the peak is shifted to z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='2 Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The core-integrated response function of the intensity to temperature exhibits even more striking variations than the response functions to velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Whereas during the um- bral flash (last three rows) they are similar, in the initial stages the response function to temperature shows a peak at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='5 Mm, indicating that at those time steps the in- formation from Stokes I contains a remarkable contribution from the high-photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The results illustrated in Figure 3 clearly show that the radiation measured in the core of the Ca ii 8542 Å line not only includes the imprint from a wide range of heights but also that those heights significantly change during the evo- lution of umbral flashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Velocity oscillations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Comparison of velocity signals at constant geometrical height and constant optical depth The velocity measured from the core of the Ca ii 8542 Å line samples atmospheric heights in the range z ∼ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='5] Mm (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Figure 4 illustrates a comparison between veloc- ity oscillations at fixed geometrical heights in that range, from z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='59 Mm (green) to z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='39 Mm (violet), with velocity fluctuations at constant optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' For the lat- ter, three different measurements are plotted: velocities ob- tained from the inversions of the synthetic profiles with one (black lines) or three (grey lines) velocity nodes, and the actual velocity directly obtained from the simulation (red lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The illustrated signals at constant optical depth cor- respond to the average velocity between log τ = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='8 and log τ = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='4 since this range provides the main contribu- tion to the core of the Ca ii 8542 Å, as given by the exam- ination of the response functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The velocity inferred from the inversions captures fairly well the actual oscillations at constant optical depth (top panel from Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Both inversions (which differ in the number of velocity nodes) show similar results, although the agreement with the actual simulated velocity of the in- version with three velocity nodes is slightly better (see the amplitude of the upflows (negative velocity) at t ∼ 600 s and t ∼ 700).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' However, there is a systematic shift in the maximum positive velocity (downflows) when the higher chromospheric layers (above z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='1 Mm) exhibit a high amplitude (wavefronts between t = 800 s and t = 1900 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In those cases, the maximum positive velocity obtained from the inversions (black and grey lines) takes place in phase with the velocity at z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='1 Mm and lags the max- imum velocity at constant optical depth from the simula- tion (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The velocity fluctuations at constant optical depth (both from inversions and actual simulated values) can hardly be associated with a single geometrical height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Instead, they exhibit contributions from different layers, which depend on the phase of the oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Article number, page 4 of 9 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Felipe and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Socas-Navarro: Opacity effects in NLTE inversions Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Comparison between velocity oscillations in geometri- cal scale, optical depth, and those inferred from the inversion of synthetic Ca ii 8542 Å profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Top panel: temporal evolution of the vertical velocity at a randomly chosen spatial position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Bot- tom panel: Power spectra of the vertical velocity averaged for all the spatial position from the numerical simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In both panels, the black lines correspond to the velocity inferred from inversions of the synthetic profiles with one node in velocity, the grey lines indicate the velocity inferred from inversions with 3 nodes in velocity averaged in log τ = [−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='8, −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='4], and the red lines correspond to the actual velocity from the simulations av- eraged in the same range of optical depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Color lines with a gradient green-blue-violet represent the velocity (top panel) and power spectra (bottom panel) at constant geometrical height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' They span from z = 590 km to z = 1390 km, which is approxi- mately the range of geometrical heights where the Ca ii 8542 Å line is sensitive to the velocity (Figure 3), with a step of 50 km between consecutive lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The legend in the bottom panel indi- cates the heights corresponding to some colors, as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The power spectra (bottom panel from Figure 4) show that Ca ii 8542 Å inversions provide an excellent charac- terization of the main frequency (period) of chromospheric oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The power of that main peak is comparable to that from oscillations at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='75 Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In contrast, dis- crepancies are found at higher frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Oscillations at constant optical depth exhibit a secondary power peak at around 13 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' This peak is absent in the power peak of oscillations at constant geometrical height (only the power spectra at z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='2 Mm shows some hints of a power en- hancement at that frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' This power excess is not an artifact from the solution of the inversion problem, but it is already present in the oscillations at a constant opti- cal depth directly extracted from the simulation (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' This fact points to the change in the layers where the spec- tral line is sensitive as a key feature to interpret oscillations in the 10-15 mHz band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In the case of even higher frequen- cies, the inversions greatly depart from the actual simulated power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The small amplitude of these fluctuations is beyond the expected precision of the inversion results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Oscillations in a randomly chosen location of the sim- ulated umbra in the high chromosphere and low chromosphere (top two panels) and the regions where the Ca ii 8542 Å inten- sity is sensitive to the velocity (bottom two panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Top panels show the temporal evolution of the vertical velocity (panel a) and temperature (middle panel) at constant geometrical heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The colors have the same meaning as in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Bottom pan- els illustrate the range of optical depths (panel c) or geometrical heights (panel d) where the Ca ii 8542 Å intensity is sensitive to the velocity, as given by the examination of the response functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The thickest lines indicate the layer where the re- sponse function is maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Thinner lines delimit successively the range of heights where the normalized response function is above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='7, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Blue-shaded regions indicate the times when the core of the line is in emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Fluctuations in the formation height of the Ca ii 8542 Å line core The response functions of intensity to velocity have been computed for all the atmospheres in the central part of the simulation, individually for each of the 660 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Fig- ure 5 illustrates the variations in the optical depths (panel c) and geometrical heights (panel d) where the line core is sensitive to velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' For each atmospheric model, we have integrated the response function in the central wavelengths and determined the height of the maximum response (thick red line) and the heights where the normalized response function exhibit some selected values (red lines with smaller thickness as lower response values are considered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The def- inition of these lines is illustrated in the bottom panels from Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The heights with higher sensitivity to the velocity ex- hibit periodic behavior, both in optical depth and geomet- rical height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' During umbral flashes (blue-shaded areas, tak- ing place at the times when the chromospheric temperature is maximum) the core of the Ca ii 8542 Å line is sensi- tive to a deeper optical depth (with the maximum response at around log τ = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Interestingly, this deeper optical Article number, page 5 of 9 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' main depth is associated with high geometrical heights (around z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='2 Mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Generally, the geometrical height where the response of the core is maximum is maintained approxi- mately constant during the entire umbral flash event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In contrast, during the quiescent phase of the oscillations, the core of the line is in absorption and the peak of its sensi- tivity is shifted to optical depths around log τ = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='7 and geometrical heights as low as z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='7 Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Power spectra of the formation height of the Ca ii 8542 Å line averaged for all locations inside the umbra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The lines rep- resent the maximum of the response function (black line), the minimum height where the response function of the intensity to velocity is above a selected threshold (red lines), and the maxi- mum height where the response function is above the threshold (blue lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The thickness of the color lines indicates the thresh- old concerning the normalized response function, corresponding to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='5, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='3 from thicker to thinner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The periodicity of the fluctuations in the formation height of the line core has been evaluated by computing the average power spectra of the geometrical heights plotted in Figure 5 that characterize the regions where the line is sen- sitive to velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Figure 6 shows that the main frequency of the oscillations is 6 mHz, in agreement with the oscillations in the 3-minute band widely reported from chromospheric umbral oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Some secondary peaks appear at 10- 16 mHz, especially for the peak of the response function (black line) and the lower height of the selected contours (red lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' These peaks coincide with the power enhance- ment found in the velocity inferred from the inversions (bot- tom panel from Figure 4) and point to fluctuations in the sensitivity of the line core to height as the origin of the measured secondary power peak in velocity at 12-13 mHz (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Velocity gradients and opacity effects The effective formation height of the Ca ii 8542 Å line core significantly changes with the atmospheric variations pro- duced by umbral oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The presence of vertical gra- dients in the stratification of the atmosphere can thus lead to spurious oscillations in the quantities inferred from the interpretation of the spectral line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In addition to the in- trinsic oscillations associated with wave travel, a second component produced by changes in the opacity can leave an imprint in the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Figure 7 illustrates the stratification of the vertical ve- locity between the high photosphere and the chromosphere during ∼18 min of simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In the top panel, the veloc- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Top panel: Vertical velocity as a function of height (verti- cal axis) and time (horizontal axis) at a randomly chosen spatial position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Red lines have the same meaning as those in the bot- tom panel from Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Bottom panel: Vertical stratification of the velocity at some selected time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The color of the line indicates the corresponding time step, as given by the dashed vertical lines in the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' ity (grey scale) is saturated outside the ±1 km s−1 range to better visualize its variations in the layers where the core of the line is formed (indicated by the red lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The bot- tom panel shows the vertical velocity at some selected time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' All of them correspond to times when the response function of the Ca ii 8542 Å line core is undergoing striking variations, with its maximum shifting from lower to higher layers (blueish lines) or vice versa (greenish lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' During those time steps, the atmospheric velocity exhibits strong vertical gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The combination of vertical gradients in the velocity with the variations in the layers where the line core is sensitive to velocity produces the aforementioned additional component in the measured velocity as a result of opacity changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Temperature oscillations Figure 3 shows that the response function of the intensity to temperature also experiences striking variations during umbral oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The range of heights where the tem- perature is probed by the core of the Ca ii 8542 Å line is even broader than that for the velocity, with the peak of the response function changing from below z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='5 Mm (t2 = 1140 s) up to z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='2 Mm (t4 = 1190 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The top panel from Figure 8 shows the periodicity of these fluctu- ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The region where the line core is sensitive to tem- perature exhibits sudden variations from the photosphere (when the temperature perturbation is low) to the chromo- sphere (at those times when the chromospheric temperature increases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In the umbral atmospheres computed during the simulation, the chromosphere is generally ∼2000 K hotter than the photosphere (bottom panel from 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' This way, the intrinsic temperature enhancements produced by the waves will be accompanied by an additional temperature increase caused by the displacement in the region where the line is sensitive to temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Article number, page 6 of 9 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Felipe and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Socas-Navarro: Opacity effects in NLTE inversions Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Top panel: Temperature as a function of height (vertical axis) and time (horizontal axis) at a randomly chosen spatial po- sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Red lines indicate the height of the peak of the response function of intensity to temperature (thickest line) and the range of heights where the normalized response function is above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='7 (medium-thick line), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='5 (thinner line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Bottom panel: Verti- cal stratification of the temperature at some selected time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The color of the line indicates the corresponding time step, as given by the dashed vertical lines in the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The comparison between the temperature oscillations at constant geometrical height and constant optical depth illustrated in Figure 9 shows clear indications of height vari- ations in the origin of the temperature signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' During the first 500 s of simulation, when the fluctuations driven under the photosphere have not yet reached upper atmospheric layers and the chromosphere is approximately at rest, the temperature at constant optical depth (both from inver- sions and directly extracted from the simulation) is con- sistent with the temperature at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='89 Mm (light-blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' However, later times exhibit oscillations with ampli- tude significantly higher than that found for that geomet- rical height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' When the chromospheric temperature is maxi- mum (during umbral flashes), the temperature at constant optical depth is comparable to the temperature at z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='2 Mm (dark-blue/violet line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In contrast, the lowest values of the temperature at constant optical depth are similar to the temperature measured at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='5 Mm (light green line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The temperature oscillations inferred from the inversions of the Ca ii 8542 Å line show a fair agreement with the real temperature oscillations directly extracted from the output of the simulation, although some discrepancies are found at the times when the temperature is maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The oscillations at constant optical depth have peak- to-peak amplitude around 2000 K, although the strongest wavefronts exhibit a change between the minimum and maximum temperature up to 4000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' This amplitude is comparable to that from oscillations at z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='3 Mm (vio- let line in Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' This is also illustrated by the power spectra (bottom panel from Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The frequency of the main power peak at 6 mHz, present in oscillations at constant geometrical height and optical depth, is perfectly reproduced by the inversion results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Its power is similar to the power from oscillations at z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='3 Mm, which is the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Comparison between temperature oscillations in geomet- rical scale, optical depth, and those inferred from the inversion of synthetic Ca ii 8542 Å profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Top panel: temporal evolution of the temperature at a randomly chosen spatial position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Bot- tom panel: Power spectra of the temperature averaged for all the spatial positions from the numerical simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The color lines have the same meaning as in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' upper limit of the region sampled by the Ca ii 8542 Å line core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In contrast, the power of the oscillations at ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='89 Mm (whose temperature is probed by the core of the line when the atmosphere is approximately at rest, first 500 s of simu- lation) is almost two orders of magnitude lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' This result indicates that opacity oscillations are a significant compo- nent of the total temperature variations measured during umbral flashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Both intrinsic and opacity oscillations take place in phase, with the core of the line shifting to upper (hotter) layers when the temperature of the intrinsic oscil- lations increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Similarly to the case of the velocity (Figure 4), the power of temperature oscillations at constant optical depth also exhibits a secondary peak at around 12-13 mHz (red line in the bottom panel from Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' However, the strength of this power peak is lower than the fluctuations that can be captured by the inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Discussion and conclusions In this paper, we have characterized the relationship be- tween the quantities inferred from inversions of the Ca ii 8542 Å line and the intrinsic oscillations at constant geo- metrical height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The former is affected by the opacity effect, which changes the response height of the spectral line and severely affects the interpretation of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' This goal has been addressed through numerical simulations of wave Article number, page 7 of 9 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' main propagation in an umbral model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The stratification of the simulated velocity and temperature fluctuations has been compared with the fluctuations of the same quantities at constant optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In addition, synthetic Ca ii 8542 Å profiles have been computed from the output of the simu- lations, and the inferences from their inversions have also been critically evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Our analysis focused on the examination of the chro- mospheric velocity and temperature fluctuations, obtained by averaging the atmosphere (both the actual simulated model and those inferred from the inversion of the syn- thetic profiles) in a range of optical depths where the Ca ii 8542 Å line is generally sensitive to changes in the atmo- sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Our results show that inversions of the Ca ii 8542 Å line provide a good characterization of the velocity os- cillations in the low chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' They capture the am- plitude (and power) of the oscillations and the frequency of the main power peak at around 6 mHz, indicating that they provide a reliable estimation of the main oscillatory period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' However, some discrepancies are also obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The time steps of the velocity maximum (downflow) inferred from the inversions lag those found in the simulations at the low chromosphere (around 60 s delay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Instead, they go in phase with the maximum velocity and higher mid- chromospheric layers (top panel from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' This points to a change in the optical depths where the Ca ii 8542 Å line is sensitive to the velocity during different phases of the os- cillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' An examination of the evolution of the response function shows that during the downflowing phase in the mid-chromosphere (z ∈ [1000, 1400]), the response of the line is shifted upward to the range log τ ∈ [−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='0, −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='1], with the maximum of the response reaching up to log τ = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='9 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The velocity power spectrum inferred from the inver- sions with three velocity nodes closely matches the sim- ulated spectrum (averaged in log τ) for frequencies lower than 8 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' For higher frequencies, the inferred power is significantly stronger than the actual simulated power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' In- terestingly, the simulated power averaged in log τ exhibits a remarkable dip at around 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='5 mHz (bottom panel from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' This dip is absent in the power computed at constant geometrical heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The dip and subsequent peak are par- tially captured by the inversions, which show a power excess between 12 and 14 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The examination of the response functions shows that a similar power peak is found in the height where the line is sensitive to the velocity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 6) and that changes in the response height take place when strong velocity gradients are present (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The presence of power peaks (or dips) at the sunspot chromosphere has been suggested to be caused by the existence of a chro- mospheric resonant cavity (Jess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Felipe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2020) or by harmonics resulting from the non-linearity of the three-minute oscillations (Chae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Felipe 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Chai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Our results suggest caution with the in- terpretation of power peaks with frequencies higher than 9 mHz in Ca ii 8542 Å observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The opacity effect also has a remarkable impact on the measured temperature fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The region where the core of the Ca ii 8542 Å line exhibits a significant response to temperature oscillates between z ∼ 500 km and z ∼ 1200 km, that is, a region with strong vertical gradients in tem- perature (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Temperature fluctuations produced by the opacity effect are in phase with the intrinsic tempera- ture oscillations associated with wave propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' When the temperature (at constant geometrical height) increases, the response of the line is shifted to upper (hotter) lay- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' This leads to stronger inferred temperature fluctua- tions, which exhibit a high power in the three-minute band (higher than the power in most of the geometrical heights where the core of the line forms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The frequency of this power peak is well captured by the inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Our goals are in line with the recent work by Keys et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' They evaluated the inversion results of os- cillations in the photospheric 6301 Å and 6302 Å lines us- ing a similar approach of comparing the output from the inversions with known simulated atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Their anal- ysis focused on short-period (24 s) waves, finding a good match between simulations and inversions except for some discrepancies during the passage of waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' These deviations are due to the small height range perturbed by the waves in comparison with the range where the lines are sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Our results for chromospheric fluctuations inferred from a line formed under NLTE conditions indicate that oscilla- tions with frequencies higher than 9 mHz (periods shorter than ∼ 110 s) are hardly well characterized by the out- put of the inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Low-frequency acoustic waves (or, more precisely, slow magnetoacoustic waves) have a longer wavelength, which is comparable to the range of heights where the Ca ii 8542 Å line is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' However, at chromo- spheric heights the temperature and velocity change at spa- tial scales much shorter than the photon free path, which makes it impossible to precisely determine their vertical stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' The inversion can only be interpreted as the average properties in a region large enough to affect the emerging radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' This way, our analysis has focused on the average chromospheric fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' A comparison be- tween the models inferred from the inversions and the ac- tual vertical stratification is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' This limi- tation could be lessened by performing multi-line inversions with several spectral lines that probe similar atmospheric layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' Financial support from grants PGC2018- 097611-A-I00 and PID2021-127487NB-I00, funded by MCIN/AEI/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='13039/501100011033 and by “ERDF A way of making Europe” is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' TF acknowledges grant RYC2020-030307-I funded by MCIN/AEI/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='13039/501100011033 and by “ESF Investing in your future”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' We acknowledge the contribution of Teide High-Performance Computing facilities to the results of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' TeideHPC facilities are provided by the Instituto Tecnológico y de Energías Renovables (ITER, SA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=', & Esteban Pozuelo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content=' 2022, ApJ, 928, 101 Article number, page 9 of 9 This figure "Orcid-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='png" is available in "png"� format from: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='org/ps/2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} +page_content='03273v1' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE1T4oBgHgl3EQfkQT7/content/2301.03273v1.pdf'} diff --git a/zNE2T4oBgHgl3EQfhwde/content/tmp_files/2301.03951v1.pdf.txt b/zNE2T4oBgHgl3EQfhwde/content/tmp_files/2301.03951v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..50864f863b12791f160d54e26d2172599291406b --- /dev/null +++ b/zNE2T4oBgHgl3EQfhwde/content/tmp_files/2301.03951v1.pdf.txt @@ -0,0 +1,1704 @@ +Modelling quantum particles falling into a black hole: the deep interior limit +Alejandro Perez, Salvatore Ribisi, and Sami Viollet +Aix Marseille Université, Université de Toulon, CNRS, CPT, Marseille, France +(Dated: January 11, 2023) +In this paper we construct a solvable toy model of the quantum dynamics of the interior of a +spherical black hole with falling spherical scalar field excitations. We first argue about how some +aspects of the quantum gravity dynamics of realistic black holes emitting Hawking radiation can be +modelled using Kantowski-Sachs solutions with a massless scalar field when one focuses on the deep +interior region r ≪ M (including the singularity). Further, we show that in the r ≪ M regime, +and in suitable variables, the KS model becomes exactly solvable at both the classical and quantum +levels. The quantum dynamics inspired by loop quantum gravity is revisited. We propose a natural +polymer-quantization where the area a of the orbits of the rotation group is quantized. The polymer +(or loop) dynamics is closely related with the Schroedinger dynamics away from the singularity with +a form of continuum limit naturally emerging from the polymer treatment. The Dirac observable +associated to the mass is quantized and shown to have an infinite degeneracy associated to the +so-called ϵ-sectors. Suitable continuum superpositions of these are well defined distributions in the +fundamental Hilbert space and satisfy the continuum Schroedinger dynamics. +I. +MOTIVATION +The fate of the singularities of general relativity is a central question for quantum gravity that concerns important +physical situations such as those arising in (big-bang) cosmologies and black hole formation and evaporation. One of +the central features of loop quantum gravity is the inherent discreteness of quantum geometry at the Planck scale. +The lack of smoothness of the geometry at the fundamental level challenges the classical view of the singularities of +general relativity as a frontier of spacetime geometry, and strongly suggests the possibility of a microscopic dynamical +description that could define dynamics beyond the limit where classical description fails. +The history of the approach starts with the discovery of Ashtekar’s connection variables which first suggested +that the quantum dynamical evolution equations of gravity might admit a background independent finite and non +perturbative formulation [1, 2]. This initial suggestion grew into the approach of loop quantum gravity (LQG) with +the contribution of many (for reviews and text books see [3–7]). The LQG approach has produced insights about the +possible nature of matter and geometry at the Planck scale and has led to new ideas about the origin of black hole +entropy, the generation of quantum effects in early cosmology, and stimulated hopes about the possible regularizing +role of Planckian granularity (for quantum field theory and gravity). However, a clear understanding of the question +of the fate of singularities in realistic physical situations has remained a difficult one, as addressing it would actually +require the (still lacking) complete dynamical control of LQG in situations involving matter and geometry degrees of +freedom in the deep ultraviolet regime in full generality. +Nevertheless, the view that the evolution across singularities should be well behaved has become consensual in the +field over time thanks to the accumulated experience in simple low dimensional as well as symmetry reduced models +of black holes and cosmology. Professor Abhay Ashtekar has been one of the key leading driving forces along this +path, and main defender of the view (to which we adhere) that dynamics across the would-be-singularity should be +well defined in the quantum theory. It is a pleasure to contribute to this special issue with this work that, we believe, +is representative of this standpoint. +The first examples of singularity avoiding models where found in the context of quantum cosmology by Martin +Bojowald [8]. This seminal work grew later into a large number of contributions in the field now known as loop +quantum cosmology [9–11]. Even in these simple models the quantum dynamics can be rather involved. However, +it was soon realized [12] that effective semiclassical equations could be used to describe the dynamics across the +singularity and that these equations were quite easy to describe. The domain of applicability of these techniques +was extended in a variety of manners to models involving black holes [13–20] (for reviews see [21–23], for quantum +modifications inspired by other approaches see [24–26]). In many of the latter cases the natural starting point has been +to consider the quantum dynamics of the interior of spherically symmetric and static spacetimes of the Kantowski- +Sachs type (the Schwarzschild black hole interior in the vacuum case). In all these cases the interior singularity is +removed and replaced by a quantum transition across what would have been the singularity in the classical description +realizing aspects of existing scenarios [27–30]. +Simple models are nice as they illustrate possibly generic features of the general situation. However, they carry the +drawback of being often removed, by the very symmetry assumptions that simplify them, from the realistic physical +situations about which one would like to gain non-trivial insights. Moreover, when it comes to black holes, most of +the studies have focused on effective dynamical descriptions, while quantum dynamics has received less attention due +arXiv:2301.03951v1 [gr-qc] 10 Jan 2023 + +2 +to its often unsurmountable complexity even in simple models. For instance, concerning the first drawback we know +that real black holes are not time translational invariant due to the expected presence of Hawking evaporation (in +contrast with the static nature of many of the quantum black hole models) and that all symmetry assumptions must +fail near the singularity when the back-reaction of Hawking particles correlated with the outside radiation would be +properly taken into account (see [31] for further discussion of this issue). When it comes to the second drawback, +even when effective descriptions can provide the dynamical evolution of the spacetime geometry with matter fields +on it, its classical nature precludes the analysis of genuine quantum phenomena such as entanglement and other +quantum information issues of highest interest from the perspective of Hawking’s evaporation (e.g. the longstanding +information puzzle or the question of the fate of unitarity in black hole evaporation). Even when our model will not +resolve the first limitation, we believe that it provides a humble small step in the right direction. Concerning the +second, we will see that the quantum dynamics is fully accessible in our simple model opening the road for exact +calculations in the quantum realm. +Quantum gravity +region +◆0 +I+ +I� +Hawking pairs +Matter +Friday, 23 September 22 +Figure 1: The Ashtekar-Bojowald paradigm +. +The interior region r < 2M of a Schwarzschild black hole of mass M can be seen as a homogeneous anisotropic +cosmological model where the r=constant surfaces (in the usual Schwarzschild coordinates) are Cauchy surfaces of +homogeneity where any two arbitrary points can be connected along orbits of the isometry group that involves spacelike +translations along the staticity Killing field ξ = ∂t and the rotations associated to spherical symmetry. Models with +these isometries will be refereed to as Kantowski-Sachs (KS) models [32]. They include not only the Schwarzschild +black hole interior geometry (vacuum case) but also the Reissner-Nordstrom black hole interior geometry (in the +Einstein-Maxwell case) and other solutions depending of the type of matter that one decides to couple to the system. +In this paper we would like to emphasize the fact that Kantowski-Sachs models (with a massless scalar field coupling) +define a natural toy model capturing some (possibly interesting) aspects of the dynamics and back-reaction of matter +near the singularity of realistic black holes that Hawking radiate and evaporate. The model can certainly not replace +the full dynamical description of a generic gravitational collapse in the full theory as it remains a toy model with +finitely many degrees of freedom. However, we will argue, it can handle in a simplistic way some dynamical aspects +that might be relevant when discussing questions in the context of evaporating black holes. +A. +Scalar excitations falling inside a Schwarzschild black hole: the deep interior regime +Let us consider a free test point particle (with no angular momentum) falling into the interior of a Schwarzschild black +hole. As the particle approaches the singularity—in a description on an r equal constant slicing of the interior—one + +3 +expects its wave function to become better and better approximated by a translational invariant wave function since +the expansion in the spacelike Killing direction ξ = ∂t diverges for r → 0. If this conclusion is correct then it means +that zero angular momentum test particles can be approximated by the type of excitations that can be accommodated +in the dynamical framework of the KS cosmologies (at least in the sense of a near singularity approximation). One +can be quantitative about this intuition as follows: free test particles with four wave vector ka on the Schwarzschild +background are associated with the conserved Killing energy E ≡ −kaξa. We are assuming that the particle has zero +angular momentum which implies that its wave function is already translational invariant in the directions transversal +to ξa on the r-slices. The wave function can only vary in the direction of the Killing ξa and the component of the +physical momentum in this direction is given by +pξ ≡ kaξa +√ξ · ξ = −E +� +r +2M − r, +(I.1) +which vanishes in the limit r → 0. The wave length of such a particle diverges, and thus particles without angular +momentum are better and better represented by translational invariant excitations as one approaches the singularity. +These are precisely the kind of homogeneous configurations that can be described in the KS framework. +This simple implication deduced from the idealized notion of test particle can be made more precise by looking +at the analogous features of scalar field excitations (solutions of the Klein-Gordon equation). Indeed, the simplistic +argument given here can be made precise in the field theoretical context as we will show in what follows. +1. +Solutions of the Klein-Gordon equation in the deep interior region +Here we argue that the Kantowski-Sachs model (described in detail in Section II) coupled to a massless scalar field +faithfully captures the dynamics of a Klein-Gordon excitation falling into the deep interior region of a Schwarzschild +black-hole. We will do this by analysing the behaviour of solutions of the Klein-Gordon equation on the Schwarzschild +background in the r ≪ 2M regime. We will focus on the spherically symmetric solutions and show that they become +homogeneous on r=constant surfaces as r → 0 and thus can be accommodated in the framework of KS configurations. +This implies that the KS system can be used to model the dynamics and (most importantly) the back-reaction of such +(zero angular momentum) scalar configurations falling into the deep interior region of spherically symmetric black +holes. +Let us first start by approximating the Schwarzschild metric in the deep interior region r ≪ 2M as +ds2 = 2M +r dt2 − +r +2M dr2 + r2dΩ2. +(I.2) +We will choose coordinates such that the time-radial part of the metric is conformally flat. Remarkably, in the deep +interior region, this is achieved by switching to area variables a = 4πr2 (the well known tortoise coordinate r∗ is +actually proportional to r2 near the singularity). With these variables the metric becomes +ds2 = +1 +16π√aHa +� +dτ 2 − da2� ++ a +4π dΩ2, +(I.3) +with aH = 16πM 2, τ = √16πaHt, and a = 4πr2. The Klein-Gordon equation for a massless scalar field then reads +□Φ = +1 +√−g ∂µ +�√−ggµν∂νΦ +� += 0 +(I.4) +⇐⇒ +� +−∂2Φ +∂a2 − 1 +a +∂Φ +∂a + ∂2Φ +∂τ 2 + +1 +4√aHaa +� +1 +sin θ +∂ +� +sin θ ∂Φ +∂θ +� +∂θ ++ +1 +sin2 θ +∂2Φ +∂ϕ2 +�� += 0. +(I.5) +To solve it, we make the usual ansatz +Φℓm = eiωtYlm(θ, ϕ)φl(a) = e +i +ωτ +√ +16πaH Ylm(θ, ϕ)φl(a) +(I.6) +which reduces the Klein-Gordon equation to +φ′′ +l + φ′ +l +a + +� +ω2 +16πaH ++ l(l + 1) +4√aHaa +� +φl = 0. +(I.7) + +4 +For the spherical modes l = 0 one obtains +φ0(a) = c1J0 +� +ωa +√16πaH +� ++ c2Y0 +� +ωa +√16πaH +� +, +(I.8) +with J0 and Y0 Bessel functions and c1 and c2 constants. As we will prove in Section II that these solutions match nicely +with solutions of the KS model (mentioned at the end of the introduction). This follows from two complementary +properties of the solutions of the Klein-Gordon equation. On the one hand, the near singularity limit of the quantity +(simply related to the KS momentum variable as we will see in Section II) +lim +a→0 a∂φ0 +∂a = −2c2 +π +(I.9) +is finite and independent of ω. On the other hand, the t dependence of the Klein-Gordon solutions is ‘ironed’ by the +infinite expansion of the geometry in the ∂t direction: the region ℓ0 ≡ ∆t = 2πω−1 where the solution has a significant +(order-one) change corresponds to a length scale ∆d ≈ 2πω−1� +M/r (in agreement with the infinite redshift effect +captured in equation (I.1)). Therefore, in a length scale ℓp ≪ ℓ ≪ ∆d along the background Killing field direction +ξ = ∂t, and in the deep interior region a ≪ M 2, the solutions of the Klein-Gordon equations can be considered as +homogeneous and therefore compatible with initial data that would be admissible in the KS model. +We will see in Section II that the momentum variable pφ in the KS model is simply related to the quantity whose +limit was considered in the previous paragraph as it is defined as +pφ ≡ −8πMℓ0r∂Φ00 +∂r += −ℓ0aHa∂Φ00 +∂a , +(I.10) +where the pre-factors arise form the hamiltonian analysis of Section II and ℓ0 is an IR cutoff scale naturally associated +in the previous discussion to the scale ∆t = 2πω−1. It follows from the previous considerations that +lim +a→0 pφ = constant +(I.11) +in the region of interest. One can relate the previous quantity to the average ‘energy density’ on the r=constant +hyper-surfaces as one approaches the singularity (this will be simply related to the KS Hamiltonian that will be +defined in the following section). Namely, +1 +ℓ0 +� t0+ℓ0 +t0 +dt +�� +dθdϕ +�� +|g|Tµν∂µ +r ∂ν +r +�� += +p2 +φ +64πM 2ℓ2 +0 +, +(I.12) +where the scale ℓ0 enters the definition of the average in the time direction. For concreteness one can match ℓ0 to the +wavelength 2πω−1 of the excitation and the previous result will already hold (of course it holds for ℓ0 > 2πω−1). +For completeness we give the limiting behaviour of solutions in the non-spherical case. For the non-spherical modes, +one can neglect the term containing the frequency in equation (I.7) in analysing the small a behaviour of solutions. +If we do so, we obtain for l ̸= 0 +φl(a) = c1J0 +� +4 +� a +π +� +l(l + 1) +M +� ++ 2c2Y0 +� +4 +� a +π +� +l(l + 1) +M +� +. +(I.13) +Solutions diverge logarithmically (as log[a]) for a → 0. This holds both for spherically symmetric as well as for +non spherically symmetric solutions as it follows from the asymptotic behaviour of the Bessel functions or from the +finiteness of pφ in the spherical case. The mild character of the divergence was emphasized in [33, 34] as an attractive +possibly interesting property when one considers the definition of the associated quantum operators in quantum field +theory (in view of a possible definition of semiclassical gravity). Here we simply point out that such simple behaviour +allows for bridging to a solvable KS model to understand aspects of the back-reaction of classical (as well as quantum) +excitations falling into a spherically symmetric black hole. +II. +THE KANTOWSKI-SACHS SPACETIME COUPLED WITH A MASSLESS SCALAR FIELD +In this section we revisit the construction of the phase space of the KS model by perforing the canonical analysis +of the associated symmetry reduced model where staticity and spherical symmetry are imposed from the onset (in +Section II A). To improve the clarity of the presentation we simply start from the vacuum case—whose solutions are + +5 +isomorphic to the interior Schwarzschild solutions—and later couple the system to a scalar field without mass. We +express variables in terms of the usual Schwarzschild-like coordinates. In Section II B we present a truncation of the +Hamiltonian and show, in Section II C, that it defines a tractable approximation of the dynamics in the r ≪ M region +of the interior of physically realistic black holes. We call this regime the deep interior dynamics. In Section II D +we show that the regime of applicability of the model includes the physically interesting situation of Hawking scalar +excitations with zero angular momentum falling inside of the black hole. +A. +Symmetry reduced covariant phase space +It is well known that for a spherically symmetric and static spacetime, the line element can be written without any +loss of generality as +ds2 = −f(r)dt2 + h(r)dr2 + r2dΩ2 . +(II.1) +It follows that the Eintein-Hilbert action (with the appropriate boundary term that renders it differentiable) becomes +Sgeo = +1 +16π +�� +R +d4x√−gR + 2 +� +∂R +K +� += ℓ0 +2ℓ2p +� +dr +� +� +fh + +� +f +h + +˙fr +√fh +� +, +(II.2) +where the dot denotes the derivative with respect to r and ℓ0 is a cut-off in the non compact spacelike ∂t direction +that regularizes the dynamical system. The cut-off will be associated a natural meaning in modelling the fate of zero +angular momentum excitations falling inside the black hole. In the deep interior region r ≪ M and we will take +ℓ0 ∼ ω−1 for ω ≈ 1/M (the typical frequency in the Hawking spectrum of a macroscopic black hole of mass M). One +can easily verify that the variations of the action lead to the Schwarzschild solutions +ds2 = −p2 +M +� +1 − 2M +r +� +dt2 + +dr2 +� +1 − 2M +r +� + r2dΩ2, +(II.3) +and the symplectic potential (stemming from the on-shell evaluation of the action variation) +θ = −ℓ0 +ℓ2p +(c1dM + 2MdpM − 2dpMr), +(II.4) +and symplectic structure +ω = ℓ0 +ℓ2p +dpM ∧ dM. +(II.5) +Instead of working directly with the physical phase space parametrized by the Dirac observables pM and M it will +be convenient for us to work with kinematical variables and constraints for the moment. This is because of the usual +difficulty in linking the timeless physical phase space with a classical intuition based on spacetime geometry. To avoid +such difficulties we would like to have a notion of parametrized ‘time evolution’ which in our context will take the form +of an area radius evolution. Thus we take the integrand of (II.2) as the Lagrangian Lgeo of the spacetime subsystem +Lgeo = ℓ0 +2ℓ2p +� +� +fh + +� +f +h + +r ˙f +√fh +� +. +(II.6) +On the other hand, we will couple the system to a massless scalar field by adding the matter action +Sm = −1 +2 +� +R +d4x√−g∂aφ∂aφ = −2πℓ0 +� +drr2 ˙φ2 +� +f +h. +(II.7) +The conjugate momenta to f, h and φ are given by +pf = ℓ0 +2ℓ2p +r +√fh +, +ph = 0 +and +pφ = −4πr2ℓ0 +� +f +h +˙φ, +(II.8) + +6 +and the primary Hamiltonian, defined by H = ˙fpf + ˙hph + ˙φpφ − Lφ − Lgeo, becomes +H1 = − ℓ0 +2ℓ2p +f(h + 1) +√fh +− +hp2 +φ +8πr2ℓ0 +√fh . +(II.9) +From the expression of the conjugate momenta (II.8) we identify the constraints +ξ ≡ pf − ℓ0 +2ℓ2p +r +√fh = 0 +and +ph = 0, +(II.10) +and the secondary Hamiltonian +H2 = H1 + λξ + ηph , +(II.11) +where λ and η are Lagrange multipliers. One can show that the stability of the two constraints (II.10) can be ensured +by fixing the associated Lagrange multipliers, i.e., the constraints (II.10) are second class and can be explicitly solved +leading to +ph = 0 +and +h = ℓ2 +0 +4ℓ4p +r2 +fp2 +f +. +(II.12) +Thus, the secondary Hamiltonian (II.11) reduces to +H2 = −1 +r +� +fpf + +1 +16πℓ2p +p2 +φ +fpf ++ ℓ2 +0 +4ℓ4p +r2 +pf +� +. +(II.13) +The previous encodes the KS dynamics of geometry coupled to a massless scalar field. The relevant solutions for phys- +ical applications correspond to small departures from the vacuum Schwarzschild solutions representing macroscopic +black holes with scalar field perturbation falling inside. We will further simplify the system by focusing on, what +we call, the deep interior region r ≪ M where M is the mass scale defined by the corresponding black hole solution +perturbed (in the sense of Sections II C and Section II D) by the presence of matter. It is in this regime where the so- +lutions of the KS system faithfully describe the dynamics of a spherically symmetric scalar perturbation (representing +for instance a Hawking particle) as it falls towards the interior singularity. The KS Hamiltonian evolution matches, in +this sense, the test-field evolution (the Klein-Gordon solutions on the Schwarzschild background fixed non dynamical +background) and incorporates, as a simplified model, aspects of the back-reaction that are expected to become more +important as one approaches the singularity. +B. +The deep interior dynamics +We are interested in the dynamical evolution in the r ≪ 2M regime. In addition we will use the present dynamical +system to model (in a suitable approximation) the back-reaction of a Hawking quantum falling into a black hole +singularity. Hawking particles do not correspond to static excitations as the one we can model with the symmetry +assumptions of the present section. However, as argued in Section I A, when spherically symmetric, these particles +look more and more static as seen by a radially freely falling observer in the limit r → 0. This is the reason why we +are interested in such regime of the present dynamical system. In the next section we will study the classical solution +of the model using perturbation theory in the parameter p2 +φ/M 2—as p2 +φ will be assumed to be much smaller to M 2 +in applications—and show that the dynamics simplifies in the deep interior region. The simplification occurs due to +the negligible effect of the last term in the expression of the hamiltonian (II.13): more precisely, in the deep interior +region, the Hamiltonian is well approximated by +Hdi = −1 +r +� +fpf + +1 +16πℓ2p +p2 +φ +fpf +� +. +(II.14) +This toy theory reflects the dynamics of the leading order in an expansion near r = 0. Order O(r) effects could be +included in perturbation theory near r = 0 in which case the term we dropped would correspond to the perturbation +Hamiltonian. The consistency of this truncation will be shown in Section II C. +The system we are dealing with has no gauge symmetries as the radial reparametrization symmetry has been gauged +fixed with the metric ansatz (II.1) by choosing the area radius as time. In order to recover the structure of a gauge + +7 +theory, with a clear analogy with the full theory of LQG, it will be convenient ‘reparametrize’ the system by promoting +the area radius r to a degree of freedom with conjugate momentum pr and add a scalar constraint C = pr − H2 = 0. +The phase space is therefore extended to (f, pf, r, pr, φ, pφ), and the number of degrees of freedom is preserved by the +inclusion of the Hamiltonian constraint +Cr = pr − 1 +r +� +fpf + +p2 +φ +16πℓ2pfpf +� +≈ 0. +(II.15) +In this approximation one can show that we have the following Dirac observables +D1 = fpf +, +D2 = fr +− +p2 +φ +4fp2 +f ++1 +, +D3 = pφ +, and +D4 = φ + pφ log(r) +2fpf +. +(II.16) +It will be convenient to make the following canonical transformation and thus introduce what we call the deep interior +variables +m = −fpf +and +pm = − log(−f) , +(II.17) +and—in trying to introduce the kinematical structure proper to loop quantum gravity—to adopt the area a of the +surfaces of constant r, namely +a = 4πr2 +and +pa = pr +8πr , +(II.18) +as new dynamical variables. With this choice the phase space is described by the geometric variables m, pm, a and pa +with Poisson brackets +{m, pm} = 1 , +{a, pa} = 1 , +and by the matter variables φ and pφ for which +{φ, pφ} = 1, +(II.19) +with all the other Poisson brackets equal to zero. +In the new variables the deep interior dynamics Hamiltonian +constraint (II.15) becomes +Ca = pa + 1 +2a +� +m + +p2 +φ +16πℓ2pm +� +≈ 0. +(II.20) +The previous constraint is central in the rest of the paper. We will see that it leads to a fully controllable dynamics +both at the classical as well as the quantum level. Indeed, the dynamics is exactly solvable in the vacuum case while +it can be dealt with in perturbation theory for the case where the scalar field is excited. In the next section we will +justify the truncation that took us from (II.13) to (II.14) (and finally to the constraint (II.20)) using perturbation +theory. In Section II D we will show that the perturbative regime is consistent with the conditions that make our +model applicable to the description of a spherically symmetry Hawking particle falling into a Schwarzschild black hole +during evaporation. +C. +Perturbative solutions in pφ/M and the dynamics in the deep interior region +Exact KS solutions with scalar fields have been studied in the past (see for instance [35]). KS coupled to scalar fields +does not lead necessarily to (asymptotically flat) back hole spacetimes globally speaking. However, we will show here +that solutions can be interpreted in terms of perturbations of a vacuum Schwarzschild solution in the deep interior +region r ≪ 2M in the regime where pφ/M ≪ 1. We will also show that in that regime the Hamiltonian (II.13), and +the equations it generates, can be well approximated by (II.14). This will lead to a simple solvable system, both at +the classical and quantum levels, which can be used to model aspects of the physics of (zero angular momentum) +scalar particles falling into a spherically symmetric black hole (possibly useful in view of describing aspects of Hawking +radiation). We will analyze the system in first order perturbation theory in pφ/M. + +8 +In order to best organise the perturbative equations we replace p2 +φ by ϵ2p2 +φ where ϵ is a smallness parameter. We +introduce the following expansion of the relevant dynamical quantities +f(r) = f0(r) + ϵ2f1(r) + O(ϵ4), +(II.21) +pf(r) = pf0(r) + ϵ2pf1(r) + O(ϵ4) , +(II.22) +and write the equations of motion for them by keeping terms up to order ϵ2. Starting from +˙f = {f, H2} = −f(r) +r ++ +ϵ2p2 +φ +16πℓ2prf(r)pf(r)2 + +ℓ2 +0r +4ℓ4ppf(r)2 , +˙pf = {pf, H2} = pf(r) +r +− +ϵ2p2 +φ +16πℓ2prf(r)2pf(r), +(II.23) +with H2 given in equation (II.13), one can solve the equations order by order with solutions +f0(r) = p2 +M +� +1 − 2M +r +� +, +f1(r) = − +ℓ2 +pp2 +φ +8πrℓ2 +0M 2 +� +2M + (M − r) log +� +r +2M − r +�� +, +(II.24) +pf0(r) = +ℓ0 +4ℓ2ppM +r, +pf1(r) = − +p2 +φr +32πℓ0M 2p3 +M +� +2M +2M − r + log +� +r +2M − r +�� +. +(II.25) +The function h(r), is recovered from the constraint (II.10) and gives +h0(r) = +1 +1 − 2M +r +, +h1(r) = − +ℓ2 +pp2 +φr log +� +r +2M−r +� +8πℓ2 +0p2 +MM(2M − r)2 . +(II.26) +Note that the Schwarzschild solution with mass M and conjugate momentum pM—as in (II.3)—is recovered in the +leading order. Consistency of the perturbative treatment requires the ratio of first-to-leading order contributions to +remain small, namely +f1(r) +f0(r) = +ℓ2 +pp2 +φ +32πℓ2 +0M 2p2 +M +log +� r2 +4M 2 +� ++ O +� r +M +� +≪ 1, +h1(r) +h0(r) = +ℓ2 +pp2 +φ +32πℓ2 +0M 2p2 +M +log +� r2 +4M 2 +� ++ O +� r +M +� +≪ 1 +(II.27) +in the regime r < M. The logarithmic behaviour of the right hand side of the previous equations is not a threat, as +the classical solutions are not to be trusted for r near the Planck scale. Therefore, assuming r > ℓp, we conclude that +our analysis is consistent as long as +ℓ2 +pp2 +φ +32πℓ2 +0M 2p2 +M +log +� +ℓ2 +p +4M 2 +� +≪ 1 +(II.28) +which is always valid in the usual macroscopic black hole regime ℓp/M ≪ 1. +Similarly, note that the ratio of +the last term in the Hamiltonian (II.13) to the leading first term is exactly h(r) which in the same regime is +O(ℓ2 +p/M 2 log(ℓp/M)) as seen from (II.26). This suggests that one can simplify the dynamics (if interested in the +deep interior region r ≪ M) and use the Hamiltonian +Hdi = −1 +r +� +fpf + +1 +16πℓ2p +p2 +φ +fpf +� +. +(II.29) +This expectation is confirmed by the analysis of the Hamiltonian flow generated by Hdi in the relevant regime. One +might be slightly uncomfortable with the UV cut-off used in devising conditions such as (II.28), after all the classical +equations predict a singular evolution when r → 0 and in this limit the right hand sides of the equations (II.27) +actually blows up invalidating in appearance the perturbative analysis. We will see that the quantum dynamics across +the classical singularity is actually well defined. Moreover, we will also see that there are exact effective equations +describing the evolution of the expectation value of semiclassical states across r = 0 as well. These equations imply +that the counterpart of the right hand side of (II.27) not only does not diverge but rather tends to zero when quantum +effects are included. + +9 +D. +Validity of perturbation theory in view of applying the model to Hawking pairs produced by a +macroscopic black hole +Here we argue for the validity of the perturbation theory analysis of the previous section in the context of applications +of the KS model to the quantum dynamical description of scalar field excitations falling inside a Schwarzschild black +hole during Hawking evaporation. +The assumptions here are: First the usual assumption that the gravitational +collapse has formed a spherical black hole with mass M ≫ mp. Second, that the Hawking pairs produced by such +a macroscopic black hole can be described as test field excitations on a stationary Schwarzschild geometry far away +from the singularity. Finally, we restrict our attentions to spherically symmetric Hawking pairs which are the only +that can be mapped to the spherically symmetric KS configurations. This is not a serious restriction given that most +of the Hawking radiation is emitted in such modes [36]. +The stationarity of the background implies that the (test field) current ja ≡ Tabξb is conserved, where ξa is the +stationarity Killing field that in the adapted coordinates that we use here is simply given by ξ = ∂t and Tab is the +energy momentum tensor of the scalar field1. Conservation of energy leads to the expectation that when a Hawking +particle is detected at I + with a given energy E = ω (which coincides with the associated Killing conserved quantity), +a Hawking partner with (Killing) energy −ω falls into the black hole singularity reducing the mass of the black hole +by ω. A more precise field theoretical expression of this is that the flux of the (expectation value of) the energy- +momentum current ja at infinity in the state of the field after the detection of the particle at I + equals minus the +flux of the same current on (for instance) a constant r < 2M hyper-surface Σr. Namely, +� +Σr +janadV +(3) = −ω. +(II.30) +Using our coordinates to write explicitly the integrand in the left hand side while pushing Σr to the region where +comparison with the KS regime is possible (a constant r such that r ≪ 2M) we get +8πM +� ℓ0 +0 +Tr0(t, r)rdt = 8πM +� ℓ0 +0 +∂tφ∂rφrdt = +ωp2 +φ +16πMℓ0 +log(r) + O(1) = −ω, +(II.31) +where the right hand side of the last equality comes from the evaluation of the energy flux of a Hawking particle, +while in the evaluation of the left hand side we used that for r ≪ 2M (see for instance (I.11)) the scalar field behaves +like +φ = Re +� +e−iωt +� +φ0 − +pφ +8πMℓ0 +log(r) +�� ++ O +� r +M +� += cos(ωt) +� +φR +0 − +pR +φ +8πMℓ0 +log(r) +� ++ sin(ωt) +� +φI +0 − +pI +φ +8πMℓ0 +log(r) +� ++ O +� r +M +� +. +(II.32) +The result in (II.31) follows also from the assumption that ℓ0 > ω−1 ≈ M. By pushing the integral to its largest +possible value when r → ℓp, we conclude that energy conservation (encoded in (II.31)) implies the bound +p2 +φ +16πM 2 log +� ℓp +M +� +< 1, +(II.33) +which is consistent with the condition for the validity of perturbation theory (II.27). We see that the physical context +provided by the problem of Hawking evaporation precisely justifies the simplifying assumptions that led to (II.20). +Notice also that the fiducial length scale ℓ0 acquires in such a physical situation an operational meaning as well. +A generalization of the analysis of the present Sections II to a wider class of close to stationary black holes is certainly +very appealing. Even though it is clear that the tools employed here would not apply to black hole spacetimes with +1 The Hawking effect is a quantum process and the relavant state describing it is a quantum state that corresponds to the vacuum in the +far past idealized by I −. Such state can be viewed in the interior as a superposition of particles. Actual particles appear inside in the +hypothetical situation of the detection of a partner at I + which, according to the standard interpretation of quantum mechanics, will +produce a collapse of the vacuum to a new state containing an actual particle falling into the black hole. The situation after such collapse +is the semiclassical situation that we model here with our classical language. A precise description of such a situation in quantum terms +is a question that can only be addressed in a full theory of quantum gravity. We argue here that our little solvable model goes a humble +little step into that direction. + +10 +i+ +I + +⌃ +Figure 2: Hawking pairs produced in a black hole spacetime with M ≫ mp are described as test field excitations on a background +spacetime that is idealized by a stationary black hole solution in the region of the Penrose diagram away from the collapsing +matter and containing i+. Stationarity implies the existence of a conserved energy-momentum current used, in this paper, to +relate the value of pφ in the KS model to the frequency of the emitted particle at I + and (via Hawking temperature) to the +black hole mass M. +. +angular momentum (due to the breaking of spherical symmetry), one could entertain the possibility of including an +electric charge without leaving the general framework of this work. However, such an apparently simple extension +will reserve challenging new aspects. The first qualitative new feature is that the singularity becomes timelike in the +unperturbed black hole. However, when a scalar field perturbation is added its back reaction is expected to produce the +phenomenon of mass inflation [37] near the Cauchy horizon of the unperturbed background. This completely changes +the global features of a realistic charged black hole interior in a way that would seem to preclude the applicability +of the present methods. More precisely, if we use the Reissner-Nordstrom background black hole geometry as a basis +for the present discussion—noting that the presence of mass inflation already implies that strong deviations from +the Reissner-Nordstrom interior solution are to be expected—the true would-be-singularity should materialize near +the location of the Cauchy horizon. However, close to the Cauchy horizon scalar field modes with frequency ω are +expected to be infinitely blue shifted precluding the type of approximation available in the present case. This makes +such exploration highly non-trivial even when certainly interesting. +III. +A NATURAL POLYMER QUANTIZATION OF THE DEEP INTERIOR DYNAMICS +In previous sections we have shown that the Hamiltonian constraint (II.20) describes the deep interior dynamics of +scalar excitations without angular momentum falling into a macroscopic Schwarzschild black hole. The approximations +used are based on assumptions that are satisfied by the (spherically symmetric) Hawking excitations produced during +black hole evaporation. Thus, the simplified dynamics in the deep interior region, which will turn out to be analytically +solvable (both at the quantum and classical level), offers a toy scenario to analyze key questions of black hole +evaporation in a controlled scenario. We will show in this section that there is natural polymer quantization of the +deep interior dynamics with remarkable simple properties such as: (like in the full theory of LQG) the discreteness +of the area of constant area surfaces, a well defined quantum dynamics across the singularity, an effective classical +description, and a direct (to our knowledge novel) link with the continuum representation. The polymer quantization +we propose does not suffer from the usual ambiguities associated to the so-called holonomy corrections [38] as the +Hamiltonian constraint evolution has (in our simple model) a clear-cut geometric interpretation that allows for a +unique polymerization that is compatible with the continuum limit (defined by the Schroedinger representation). +This special geometric property arises from the fact that the Hamiltonian constraint is linear in the momentum +conjugate to the area of the spheres whose spectrum is discrete in our polymer representation. Ambiguities remain in +the form of the so-called inverse volume corrections which are necessary if one defines the quantum dynamics across +the singularity. + +11 +A. +Sketch of the Schrodinger quantization +In the standard Schrodinger representation one would quantize the phase space of Section II A by promoting the +variables a, m, pa, pm to self adjoint operators +�m ψ(m, pφ, a) = mψ(m, pφ, a), +�pmψ(m, pφ, a) = −i∂mψ(m, pφ, a), +�a ψ(m, pφ, a) = aψ(m, pφ, a), +�paψ(m, pφ, a) = −i∂aψ(m, pφ, a), +�pφψ(m, pφ, a) = pφψ(m, pφ, a), +�φ ψ(m, pφ, a) = i∂φψ(m, pφ, a), +in the kinematical Hilbert space is HS = L 2(R3), equipped with the usual inner product +⟨ψ1, ψ2⟩ = +� +∞ +−∞ +� +∞ +−∞ +� +∞ +−∞ +ψ1(m, pφ, a)ψ2(m, pφ, a)dmdpφda, +(III.1) +where we have chosen the momentum representation for the scalar field for convenience (as pφ is one of the constants +of motion of the system). Eigenstates of the �a operator are interpreted as distributions (they are not in the Hilbert +space) and one usually writes +�a |a⟩ = a |a⟩ +(III.2) +with a ∈ R and form an orthonormal basis +⟨a, a′⟩ = δ(a, a′). +(III.3) +The dynamics is imposed by solving the Hamiltonian constraint (II.20) which, in the present representation, takes +the precise form of a Schrodinger equation in the area variable a, namely +� +−iℏ ∂ +∂a + 1 +2a +� +m + +p2 +φ +16πℓ2pm +�� +ψ(m, pφ, a) = 0. +(III.4) +As usual, solutions of the constraint are certainly not square integrable in the a-direction, thus physical states are +outside of the kinematical Hilbert. The physical Hilbert space is defined as the space of square integrable functions +of m and pφ at fixed time a—Hphys = L 2(R2)—with inner product +⟨ψ1(a), ψ2(a)⟩phys = +� +∞ +−∞ +� +∞ +−∞ +ψ1(m, pφ, a)ψ2(m, pφ, a)dmdpφ, +(III.5) +which is preserved, i.e. it is independent of a, by the Schrodinger equation (evolution is unitary in a). +Two important remarks are in order: First note that we are formulating in detail the dynamics of the system +in the near singularity approximation. +The physical reason for this is that (as argued previously) it is only in +this approximation that the system can be compared with a (spherically symmetric) black hole with spherically +symmetric excitations falling inside. A side gain is also the simplification of the dynamics which will allow us for +a simpler quantization and the analysis of the possibility of a well defined dynamics across the singularity when we +undergo the LQG inspired quantization. One could however consider the quantization of the minisuperspace system +without the near singularity approximation. In that case one would need to write a Schroedinger equation using +the Hamiltonian (II.13), now genuinely time-dependent (r-dependent), for which unitary evolution would involve +path ordered exponentials (as the Hamiltonian does not commute with itself at different r values). In addition one +would need to work with either r, f, pr, pf variables or a, f, pa, pf variables without the luxury of the simplifications +introduced by the use of the near singularity variables (II.17). +B. +The polymer quantization +We define now a representation of the phase space variables that incorporates a key feature of the full theory of LQG: +the area quantization. Such representation closely mimics the structure of the quantum theory in the fundamental +theory in such a way that the area variable a acquires a discrete spectrum. Mathematically, this is achieved by +replacing the L 2 structure of the inner product in the variable pa by the inner product of the Bohr compactification +of the pa phase space dimension. More precisely one substitutes the kinematical inner product in the Schroedinger +representation (III.1) by +⟨ψ1, ψ2⟩ = +lim +∆→+∞ +1 +2∆ +� +∆ +−∆ +�� +∞ +−∞ +ψ1(m, pφ, pa)ψ2(m, pφ, pa)dmdpφ +� +dpa . +(III.6) + +12 +With this inner product, periodic functions of pa with arbitrary period are normalizable and the conjugate a- +representation acquires the property of discreteness in a way that closely mimics the structure of the fundamental +theory of loop quantum gravity [11]. In particular eigenstates of �a exist +�a |a⟩ = a |a⟩ +(III.7) +with a ∈ R. These state form an orthonormal basis with inner product +⟨a, a′⟩ = δa,a′, +(III.8) +in contrast with (III.3). Discreteness of the spectrum of �a comes at the prize of changing the kinematical Hilbert space +structure in a way that precludes the infinitesimal translation operator �pa to exist. Instead only finite translations +(quasi periodic functions of pa) can be represented as unitary operators in the polymer Hilbert space. Their action +on the a-basis is given by +� +eiλpaψ(m, pφ, a) = ψ(m, pφ, a + λℓp). +(III.9) +Eigenstates of the finite translations (or shift operators) exist and are given by wave functions supported on discrete +a-lattices. Namely, +ψk,ϵ(a) ≡ +� +exp(ika) +if +a ∈ Γϵ,λ ≡ {(ϵ + nλ)ℓ2 +p ∈ R}n∈Z +0 +otherwise +(III.10) +where the parameter ϵ ∈ [0, λ) ∈ R. The discrete lattices denoted Γϵ,λ are the analog of the spin-network graphs in +LQG with the values of a on lattice sites the analog of the corresponding spin labels. With all this one has (using +(III.9)) that +� +eiλpaψk,ϵ(a) = eiλkψk,ϵ(a). +(III.11) +Note that, unlike the Schroedinger representation where the eigen-space of the momentum operator is one dimensional, +the eigen-spaces of the translation operator (labelled by the eigenvalue eiλk) are infinite dimensional and non separable. +This is explicit from the independence of the eigen-values of the continuous parameter ϵ ∈ [0, λ) labelling eigenstates. +Such huge added degeneracy in the spectrum of the shift operators is a general feature of the polymer representation. +We will show that this degeneracy can show up in Dirac observables of central physical importance such as the mass +operator in Section III G. +C. +Quantum dynamics +The dynamics is dictated by the quantization and imposition of the constraint (II.20). As the operator corresponding +to pa does not exist in our kinematical Hilbert space we introduce a polymerized version. Traditionally, this is achieved +by replacing the infinitesimal translation operator +λ�pa +−→ +traditional +� +sin λpa. +(III.12) +The rule consists of making some ‘minimal’ substitution of pa by a periodic regularization satisfying that in the limit +λ → 0 the functional choice will approximate the original function. Such rule is intrinsically ambiguous and, it opens +in general the door for an infinite set of possibilities. Such choices are to be interpreted as quantization ambiguities +of the Hamiltonian constraint with potential quantitative dynamical consequences (for a general discussion see [39]). +Dynamical implications of these ambiguities can be analysed in detail in simple models of quantum cosmology [38] +and black holes [40]. +In the full theory a new perspective on the regularization issue has been introduced motivated by the novel math- +ematical notion of generalized gauge covariant Lie derivatives [41] and their geometric interpretation allowing for the +introduction of a natural regularization (and subsequent) anomaly free quantization of the Hamiltonian constraint +[42]. Even when the procedure does not eliminate all ambiguities of quantization (choices are available in the part of +the quantum constraint responsible for propagation [43]), the new technique reduces drastically some of them in the +part of the Hamiltonian that is more stringently constrained by the quantum algebra of surface deformations. +What we want to emphasize here is that the analogous procedure in the case of our symmetry reduced Hamiltoinian +has a similar effect. Indeed, because our classical Hamiltonian constraint is linear in the variable pa (whose associated +Hamiltonian vector field has a crystal-clear geometric interpretation of infinitesimal translations in a), one has an + +13 +unambiguous choice of quantization: the obvious choice to replace infinitesimal translations (which do not exist in +the polymer representation) by finite translations or shifts. From this geometric perspective the right polymerization +is the regularization that make the replacement +λ�pa +−→ +� +eiλpa. +(III.13) +In other words the differential time evolution in the Schrodinger equation must be represented in the polymer Hilbert +space by a finite translation with a polymerization scale λ. However, as such an action is associated with a clear +geometric meaning, the geometric compatibility with the Schrodinger equation can be preserved if the second term in +the classical Hamiltonian (II.20) is exponentiated too in order to produce the well known unitary evolution operator +that produces finite area evolution. Disregarding for the moment quantum corrections that will have to be included +near the a = 0 region (see Section III E), the quantum constraint is taken to be +exp(iλpa) +� +�� +� +finite areatime +translation +|ψ⟩ − +finite areatime +unitary evolution operator +� +�� +� +exp +� +i +2 log +� +a + λℓ2 +p +a +� � +m + +p2 +φ +16πℓ2pm +�� +|ψ⟩ = 0, +(III.14) +whose action is well defined in the polymer representation and whose solutions are easily found (by acting on the left +with ⟨m, pφ, a|) to be wave functions satisfying the discrete dynamics given by +ψ(m, pφ, a + λℓ2 +p) = e +i +2 log +� +a+λℓ2 +p +a +�� +m+ +p2 +φ +16πℓ2pm +� +ψ(m, pφ, a). +(III.15) +The physical Hilbert space is defined via the usual inner product at fixed (discrete) time a via +⟨ψ1(a), ψ2(a)⟩phys = +� +∞ +−∞ +� +∞ +−∞ +ψ1(m, pφ, a)ψ2(m, pφ, a)dmdpφ, +(III.16) +which independent of the lattice sites as required (a property that we could identify with the intrinsic unitarity of the +quantum constraint kernel). More precisely the physical inner product is a constant of the quantum motion associated +to the full history represented by the lattice Γϵ,λ as implied by unitarity. Explicitly one has +⟨ψ1(a), ψ2(a)⟩phys = ⟨ψ1(a + λ), ψ2(a + λ)⟩phys +∀ +a ∈ Γϵ,λ. +(III.17) +Ambiguities of regularization that are usually associated with the polymerization procedure are thus completely absent +in this model. The reason is the linear dependence of the Hamiltonian constraint in the polymerized variable which +allows for a regularization fixed by the geometric interpretation of the classical Hamiltonian vector field associated +to the corresponding variable. +However, ambiguities remain when one studies the evolution across the would-be- +singularity of the Kantowski-Sachs model at a = 0. We will study this in the next section. +Now we would like to concentrate on the evolution when we are away from the a = 0. In such regime the one step +evolution (III.15) can be composed to produce the arbitrary initial to final area evolution +ψ(m, pφ, ϵ + nλ) = +�ϵ + nλ +ϵ + qλ +� im +2 +� +1+ +p2 +φ +16πℓ2pm2 +� +ψ(m, pφ, ϵ + qλ), +(III.18) +for arbitrary integers n, q > 1. Evolving across the a = 0 point will be discussed later. +D. +The continuum versus the polymer dynamics +The polymer dynamics that arises from the geometric action of the quantum constraint (III.14) enjoys of the +appealing feature of being closely related to the dynamics that one would obtain in the continuum Schroedinger +representation. This statement can be made precise as follows: any solution of the Schroedinger equation (III.4) +induces on any given lattice Γϵ,λ a solution of (III.14). Conversely, physical states of the polymer theory represent +a discrete sampling of the continuum solutions of (III.4). +However, the Schroedinger evolution is ill defined at + +14 +the singularity a = 0 due to the divergence of the 1/a factor in front of the second term of (III.4). The polymer +representation allows for a well defined evolution across the singularity thanks to the deviations from the 1/a behaviour +introduced by the analog of the ‘inverse-volume’ corrections (see next section). Nevertheless, with the appropriate +modification of the 1/a factor in the Schroedinger equation the correspondence between the discrete and continuum +continues to hold. +E. +Quantum evolution across the classical singularity +Quantum evolution in a for all values of a including the singularity is dictated by the quantum corrected version of +the constraint (III.14) given by +ψ(m, pφ, a) = e +i +2 +� +a� +a0[ 1 +a]qda +�� +m+ +p2 +φ +16πℓ2pm +� +ψ(m, pφ, a0) +(III.19) += e +i +2 [τ(a)−τ(a0)] +� +m+ +p2 +φ +16πℓ2pm +� +ψ(m, pφ, a0), +(III.20) +where +�1 +a +� +q +(III.21) +denotes the quantum corrected expression for the operator a−1 (the analog of inverse volume correction in cosmology) +that can be implemented in various ways due to inherent ambiguities associated to the polymer quantization. In +general this will give deviations of the a−1 behaviour in the region a ∼ ℓ2 +p. This modifies the integral of a−1 in a way +charaterized by the (to a large extend arbitrary [38]) function τ(a) introduced in the second line. One, among the +many possibilities, is the one that follows from the so-called Thiemann’s trick whose most elementary form is (see [11] +for its application in cosmology, see [38] for a discussion of the multiplicity of variants) +�1 +a +� +Thm +≡ sign(a) +� +� +� +|a + ℓ2p| − +� +|a − ℓ2p| +ℓ2p +� +� +2 += 1 +a + O +� +a−3� +. +(III.22) +Integration leads to the following τ(a) function in (III.19) +τ(a) = +� +� +� +|a| +� +|a| − +� +|a|2 − 1 +� ++ log +�� +|a|2 − 1 + |a| +� +− π +2 + 1 +1 ≤ |a| +−|a| +�� +1 − |a|2 − 2 +� +− sin−1(|a|) +|a| ≤ 1 +, +(III.23) +whose graph is presented in Figure 3. +The classical theory does not fix the evolution uniquely in this high curvature regime where quantum geometry +effects cannot be neglected. Quantum geometry effects regularize the dynamics near the a = 0 singularity; one way +of seeing this is that the factor log(a) in the quantum evolution away from the singularity in (III.14) receives inverse +volume quantum geometry corrections. As discussed in [38] these corrections are ambiguous (as fact that should not +be surprising given the expectation that the classical theory cannot guide us all the way to the deep UV in QFT). +Instead of proposing one particular UV extension, as in the example shown where Thiemann regularization was used, +one might simply keep all possibilities open and assume that the corresponding operator is regularized in the relevant +region by some arbitrary function log(a) → τ(a). In regions where τ(a) = log(a) the quantum evolution leads to +semiclassical equations that match exactly Einstein’s equations in the KS sector (more details in Section III F) +F. +Dynamics of semiclassical states and tunelling across the singularity +Let us first consider the vacuum pφ = 0 case. +This case is important because it should correspond to the +Schrwarzschild interior in the region r ≪ M (or a ≪ M 2). The dynamical evolution (III.19) becomes +ψ(m, a) = e +i +2 m(τ(a)−τ(a0))ψ(m, a0). +(III.24) + +15 +a +⌧(a) +-10 +-5 +5 +10 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Figure 3: The dynamics across the singularity is regular due to inverse volume corrections. In this picture we show the function +τ(a), defining the dynamical evolution across the singularity, in the case of Thiemann’s regularization of the inverse-area- +operator. +. +In the pm representation the previous equation simply reads +ψ(pm, a) = ψ(pm + τ(a) − τ(a0) +2 +, a0), +(III.25) +i.e., a simple translation in momentum space. Such dynamical evolution in a implies in an obvious manner that +semiclassical states peaked at classical values (m, pm) at area a0 with some given fluctuations will be simply evolved +into the translated state with the very same fluctuation properties peaked at (m, pm + (τ(a) − τ(a0))/2) at area +a. This is a remarkable property of the quantum system: independently of the quantum gravity effects encoded in +the precise form of τ(a), expectation values satisfy well defined effective dynamical equations which are exact (not +an approximation), and semiclassical states are not spread by the dynamical evolution (as in the simple case of the +harmonic oscillator). Notice that the validity of effective dynamical equations in the context of black hole models +have remained a conjecture in other formulations [44]. +In regions where τ(a) = log a the previous evolution gives +pm(a) = log(r) + pm(a0), +m(a) = m(a0). +(III.26) +From the definitions (II.17) we have that fpf = m(a) and f = − exp[−pm(a)]. The constraint (II.12), gives us h(r) +and the metric becomes +ds2 = e−pm(a)dt2 − +ℓ2 +0 +(4ℓp)4π2 +e−pm(a) +m(a)2 da2 + a +4π dΩ2 +(III.27) +which corresponds to the Schrwarzschild solution with the two Dirac observables M and pM given in terms of the +initial conditions by +M = 2 +√ +4πℓ4 +p +ℓ2 +0 +√a0 +m(a0)2epm(a0), +pM = ℓ0 +√ +4π +2ℓp√a0 +e−pm(a0) +m(a0) . +(III.28) +When quantum inverse volume corrections are taken into account then the quantum evolution is perfectly well defined +across the classical singularity. The evolution of the mean values of a semiclassical state is also well defined and given +by +pm(a) = pm(a0) + 1 +2(τ(a) − τ(a0)), +m(a) = m(a0). +(III.29) + +16 +Such solutions can also be obtained from the Hamiltonian constraint given that one replaces the operator a−1 by +its quantum regularization (effective equations are in this sense exact). The metric for all values of |a| ≪ M 2 but +otherwise arbitrary becomes +ds2 = exp +� +−pm(a0) − 1 +2(τ(a) − τ(a0)) +� � +dt2 − +ℓ2 +0 +(4ℓp)4π2 +da2 +m(a0)2 +� ++ a +4π dΩ2. +(III.30) +Since the Thiemann regularization produces a τ(a) ∼ a2 ∼ r4 near a = 0 we see that the previous metric is just given +by a two dimensional Minkowski metric fibrated with two dimensional sphere with time dependent radius r. The +shrinking of the spheres leads to a singularity at a = 0 where the spheres collapse and the spacetime geometry (as +described by the effective line element) becomes a two dimensional flat one at the singularity in the a-t plane. Despite +the singular nature of the effective metric the fundamental quantum evolution is well defined across the singularity. +In the presence of matter the situation is a bit more involved due to the factor pφ/m appearing in matter contribution +to the Hamiltonian constraint (III.14). However, in the spirit of applying this analysis to macroscopic black holes and +modelling the dynamics of a weak scalar excitation (a Hawking particle) falling into the singularity, it is natural to +focus on semiclassical states (Gaussian) peaked on values such that m ≫ pφ with fluctuations σm ≪ m. As in the +vacuum case m = m0, i.e., m is a constant of motion as well as its spread σm. The dynamics of the conjugate variable +(the mean value pm of the variable pm) can be evaluated using stationary phase methods and the result is +pm(a) = pm(a0) + 1 +2(τ(a) − τ(a0)) +� +1 + +p2 +φ +m(a0)2 +� +1 + 3 +4 +σ2 +m +m(a0)2 +�� ++ O +� +σ3 +m +m(a0)3 +� +, +(III.31) +which can be seen to correspond to the classical solutions found in Section II C. Notice that, as expected from the +form of the matter coupling, there are here quantum corrections characterized by terms proportional to σ2 +m/m(a0)2. +The spread in the variable pm is not time independent if we take into account higher order corrections, namely +σ2 +pm(a) = +1 +σ2m ++ +σ2 +mp4 +φ +4m(a0)4 (τ(a) − τ(a0))2. +(III.32) +The previous equations are derived assuming that the scalar field is in an eigenstate of the momentum pφ. This is an +idealization that simplifies the analysis of the dynamical evolution of the geometry. Similarly, if we assume that the +geometry state was in an eigenstate of m then we can easily analyze the dynamics of the scalar field assuming that it +is initially in a gaussian semiclassical state picked about pφ(a0) and φ(a0). In accordance with the classical solutions +we get +pφ(a) = pφ(a0) +φ(a) = φ(a0) + (τ(a) − τ(a0)) pφ(a0) +16πℓ2pm. +(III.33) +One way to quickly derive these equations by inspection is to realize that the Hamiltonian constraint (II.20) is that +of a non relativistic point particle with mass proportional to our geometric variable m evolving in dτ = [1/a]qda. +Note that (III.31) implies that the back-reaction of the scalar field enters only through a simple modification of the +exponential conformal factor in front of the 2-metric in the a-t ‘plane’ in equation (III.30). +Unlike the geometry degrees of freedom, the fluctuations in the scalar field grow as one approaches the would-be- +singularity: for a given geometry semiclassical state picked around the mass M, the spread of the scalar field σφ in +an initially eigenstate of φ at area a grows to a maximum value close to the would-be-singularity such that +Mσφ < +� +M log +� +a/ℓ2p +� +16πℓ0 +, +σpφ < +� +16πM +ℓ0 log +� +a/ℓ2p +� +(III.34) +which are small in the interior if we take ℓ0 ≫ M as expected from appearance of ℓ0 in the fundamental commutation +relations [45]. Note that, if we take into account inverse volume corrections of the type suggested by LQG (see Figure +3), the scalar field reaches a critical point at a = 0 where its area velocity vanishes. +G. +The mass operator (in the vacuum case) +In this section we concentrate in the vacuum case for simplicity as the mass can be directly read off the form of the +metric in this case via a simple comparison with the classical Schwarzschild solution. In this case the result is +M = 2 +√ +4πℓ4 +p +ℓ2 +0 +√a m2epm. +(III.35) + +17 +where we used the vacuum solution (II.3) (in its r ≪ M approximation) and (II.17). It is easy to verify that the +previous is indeed a Dirac observable by showing that it commutes with Hamiltonian constraint (II.20). Its non linear +dependence on the basic variables anticipates factor ordering ambiguities when it comes to promoting the mass to a +quantum operator. Here we focus on the choice +� +M = α(�a) +� +�me�pm �m +� +, +(III.36) +where α(a) ≡ 2 +√ +4πℓ4 +p/(ℓ2 +0 +√a). The eigenstates equation +� +M|φM⟩ = M|φM⟩, +(III.37) +turns into the differential equation +α(a)epm ∂φM(pm, a) +∂pm ++ α(a)epm ∂2φM(pm, a) +∂p2m ++ MφM(pm, a) = 0, +(III.38) +if we expand the eigenstate in the pm, a basis, namely +|M⟩ = +� +a∈Γϵ,λ +� +φM(pm, a)|pm⟩|a⟩dpm, +(III.39) +where the sum runs over the discrete lattice Γϵ,λ defined in (III.10) when introducing the dynamical constraint (III.14). +This differential eigenvalue equation is solved by +φM(pm, a) ≡ ⟨pm, a| M⟩ = +� +2 +√ +M +α(a) e−pm/2J1 +� +2 +� +M +α(a)e−pm/2 +� +, +(III.40) +where J1 is a Bessel function. One can explicitly verify that the quantum dynamics (III.14) preserves the eigenstates +by explicitly showing that the evolution between arbitrary lattice points a1, a2 ∈ Γλ,ϵ sends the wave function of the +eigenstate at the a1 lattice point to the a2 lattice point (as expected for a Dirac observable); or equivalently, the +eigenstates of the mass are physical states solving (III.14). Explicitly, +� +e +i +2 (log(a2)−log(a1))mφM(pm, a1) = φM +� +pm + 1 +2(log(a2) − log(a1)), a0 +� += φM (p, a2) . +(III.41) +Now the evolution across a = 0 requires inverse volume corrections which modifies the previous dynamical law by +replacing log(a) → τ(a). The mass Dirac observable still exists once inverse volume corrections are turned on. It +corresponds to the modification of (III.36) via the substitution �a → exp(�τ(a)). Eigenstates are also obtained by the +same substitution in (III.40) and satisfy the expected Dirac observable condition (which now holds for lattice points +at different sides across the singularity) +� +e +i +2 (τ(a2)−τ(a1))mφM(pm, a1) = φM +� +pm + 1 +2(τ(a2) − τ(a1)), a0 +� += φM (p, a2) . +(III.42) +When supported on the same lattice, one can show that they satisfy the orthogonality relation +⟨M|M ′⟩phys = δ(M, M ′), +(III.43) +where the inner product is computed with the physical inner product (III.16). Thus the spectrum of the mass operator +is continuous. It was argued in the context of the full LQG theory in [31, 46, 47] that the eigenspaces of the mass +should be infinite degenerate due to the underlying discrete structure of the fundamental theory and the existence of +defects that would not be registered in the ADM mass operator. Interestingly, the conjectured property is illustrated +explicitly in our simple toy model as the eigenvectors (III.39) for a given eigenvalue M there are infinitely many +and labelled by a continuum parameter. More precisely they are associated with wave functions of the form (III.40) +supported on lattices with different values of ϵ. Thus eigenstates of the mass should then be denoted |M, ϵ⟩ with +orthogonality relation +⟨M, ϵ|M ′, ϵ′⟩phys = δ(M, M ′)δϵ,ϵ′, +(III.44) + +18 +where δϵ,ϵ′ is the Kronecker delta symbol. The existence of such a large degeneracy is a generic feature of the polymer +representation. Even when this is a toy model of quantum gravity, this feature is likely to reflect a basic property of +the representation of the algebra of observables in the full LQG context. Here we are showing that the mass operator +is hugely degenerate suggesting that the usual assumption of the uniqueness of the vacuum in background dependent +treatments of quantum field theory might fail in a full loop quantum gravity context. +Alternative factor orderings of the quantum operator M could be treated similarly (some simple choices lead to +slightly different eigenvectors written also in terms of Bessel functions). Such an ambiguity is not relevant for our +purposes (and it does not change the key fact that the spectrum of M is infinitely degenerate) as the aim of the model +is not to construct any quantitative physical prediction but rather to use it as a toy model to investigate possibly +sufficiently generic features that could actually survive in the full theory. The large degeneracy of the mass spectrum +is, in our view, an interesting example of one such feature. +IV. +DISCUSSION +We have shown that test field solutions of the Klein-Gordon equation with zero angular momentum behave like +solutions of the KS symmetry reduced model in the deep interior region r ≪ M defined in terms of the background +Schwarzschild spacetime. This implies that, spherically symmetric scalar matter falling into a spherical black hole can +be modelled by the KS solutions near the singularity. Despite the expected limitations of symmetry reduced models +in capturing the full physics in the UV regime, the model includes back-reaction of the scalar matter. Focusing in the +deep interior region and using perturbation theory in pφ/M we show that it is possible to interpret the solutions of KS +with matter as Schwarzschild solutions with matter excitations falling towards the r = 0 singularity (this interpretation +is not global but it shown to be correct in the deep interior region). The Hamiltonian dynamics simplifies considerably +in that regime becoming tractable both at the classical as well as the quantum level. Perturbation theory applies (we +have shown) to the situation involving Hawking particles falling into the singularity. +In close analogy to LQG, we define a quantization of the system describing the deep interior region where the area of +the r =constant spheres has a discrete spectrum. This leads to the polymer representation of the area of the orbits of +the rotation group and its conjugate momentum that allows for a well defined quantum evolution across the singularity +if one introduces customary ‘inverse-volume’ corrections to the quantum scalar constraint. The Hamiltonian constraint +admits a simple geometric interpretation in the to-be-polymerized sector due to the linearity of the Hamiltonian +constraint in the momentum variable conjugated to the area of the r =constant spheres. The geometric nature of +the action of the classical constraint allows for the introduction of a unique polymerization prescription respecting +this action at the quantum level. This reduces the ambiguity usually associated to the procedure of quantization of +the dynamical constraints for reasons that resonate with the ones that lead to similar advantages in the full theory +[41]. Remarkably, the dynamics is exactly solvable at the quantum level. In the vacuum case, the mass operator is +a Dirac observable that we quantize and whose spectrum is given explicitly. Semiclassical states dynamics leads to +effective evolution equations that can be characterized exactly in the vacuum case and using suitable stationary phase +approximations in the case where matter is present. These effective equations coincide with Einsteins equations in +regions where the inverse volume corrections can be neglected. +An important formal aspect of the model is that it presents a concrete example of violation of the ‘unicity of +the vacuum’ assumption that permeates discussions of Hawking’s information puzzle for over 40 years. +In loop +quantum gravity the discrete structure of the theory at the Planck scale suggests that a given (macroscopic) ADM +mass configuration need not correspond to a unique quantum state. High degeneracy due to the contribution of +microscopic degrees of freedom is expected but hard to prove at the present stage. This leads to a certain degree of +disagreement on the status of such statements in the field at large. Although, a key instance where such degeneracy +is accepted with little controversy is in the loop quantum gravity models designed to calculate black hole entropy (for +reviews and references see [48, 49]) where the statistical origin of the entropy lies precisely in the large multiplicity +of underlying microscopic states. Our simple model might still be too simple to represent definite evidence in this +direction. Nevertheless, the results of Section III G do provide a toy model to eventually study the implications of the +large degeneracy of the mass spectrum in discussion of the fate of information in black hole evaporation. +The model we introduce here is simple and workable, we hope it could provide potentially useful insights in dealing +with qualitative questions concerning black hole evaporation. The investigation of these interesting possibilities is left +for the future. + +19 +V. +ACKNOWLEDGEMENTS +We thank the interaction with Simone Speziale for valuable insights and specially for discussion on the hamiltonian +analysis of the system. +References +[1] A. Ashtekar, “New Variables for Classical and Quantum Gravity,” Phys. Rev. Lett. 57 (1986) 2244–2247. +[2] A. Ashtekar, “New Hamiltonian Formulation of General Relativity,” Phys. Rev. D 36 (1987) 1587–1602. +[3] A. Perez, “Introduction to loop quantum gravity and spin foams,” gr-qc/0409061, published in 2nd International +Conference on Fundamental Interactions (ICFI 2004) 6-12 Jun 2004. Domingos Martins, Espirito Santo, Brazil. +[4] A. Ashtekar and J. Lewandowski, “Background independent quantum gravity: A Status report,” Class. Quant. Grav. 21 +(2004) R53, arXiv:gr-qc/0404018. +[5] C. Rovelli, Quantum Gravity. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2004. +[6] T. Thiemann, Modern Canonical Quantum General Relativity. Cambridge Monographs on Mathematical Physics. +Cambridge University Press, 2007. +[7] I. Agullo and P. Singh, Loop Quantum Cosmology, pp. 183–240. WSP, 2017. arXiv:1612.01236. +[8] M. Bojowald, “Absence of singularity in loop quantum cosmology,” Phys.Rev.Lett. 86 (2001) 5227–5230, +arXiv:gr-qc/0102069. +[9] M. Bojowald, “Loop quantum cosmology,” Living Rev. Rel. 8 (2005) 11, arXiv:gr-qc/0601085. +[10] A. Ashtekar, M. Bojowald, and J. Lewandowski, “Mathematical structure of loop quantum cosmology,” Adv. Theor. +Math. Phys. 7 (2003), no. 2, 233–268, arXiv:gr-qc/0304074. +[11] A. Ashtekar and P. Singh, “Loop Quantum Cosmology: A Status Report,” Class. Quant. Grav. 28 (2011) 213001, +arXiv:1108.0893. +[12] V. Taveras, “Corrections to the Friedmann Equations from LQG for a Universe with a Free Scalar Field,” Phys. Rev. D +78 (2008) 064072, arXiv:0807.3325. +[13] L. 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Singh, “Quantum Transfiguration of Kruskal Black Holes,” Phys. Rev. Lett. 121 (2018), +no. 24, 241301, arXiv:1806.00648. +[21] J. Olmedo, “Brief review on black hole loop quantization,” Universe 2 (2016), no. 2, 12, arXiv:1606.01429. +[22] R. Gambini, J. Olmedo, and J. Pullin, “Quantum geometry and black holes,” arXiv:2211.05621. +[23] A. Ashtekar, J. Olmedo, and P. Singh, “Regular black holes from Loop Quantum Gravity,” arXiv:2301.01309. +[24] P. Nicolini, E. Spallucci, and M. F. Wondrak, “Quantum Corrected Black Holes from String T-Duality,” Phys. Lett. B +797 (2019) 134888, arXiv:1902.11242. +[25] B. Koch and F. Saueressig, “Black holes within Asymptotic Safety,” Int. J. Mod. Phys. A 29 (2014), no. 8, 1430011, +arXiv:1401.4452. +[26] F. Saueressig, N. Alkofer, G. D’Odorico, and F. Vidotto, “Black holes in Asymptotically Safe Gravity,” PoS FFP14 +(2016) 174, arXiv:1503.06472. +[27] J. M. Bardeen, “Non-singular general-relativistic gravitational collapse,” in Proc. Int. Conf. GR5, Tbilisi, vol. 174, p. 174. +1968. +[28] S. A. Hayward, “Formation and evaporation of regular black holes,” Phys. Rev. Lett. 96 (2006) 031103, +arXiv:gr-qc/0506126. + +20 +[29] T. De Lorenzo, C. Pacilio, C. Rovelli, and S. Speziale, “On the Effective Metric of a Planck Star,” Gen. Rel. Grav. 47 +(2015), no. 4, 41, arXiv:1412.6015. +[30] V. P. Frolov, “Notes on nonsingular models of black holes,” Phys. Rev. D 94 (2016), no. 10, 104056, arXiv:1609.01758. +[31] A. Perez and D. Sudarsky, “A dialog on the fate of information in black hole evaporation,” arXiv:2205.08469. +[32] R. Kantowski and R. K. Sachs, “Some spatially homogeneous anisotropic relativistic cosmological models,” Journal of +Mathematical Physics 7 (1966), no. 3, 443–446. +[33] A. Ashtekar and A. del Río, “Probing cosmological singularities with quantum fields: Open and closed FLRW universes,” +arXiv:2209.09922. +[34] A. Ashtekar, A. del Río, and M. Schneider, “Space-like Singularities of General Relativity: A Phantom menace?,” Gen. +Rel. Grav. 54 (2022) 45, arXiv:2205.00298. +[35] B. C. Xanthopoulos and T. Zannias, “Kantowski‚ÄìSachs metrics with source: A massless scalar field,” Journal of +Mathematical Physics 33 (1992), no. 4, 1415–1419, arXiv:https://doi.org/10.1063/1.529717. +[36] D. N. Page, “Particle Emission Rates from a Black Hole: Massless Particles from an Uncharged, Nonrotating Hole,” Phys. +Rev. D 13 (1976) 198–206. +[37] E. Poisson and W. Israel, “Internal structure of black holes,” Phys. Rev. D 41 (1990) 1796–1809. +[38] L. Amadei, A. Perez, and S. Ribisi, “The landscape of polymer quantum cosmology,” arXiv:2203.07044. +[39] A. Perez, “On the regularization ambiguities in loop quantum gravity,” Phys. Rev. D 73 (2006) 044007, +arXiv:gr-qc/0509118. +[40] J. Münch, A. Perez, S. Speziale, and S. Viollet, “Generic features of a polymer quantum black hole,” arXiv:2212.06708. +[41] A. Ashtekar and M. Varadarajan, “Gravitational Dynamics—A Novel Shift in the Hamiltonian Paradigm,” Universe 7 +(2021), no. 1, 13, arXiv:2012.12094. +[42] M. Varadarajan, “Anomaly free quantum dynamics for Euclidean LQG,” arXiv:2205.10779. +[43] M. Varadarajan and A. Perez, “public and private discussion during LOOPs 22 conference,” July 2022. +[44] A. Ashtekar, J. Olmedo, and P. Singh, “Quantum extension of the Kruskal spacetime,” Phys. Rev. D 98 (2018), no. 12, +126003, arXiv:1806.02406. +[45] C. Rovelli and E. Wilson-Ewing, “Why are the effective equations of loop quantum cosmology so accurate?,” Phys. Rev. +D 90 (2014), no. 2, 023538, arXiv:1310.8654. +[46] A. Perez, “No firewalls in quantum gravity: the role of discreteness of quantum geometry in resolving the information loss +paradox,” Class. Quant. Grav. 32 (2015), no. 8, 084001, arXiv:1410.7062. +[47] L. Amadei, H. Liu, and A. Perez, “Unitarity and information in quantum gravity: a simple example,” Front. Astron. +Space Sci. 8 (2021) 46, arXiv:1912.09750. +[48] J. F. Barbero G. and A. Perez, Quantum Geometry and Black Holes, pp. 241–279. WSP, 2017. arXiv:1501.02963. +[49] A. Perez, “Black Holes in Loop Quantum Gravity,” Rept. Prog. Phys. 80 (2017), no. 12, 126901, arXiv:1703.09149. + diff --git a/zNE2T4oBgHgl3EQfhwde/content/tmp_files/load_file.txt b/zNE2T4oBgHgl3EQfhwde/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c8f160a08f6db3d2cb569322a480c4cd20511f60 --- /dev/null +++ b/zNE2T4oBgHgl3EQfhwde/content/tmp_files/load_file.txt @@ -0,0 +1,871 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf,len=870 +page_content='Modelling quantum particles falling into a black hole: the deep interior limit Alejandro Perez, Salvatore Ribisi, and Sami Viollet Aix Marseille Université, Université de Toulon, CNRS, CPT, Marseille, France (Dated: January 11, 2023) In this paper we construct a solvable toy model of the quantum dynamics of the interior of a spherical black hole with falling spherical scalar field excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' We first argue about how some aspects of the quantum gravity dynamics of realistic black holes emitting Hawking radiation can be modelled using Kantowski-Sachs solutions with a massless scalar field when one focuses on the deep interior region r ≪ M (including the singularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Further, we show that in the r ≪ M regime, and in suitable variables, the KS model becomes exactly solvable at both the classical and quantum levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The quantum dynamics inspired by loop quantum gravity is revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' We propose a natural polymer-quantization where the area a of the orbits of the rotation group is quantized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The polymer (or loop) dynamics is closely related with the Schroedinger dynamics away from the singularity with a form of continuum limit naturally emerging from the polymer treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The Dirac observable associated to the mass is quantized and shown to have an infinite degeneracy associated to the so-called ϵ-sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Suitable continuum superpositions of these are well defined distributions in the fundamental Hilbert space and satisfy the continuum Schroedinger dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' MOTIVATION The fate of the singularities of general relativity is a central question for quantum gravity that concerns important physical situations such as those arising in (big-bang) cosmologies and black hole formation and evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' One of the central features of loop quantum gravity is the inherent discreteness of quantum geometry at the Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The lack of smoothness of the geometry at the fundamental level challenges the classical view of the singularities of general relativity as a frontier of spacetime geometry, and strongly suggests the possibility of a microscopic dynamical description that could define dynamics beyond the limit where classical description fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The history of the approach starts with the discovery of Ashtekar’s connection variables which first suggested that the quantum dynamical evolution equations of gravity might admit a background independent finite and non perturbative formulation [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This initial suggestion grew into the approach of loop quantum gravity (LQG) with the contribution of many (for reviews and text books see [3–7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The LQG approach has produced insights about the possible nature of matter and geometry at the Planck scale and has led to new ideas about the origin of black hole entropy, the generation of quantum effects in early cosmology, and stimulated hopes about the possible regularizing role of Planckian granularity (for quantum field theory and gravity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' However, a clear understanding of the question of the fate of singularities in realistic physical situations has remained a difficult one, as addressing it would actually require the (still lacking) complete dynamical control of LQG in situations involving matter and geometry degrees of freedom in the deep ultraviolet regime in full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Nevertheless, the view that the evolution across singularities should be well behaved has become consensual in the field over time thanks to the accumulated experience in simple low dimensional as well as symmetry reduced models of black holes and cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Professor Abhay Ashtekar has been one of the key leading driving forces along this path, and main defender of the view (to which we adhere) that dynamics across the would-be-singularity should be well defined in the quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' It is a pleasure to contribute to this special issue with this work that, we believe, is representative of this standpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The first examples of singularity avoiding models where found in the context of quantum cosmology by Martin Bojowald [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This seminal work grew later into a large number of contributions in the field now known as loop quantum cosmology [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Even in these simple models the quantum dynamics can be rather involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' However, it was soon realized [12] that effective semiclassical equations could be used to describe the dynamics across the singularity and that these equations were quite easy to describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The domain of applicability of these techniques was extended in a variety of manners to models involving black holes [13–20] (for reviews see [21–23], for quantum modifications inspired by other approaches see [24–26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In many of the latter cases the natural starting point has been to consider the quantum dynamics of the interior of spherically symmetric and static spacetimes of the Kantowski- Sachs type (the Schwarzschild black hole interior in the vacuum case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In all these cases the interior singularity is removed and replaced by a quantum transition across what would have been the singularity in the classical description realizing aspects of existing scenarios [27–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Simple models are nice as they illustrate possibly generic features of the general situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' However, they carry the drawback of being often removed, by the very symmetry assumptions that simplify them, from the realistic physical situations about which one would like to gain non-trivial insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Moreover, when it comes to black holes, most of the studies have focused on effective dynamical descriptions, while quantum dynamics has received less attention due arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='03951v1 [gr-qc] 10 Jan 2023 2 to its often unsurmountable complexity even in simple models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' For instance, concerning the first drawback we know that real black holes are not time translational invariant due to the expected presence of Hawking evaporation (in contrast with the static nature of many of the quantum black hole models) and that all symmetry assumptions must fail near the singularity when the back-reaction of Hawking particles correlated with the outside radiation would be properly taken into account (see [31] for further discussion of this issue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' When it comes to the second drawback, even when effective descriptions can provide the dynamical evolution of the spacetime geometry with matter fields on it, its classical nature precludes the analysis of genuine quantum phenomena such as entanglement and other quantum information issues of highest interest from the perspective of Hawking’s evaporation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' the longstanding information puzzle or the question of the fate of unitarity in black hole evaporation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Even when our model will not resolve the first limitation, we believe that it provides a humble small step in the right direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Concerning the second, we will see that the quantum dynamics is fully accessible in our simple model opening the road for exact calculations in the quantum realm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Quantum gravity region ◆0 I+ I� Hawking pairs Matter Friday, 23 September 22 Figure 1: The Ashtekar-Bojowald paradigm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The interior region r < 2M of a Schwarzschild black hole of mass M can be seen as a homogeneous anisotropic cosmological model where the r=constant surfaces (in the usual Schwarzschild coordinates) are Cauchy surfaces of homogeneity where any two arbitrary points can be connected along orbits of the isometry group that involves spacelike translations along the staticity Killing field ξ = ∂t and the rotations associated to spherical symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Models with these isometries will be refereed to as Kantowski-Sachs (KS) models [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' They include not only the Schwarzschild black hole interior geometry (vacuum case) but also the Reissner-Nordstrom black hole interior geometry (in the Einstein-Maxwell case) and other solutions depending of the type of matter that one decides to couple to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In this paper we would like to emphasize the fact that Kantowski-Sachs models (with a massless scalar field coupling) define a natural toy model capturing some (possibly interesting) aspects of the dynamics and back-reaction of matter near the singularity of realistic black holes that Hawking radiate and evaporate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The model can certainly not replace the full dynamical description of a generic gravitational collapse in the full theory as it remains a toy model with finitely many degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' However, we will argue, it can handle in a simplistic way some dynamical aspects that might be relevant when discussing questions in the context of evaporating black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Scalar excitations falling inside a Schwarzschild black hole: the deep interior regime Let us consider a free test point particle (with no angular momentum) falling into the interior of a Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' As the particle approaches the singularity—in a description on an r equal constant slicing of the interior—one 3 expects its wave function to become better and better approximated by a translational invariant wave function since the expansion in the spacelike Killing direction ξ = ∂t diverges for r → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' If this conclusion is correct then it means that zero angular momentum test particles can be approximated by the type of excitations that can be accommodated in the dynamical framework of the KS cosmologies (at least in the sense of a near singularity approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' One can be quantitative about this intuition as follows: free test particles with four wave vector ka on the Schwarzschild background are associated with the conserved Killing energy E ≡ −kaξa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' We are assuming that the particle has zero angular momentum which implies that its wave function is already translational invariant in the directions transversal to ξa on the r-slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The wave function can only vary in the direction of the Killing ξa and the component of the physical momentum in this direction is given by pξ ≡ kaξa √ξ · ξ = −E � r 2M − r, (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='1) which vanishes in the limit r → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The wave length of such a particle diverges, and thus particles without angular momentum are better and better represented by translational invariant excitations as one approaches the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' These are precisely the kind of homogeneous configurations that can be described in the KS framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This simple implication deduced from the idealized notion of test particle can be made more precise by looking at the analogous features of scalar field excitations (solutions of the Klein-Gordon equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Indeed, the simplistic argument given here can be made precise in the field theoretical context as we will show in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Solutions of the Klein-Gordon equation in the deep interior region Here we argue that the Kantowski-Sachs model (described in detail in Section II) coupled to a massless scalar field faithfully captures the dynamics of a Klein-Gordon excitation falling into the deep interior region of a Schwarzschild black-hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' We will do this by analysing the behaviour of solutions of the Klein-Gordon equation on the Schwarzschild background in the r ≪ 2M regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' We will focus on the spherically symmetric solutions and show that they become homogeneous on r=constant surfaces as r → 0 and thus can be accommodated in the framework of KS configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This implies that the KS system can be used to model the dynamics and (most importantly) the back-reaction of such (zero angular momentum) scalar configurations falling into the deep interior region of spherically symmetric black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Let us first start by approximating the Schwarzschild metric in the deep interior region r ≪ 2M as ds2 = 2M r dt2 − r 2M dr2 + r2dΩ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='2) We will choose coordinates such that the time-radial part of the metric is conformally flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Remarkably, in the deep interior region, this is achieved by switching to area variables a = 4πr2 (the well known tortoise coordinate r∗ is actually proportional to r2 near the singularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' With these variables the metric becomes ds2 = 1 16π√aHa � dτ 2 − da2� + a 4π dΩ2, (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='3) with aH = 16πM 2, τ = √16πaHt, and a = 4πr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The Klein-Gordon equation for a massless scalar field then reads □Φ = 1 √−g ∂µ �√−ggµν∂νΦ � = 0 (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='4) ⇐⇒ � −∂2Φ ∂a2 − 1 a ∂Φ ∂a + ∂2Φ ∂τ 2 + 1 4√aHaa � 1 sin θ ∂ � sin θ ∂Φ ∂θ � ∂θ + 1 sin2 θ ∂2Φ ∂ϕ2 �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='5) To solve it, we make the usual ansatz Φℓm = eiωtYlm(θ, ϕ)φl(a) = e i ωτ √ 16πaH Ylm(θ, ϕ)φl(a) (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='6) which reduces the Klein-Gordon equation to φ′′ l + φ′ l a + � ω2 16πaH + l(l + 1) 4√aHaa � φl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='7) 4 For the spherical modes l = 0 one obtains φ0(a) = c1J0 � ωa √16πaH � + c2Y0 � ωa √16πaH � , (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='8) with J0 and Y0 Bessel functions and c1 and c2 constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' As we will prove in Section II that these solutions match nicely with solutions of the KS model (mentioned at the end of the introduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This follows from two complementary properties of the solutions of the Klein-Gordon equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' On the one hand, the near singularity limit of the quantity (simply related to the KS momentum variable as we will see in Section II) lim a→0 a∂φ0 ∂a = −2c2 π (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='9) is finite and independent of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' On the other hand, the t dependence of the Klein-Gordon solutions is ‘ironed’ by the infinite expansion of the geometry in the ∂t direction: the region ℓ0 ≡ ∆t = 2πω−1 where the solution has a significant (order-one) change corresponds to a length scale ∆d ≈ 2πω−1� M/r (in agreement with the infinite redshift effect captured in equation (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Therefore, in a length scale ℓp ≪ ℓ ≪ ∆d along the background Killing field direction ξ = ∂t, and in the deep interior region a ≪ M 2, the solutions of the Klein-Gordon equations can be considered as homogeneous and therefore compatible with initial data that would be admissible in the KS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' We will see in Section II that the momentum variable pφ in the KS model is simply related to the quantity whose limit was considered in the previous paragraph as it is defined as pφ ≡ −8πMℓ0r∂Φ00 ∂r = −ℓ0aHa∂Φ00 ∂a , (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='10) where the pre-factors arise form the hamiltonian analysis of Section II and ℓ0 is an IR cutoff scale naturally associated in the previous discussion to the scale ∆t = 2πω−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' It follows from the previous considerations that lim a→0 pφ = constant (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='11) in the region of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' One can relate the previous quantity to the average ‘energy density’ on the r=constant hyper-surfaces as one approaches the singularity (this will be simply related to the KS Hamiltonian that will be defined in the following section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Namely, 1 ℓ0 � t0+ℓ0 t0 dt �� dθdϕ �� |g|Tµν∂µ r ∂ν r �� = p2 φ 64πM 2ℓ2 0 , (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='12) where the scale ℓ0 enters the definition of the average in the time direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' For concreteness one can match ℓ0 to the wavelength 2πω−1 of the excitation and the previous result will already hold (of course it holds for ℓ0 > 2πω−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' For completeness we give the limiting behaviour of solutions in the non-spherical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' For the non-spherical modes, one can neglect the term containing the frequency in equation (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='7) in analysing the small a behaviour of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' If we do so, we obtain for l ̸= 0 φl(a) = c1J0 � 4 � a π � l(l + 1) M � + 2c2Y0 � 4 � a π � l(l + 1) M � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='13) Solutions diverge logarithmically (as log[a]) for a → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This holds both for spherically symmetric as well as for non spherically symmetric solutions as it follows from the asymptotic behaviour of the Bessel functions or from the finiteness of pφ in the spherical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The mild character of the divergence was emphasized in [33, 34] as an attractive possibly interesting property when one considers the definition of the associated quantum operators in quantum field theory (in view of a possible definition of semiclassical gravity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Here we simply point out that such simple behaviour allows for bridging to a solvable KS model to understand aspects of the back-reaction of classical (as well as quantum) excitations falling into a spherically symmetric black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' THE KANTOWSKI-SACHS SPACETIME COUPLED WITH A MASSLESS SCALAR FIELD In this section we revisit the construction of the phase space of the KS model by perforing the canonical analysis of the associated symmetry reduced model where staticity and spherical symmetry are imposed from the onset (in Section II A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' To improve the clarity of the presentation we simply start from the vacuum case—whose solutions are 5 isomorphic to the interior Schwarzschild solutions—and later couple the system to a scalar field without mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' We express variables in terms of the usual Schwarzschild-like coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In Section II B we present a truncation of the Hamiltonian and show, in Section II C, that it defines a tractable approximation of the dynamics in the r ≪ M region of the interior of physically realistic black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' We call this regime the deep interior dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In Section II D we show that the regime of applicability of the model includes the physically interesting situation of Hawking scalar excitations with zero angular momentum falling inside of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Symmetry reduced covariant phase space It is well known that for a spherically symmetric and static spacetime, the line element can be written without any loss of generality as ds2 = −f(r)dt2 + h(r)dr2 + r2dΩ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='1) It follows that the Eintein-Hilbert action (with the appropriate boundary term that renders it differentiable) becomes Sgeo = 1 16π �� R d4x√−gR + 2 � ∂R K � = ℓ0 2ℓ2p � dr � � fh + � f h + ˙fr √fh � , (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='2) where the dot denotes the derivative with respect to r and ℓ0 is a cut-off in the non compact spacelike ∂t direction that regularizes the dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The cut-off will be associated a natural meaning in modelling the fate of zero angular momentum excitations falling inside the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In the deep interior region r ≪ M and we will take ℓ0 ∼ ω−1 for ω ≈ 1/M (the typical frequency in the Hawking spectrum of a macroscopic black hole of mass M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' One can easily verify that the variations of the action lead to the Schwarzschild solutions ds2 = −p2 M � 1 − 2M r � dt2 + dr2 � 1 − 2M r � + r2dΩ2, (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='3) and the symplectic potential (stemming from the on-shell evaluation of the action variation) θ = −ℓ0 ℓ2p (c1dM + 2MdpM − 2dpMr), (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='4) and symplectic structure ω = ℓ0 ℓ2p dpM ∧ dM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='5) Instead of working directly with the physical phase space parametrized by the Dirac observables pM and M it will be convenient for us to work with kinematical variables and constraints for the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This is because of the usual difficulty in linking the timeless physical phase space with a classical intuition based on spacetime geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' To avoid such difficulties we would like to have a notion of parametrized ‘time evolution’ which in our context will take the form of an area radius evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Thus we take the integrand of (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='2) as the Lagrangian Lgeo of the spacetime subsystem Lgeo = ℓ0 2ℓ2p � � fh + � f h + r ˙f √fh � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='6) On the other hand, we will couple the system to a massless scalar field by adding the matter action Sm = −1 2 � R d4x√−g∂aφ∂aφ = −2πℓ0 � drr2 ˙φ2 � f h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='7) The conjugate momenta to f, h and φ are given by pf = ℓ0 2ℓ2p r √fh , ph = 0 and pφ = −4πr2ℓ0 � f h ˙φ, (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='8) 6 and the primary Hamiltonian, defined by H = ˙fpf + ˙hph + ˙φpφ − Lφ − Lgeo, becomes H1 = − ℓ0 2ℓ2p f(h + 1) √fh − hp2 φ 8πr2ℓ0 √fh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='9) From the expression of the conjugate momenta (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='8) we identify the constraints ξ ≡ pf − ℓ0 2ℓ2p r √fh = 0 and ph = 0, (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='10) and the secondary Hamiltonian H2 = H1 + λξ + ηph , (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='11) where λ and η are Lagrange multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' One can show that the stability of the two constraints (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='10) can be ensured by fixing the associated Lagrange multipliers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=', the constraints (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='10) are second class and can be explicitly solved leading to ph = 0 and h = ℓ2 0 4ℓ4p r2 fp2 f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='12) Thus, the secondary Hamiltonian (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='11) reduces to H2 = −1 r � fpf + 1 16πℓ2p p2 φ fpf + ℓ2 0 4ℓ4p r2 pf � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='13) The previous encodes the KS dynamics of geometry coupled to a massless scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The relevant solutions for phys- ical applications correspond to small departures from the vacuum Schwarzschild solutions representing macroscopic black holes with scalar field perturbation falling inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' We will further simplify the system by focusing on, what we call, the deep interior region r ≪ M where M is the mass scale defined by the corresponding black hole solution perturbed (in the sense of Sections II C and Section II D) by the presence of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' It is in this regime where the so- lutions of the KS system faithfully describe the dynamics of a spherically symmetric scalar perturbation (representing for instance a Hawking particle) as it falls towards the interior singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The KS Hamiltonian evolution matches, in this sense, the test-field evolution (the Klein-Gordon solutions on the Schwarzschild background fixed non dynamical background) and incorporates, as a simplified model, aspects of the back-reaction that are expected to become more important as one approaches the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The deep interior dynamics We are interested in the dynamical evolution in the r ≪ 2M regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In addition we will use the present dynamical system to model (in a suitable approximation) the back-reaction of a Hawking quantum falling into a black hole singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Hawking particles do not correspond to static excitations as the one we can model with the symmetry assumptions of the present section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' However, as argued in Section I A, when spherically symmetric, these particles look more and more static as seen by a radially freely falling observer in the limit r → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This is the reason why we are interested in such regime of the present dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In the next section we will study the classical solution of the model using perturbation theory in the parameter p2 φ/M 2—as p2 φ will be assumed to be much smaller to M 2 in applications—and show that the dynamics simplifies in the deep interior region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The simplification occurs due to the negligible effect of the last term in the expression of the hamiltonian (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='13): more precisely, in the deep interior region, the Hamiltonian is well approximated by Hdi = −1 r � fpf + 1 16πℓ2p p2 φ fpf � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='14) This toy theory reflects the dynamics of the leading order in an expansion near r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Order O(r) effects could be included in perturbation theory near r = 0 in which case the term we dropped would correspond to the perturbation Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The consistency of this truncation will be shown in Section II C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The system we are dealing with has no gauge symmetries as the radial reparametrization symmetry has been gauged fixed with the metric ansatz (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='1) by choosing the area radius as time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In order to recover the structure of a gauge 7 theory, with a clear analogy with the full theory of LQG, it will be convenient ‘reparametrize’ the system by promoting the area radius r to a degree of freedom with conjugate momentum pr and add a scalar constraint C = pr − H2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The phase space is therefore extended to (f, pf, r, pr, φ, pφ), and the number of degrees of freedom is preserved by the inclusion of the Hamiltonian constraint Cr = pr − 1 r � fpf + p2 φ 16πℓ2pfpf � ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='15) In this approximation one can show that we have the following Dirac observables D1 = fpf , D2 = fr − p2 φ 4fp2 f +1 , D3 = pφ , and D4 = φ + pφ log(r) 2fpf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='16) It will be convenient to make the following canonical transformation and thus introduce what we call the deep interior variables m = −fpf and pm = − log(−f) , (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='17) and—in trying to introduce the kinematical structure proper to loop quantum gravity—to adopt the area a of the surfaces of constant r, namely a = 4πr2 and pa = pr 8πr , (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='18) as new dynamical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' With this choice the phase space is described by the geometric variables m, pm, a and pa with Poisson brackets {m, pm} = 1 , {a, pa} = 1 , and by the matter variables φ and pφ for which {φ, pφ} = 1, (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='19) with all the other Poisson brackets equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In the new variables the deep interior dynamics Hamiltonian constraint (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='15) becomes Ca = pa + 1 2a � m + p2 φ 16πℓ2pm � ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='20) The previous constraint is central in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' We will see that it leads to a fully controllable dynamics both at the classical as well as the quantum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Indeed, the dynamics is exactly solvable in the vacuum case while it can be dealt with in perturbation theory for the case where the scalar field is excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In the next section we will justify the truncation that took us from (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='13) to (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='14) (and finally to the constraint (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='20)) using perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In Section II D we will show that the perturbative regime is consistent with the conditions that make our model applicable to the description of a spherically symmetry Hawking particle falling into a Schwarzschild black hole during evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Perturbative solutions in pφ/M and the dynamics in the deep interior region Exact KS solutions with scalar fields have been studied in the past (see for instance [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' KS coupled to scalar fields does not lead necessarily to (asymptotically flat) back hole spacetimes globally speaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' However, we will show here that solutions can be interpreted in terms of perturbations of a vacuum Schwarzschild solution in the deep interior region r ≪ 2M in the regime where pφ/M ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' We will also show that in that regime the Hamiltonian (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='13), and the equations it generates, can be well approximated by (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This will lead to a simple solvable system, both at the classical and quantum levels, which can be used to model aspects of the physics of (zero angular momentum) scalar particles falling into a spherically symmetric black hole (possibly useful in view of describing aspects of Hawking radiation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' We will analyze the system in first order perturbation theory in pφ/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' 8 In order to best organise the perturbative equations we replace p2 φ by ϵ2p2 φ where ϵ is a smallness parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' We introduce the following expansion of the relevant dynamical quantities f(r) = f0(r) + ϵ2f1(r) + O(ϵ4), (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='21) pf(r) = pf0(r) + ϵ2pf1(r) + O(ϵ4) , (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='22) and write the equations of motion for them by keeping terms up to order ϵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Starting from ˙f = {f, H2} = −f(r) r + ϵ2p2 φ 16πℓ2prf(r)pf(r)2 + ℓ2 0r 4ℓ4ppf(r)2 , ˙pf = {pf, H2} = pf(r) r − ϵ2p2 φ 16πℓ2prf(r)2pf(r), (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='23) with H2 given in equation (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='13), one can solve the equations order by order with solutions f0(r) = p2 M � 1 − 2M r � , f1(r) = − ℓ2 pp2 φ 8πrℓ2 0M 2 � 2M + (M − r) log � r 2M − r �� , (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='24) pf0(r) = ℓ0 4ℓ2ppM r, pf1(r) = − p2 φr 32πℓ0M 2p3 M � 2M 2M − r + log � r 2M − r �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='25) The function h(r), is recovered from the constraint (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='10) and gives h0(r) = 1 1 − 2M r , h1(r) = − ℓ2 pp2 φr log � r 2M−r � 8πℓ2 0p2 MM(2M − r)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='26) Note that the Schwarzschild solution with mass M and conjugate momentum pM—as in (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='3)—is recovered in the leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Consistency of the perturbative treatment requires the ratio of first-to-leading order contributions to remain small, namely f1(r) f0(r) = ℓ2 pp2 φ 32πℓ2 0M 2p2 M log � r2 4M 2 � + O � r M � ≪ 1, h1(r) h0(r) = ℓ2 pp2 φ 32πℓ2 0M 2p2 M log � r2 4M 2 � + O � r M � ≪ 1 (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='27) in the regime r < M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The logarithmic behaviour of the right hand side of the previous equations is not a threat, as the classical solutions are not to be trusted for r near the Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Therefore, assuming r > ℓp, we conclude that our analysis is consistent as long as ℓ2 pp2 φ 32πℓ2 0M 2p2 M log � ℓ2 p 4M 2 � ≪ 1 (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='28) which is always valid in the usual macroscopic black hole regime ℓp/M ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Similarly, note that the ratio of the last term in the Hamiltonian (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='13) to the leading first term is exactly h(r) which in the same regime is O(ℓ2 p/M 2 log(ℓp/M)) as seen from (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This suggests that one can simplify the dynamics (if interested in the deep interior region r ≪ M) and use the Hamiltonian Hdi = −1 r � fpf + 1 16πℓ2p p2 φ fpf � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='29) This expectation is confirmed by the analysis of the Hamiltonian flow generated by Hdi in the relevant regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' One might be slightly uncomfortable with the UV cut-off used in devising conditions such as (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='28), after all the classical equations predict a singular evolution when r → 0 and in this limit the right hand sides of the equations (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='27) actually blows up invalidating in appearance the perturbative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' We will see that the quantum dynamics across the classical singularity is actually well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Moreover, we will also see that there are exact effective equations describing the evolution of the expectation value of semiclassical states across r = 0 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' These equations imply that the counterpart of the right hand side of (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='27) not only does not diverge but rather tends to zero when quantum effects are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' 9 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Validity of perturbation theory in view of applying the model to Hawking pairs produced by a macroscopic black hole Here we argue for the validity of the perturbation theory analysis of the previous section in the context of applications of the KS model to the quantum dynamical description of scalar field excitations falling inside a Schwarzschild black hole during Hawking evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The assumptions here are: First the usual assumption that the gravitational collapse has formed a spherical black hole with mass M ≫ mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Second, that the Hawking pairs produced by such a macroscopic black hole can be described as test field excitations on a stationary Schwarzschild geometry far away from the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Finally, we restrict our attentions to spherically symmetric Hawking pairs which are the only that can be mapped to the spherically symmetric KS configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This is not a serious restriction given that most of the Hawking radiation is emitted in such modes [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The stationarity of the background implies that the (test field) current ja ≡ Tabξb is conserved, where ξa is the stationarity Killing field that in the adapted coordinates that we use here is simply given by ξ = ∂t and Tab is the energy momentum tensor of the scalar field1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Conservation of energy leads to the expectation that when a Hawking particle is detected at I + with a given energy E = ω (which coincides with the associated Killing conserved quantity), a Hawking partner with (Killing) energy −ω falls into the black hole singularity reducing the mass of the black hole by ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' A more precise field theoretical expression of this is that the flux of the (expectation value of) the energy- momentum current ja at infinity in the state of the field after the detection of the particle at I + equals minus the flux of the same current on (for instance) a constant r < 2M hyper-surface Σr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Namely, � Σr janadV (3) = −ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='30) Using our coordinates to write explicitly the integrand in the left hand side while pushing Σr to the region where comparison with the KS regime is possible (a constant r such that r ≪ 2M) we get 8πM � ℓ0 0 Tr0(t, r)rdt = 8πM � ℓ0 0 ∂tφ∂rφrdt = ωp2 φ 16πMℓ0 log(r) + O(1) = −ω, (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='31) where the right hand side of the last equality comes from the evaluation of the energy flux of a Hawking particle, while in the evaluation of the left hand side we used that for r ≪ 2M (see for instance (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='11)) the scalar field behaves like φ = Re � e−iωt � φ0 − pφ 8πMℓ0 log(r) �� + O � r M � = cos(ωt) � φR 0 − pR φ 8πMℓ0 log(r) � + sin(ωt) � φI 0 − pI φ 8πMℓ0 log(r) � + O � r M � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='32) The result in (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='31) follows also from the assumption that ℓ0 > ω−1 ≈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' By pushing the integral to its largest possible value when r → ℓp, we conclude that energy conservation (encoded in (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='31)) implies the bound p2 φ 16πM 2 log � ℓp M � < 1, (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='33) which is consistent with the condition for the validity of perturbation theory (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' We see that the physical context provided by the problem of Hawking evaporation precisely justifies the simplifying assumptions that led to (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Notice also that the fiducial length scale ℓ0 acquires in such a physical situation an operational meaning as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' A generalization of the analysis of the present Sections II to a wider class of close to stationary black holes is certainly very appealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Even though it is clear that the tools employed here would not apply to black hole spacetimes with 1 The Hawking effect is a quantum process and the relavant state describing it is a quantum state that corresponds to the vacuum in the far past idealized by I −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Such state can be viewed in the interior as a superposition of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Actual particles appear inside in the hypothetical situation of the detection of a partner at I + which, according to the standard interpretation of quantum mechanics, will produce a collapse of the vacuum to a new state containing an actual particle falling into the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The situation after such collapse is the semiclassical situation that we model here with our classical language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' A precise description of such a situation in quantum terms is a question that can only be addressed in a full theory of quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' We argue here that our little solvable model goes a humble little step into that direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' 10 i+ I + ⌃ Figure 2: Hawking pairs produced in a black hole spacetime with M ≫ mp are described as test field excitations on a background spacetime that is idealized by a stationary black hole solution in the region of the Penrose diagram away from the collapsing matter and containing i+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Stationarity implies the existence of a conserved energy-momentum current used, in this paper, to relate the value of pφ in the KS model to the frequency of the emitted particle at I + and (via Hawking temperature) to the black hole mass M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' angular momentum (due to the breaking of spherical symmetry), one could entertain the possibility of including an electric charge without leaving the general framework of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' However, such an apparently simple extension will reserve challenging new aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The first qualitative new feature is that the singularity becomes timelike in the unperturbed black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' However, when a scalar field perturbation is added its back reaction is expected to produce the phenomenon of mass inflation [37] near the Cauchy horizon of the unperturbed background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This completely changes the global features of a realistic charged black hole interior in a way that would seem to preclude the applicability of the present methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' More precisely, if we use the Reissner-Nordstrom background black hole geometry as a basis for the present discussion—noting that the presence of mass inflation already implies that strong deviations from the Reissner-Nordstrom interior solution are to be expected—the true would-be-singularity should materialize near the location of the Cauchy horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' However, close to the Cauchy horizon scalar field modes with frequency ω are expected to be infinitely blue shifted precluding the type of approximation available in the present case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This makes such exploration highly non-trivial even when certainly interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' A NATURAL POLYMER QUANTIZATION OF THE DEEP INTERIOR DYNAMICS In previous sections we have shown that the Hamiltonian constraint (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='20) describes the deep interior dynamics of scalar excitations without angular momentum falling into a macroscopic Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The approximations used are based on assumptions that are satisfied by the (spherically symmetric) Hawking excitations produced during black hole evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Thus, the simplified dynamics in the deep interior region, which will turn out to be analytically solvable (both at the quantum and classical level), offers a toy scenario to analyze key questions of black hole evaporation in a controlled scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' We will show in this section that there is natural polymer quantization of the deep interior dynamics with remarkable simple properties such as: (like in the full theory of LQG) the discreteness of the area of constant area surfaces, a well defined quantum dynamics across the singularity, an effective classical description, and a direct (to our knowledge novel) link with the continuum representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The polymer quantization we propose does not suffer from the usual ambiguities associated to the so-called holonomy corrections [38] as the Hamiltonian constraint evolution has (in our simple model) a clear-cut geometric interpretation that allows for a unique polymerization that is compatible with the continuum limit (defined by the Schroedinger representation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This special geometric property arises from the fact that the Hamiltonian constraint is linear in the momentum conjugate to the area of the spheres whose spectrum is discrete in our polymer representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Ambiguities remain in the form of the so-called inverse volume corrections which are necessary if one defines the quantum dynamics across the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' 11 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Sketch of the Schrodinger quantization In the standard Schrodinger representation one would quantize the phase space of Section II A by promoting the variables a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' pa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' pm to self adjoint operators �m ψ(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' pφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' a) = mψ(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' pφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' �pmψ(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' pφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' a) = −i∂mψ(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' pφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' �a ψ(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' pφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' a) = aψ(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' pφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' �paψ(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' pφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' a) = −i∂aψ(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' pφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' �pφψ(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' pφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' a) = pφψ(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' pφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' �φ ψ(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' pφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' a) = i∂φψ(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' pφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' in the kinematical Hilbert space is HS = L 2(R3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' equipped with the usual inner product ⟨ψ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' ψ2⟩ = � +∞ −∞ � +∞ −∞ � +∞ −∞ ψ1(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' pφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' a)ψ2(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' pφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' a)dmdpφda,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='1) where we have chosen the momentum representation for the scalar field for convenience (as pφ is one of the constants of motion of the system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Eigenstates of the �a operator are interpreted as distributions (they are not in the Hilbert space) and one usually writes �a |a⟩ = a |a⟩ (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='2) with a ∈ R and form an orthonormal basis ⟨a, a′⟩ = δ(a, a′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='3) The dynamics is imposed by solving the Hamiltonian constraint (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='20) which, in the present representation, takes the precise form of a Schrodinger equation in the area variable a, namely � −iℏ ∂ ∂a + 1 2a � m + p2 φ 16πℓ2pm �� ψ(m, pφ, a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='4) As usual, solutions of the constraint are certainly not square integrable in the a-direction, thus physical states are outside of the kinematical Hilbert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The physical Hilbert space is defined as the space of square integrable functions of m and pφ at fixed time a—Hphys = L 2(R2)—with inner product ⟨ψ1(a), ψ2(a)⟩phys = � +∞ −∞ � +∞ −∞ ψ1(m, pφ, a)ψ2(m, pφ, a)dmdpφ, (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='5) which is preserved, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' it is independent of a, by the Schrodinger equation (evolution is unitary in a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Two important remarks are in order: First note that we are formulating in detail the dynamics of the system in the near singularity approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The physical reason for this is that (as argued previously) it is only in this approximation that the system can be compared with a (spherically symmetric) black hole with spherically symmetric excitations falling inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' A side gain is also the simplification of the dynamics which will allow us for a simpler quantization and the analysis of the possibility of a well defined dynamics across the singularity when we undergo the LQG inspired quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' One could however consider the quantization of the minisuperspace system without the near singularity approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In that case one would need to write a Schroedinger equation using the Hamiltonian (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='13), now genuinely time-dependent (r-dependent), for which unitary evolution would involve path ordered exponentials (as the Hamiltonian does not commute with itself at different r values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In addition one would need to work with either r, f, pr, pf variables or a, f, pa, pf variables without the luxury of the simplifications introduced by the use of the near singularity variables (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The polymer quantization We define now a representation of the phase space variables that incorporates a key feature of the full theory of LQG: the area quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Such representation closely mimics the structure of the quantum theory in the fundamental theory in such a way that the area variable a acquires a discrete spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Mathematically, this is achieved by replacing the L 2 structure of the inner product in the variable pa by the inner product of the Bohr compactification of the pa phase space dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' More precisely one substitutes the kinematical inner product in the Schroedinger representation (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='1) by ⟨ψ1, ψ2⟩ = lim ∆→+∞ 1 2∆ � +∆ −∆ �� +∞ −∞ ψ1(m, pφ, pa)ψ2(m, pφ, pa)dmdpφ � dpa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='6) 12 With this inner product, periodic functions of pa with arbitrary period are normalizable and the conjugate a- representation acquires the property of discreteness in a way that closely mimics the structure of the fundamental theory of loop quantum gravity [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In particular eigenstates of �a exist �a |a⟩ = a |a⟩ (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='7) with a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' These state form an orthonormal basis with inner product ⟨a, a′⟩ = δa,a′, (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='8) in contrast with (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Discreteness of the spectrum of �a comes at the prize of changing the kinematical Hilbert space structure in a way that precludes the infinitesimal translation operator �pa to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Instead only finite translations (quasi periodic functions of pa) can be represented as unitary operators in the polymer Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Their action on the a-basis is given by � eiλpaψ(m, pφ, a) = ψ(m, pφ, a + λℓp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='9) Eigenstates of the finite translations (or shift operators) exist and are given by wave functions supported on discrete a-lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Namely, ψk,ϵ(a) ≡ � exp(ika) if a ∈ Γϵ,λ ≡ {(ϵ + nλ)ℓ2 p ∈ R}n∈Z 0 otherwise (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='10) where the parameter ϵ ∈ [0, λ) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The discrete lattices denoted Γϵ,λ are the analog of the spin-network graphs in LQG with the values of a on lattice sites the analog of the corresponding spin labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' With all this one has (using (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='9)) that � eiλpaψk,ϵ(a) = eiλkψk,ϵ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='11) Note that, unlike the Schroedinger representation where the eigen-space of the momentum operator is one dimensional, the eigen-spaces of the translation operator (labelled by the eigenvalue eiλk) are infinite dimensional and non separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This is explicit from the independence of the eigen-values of the continuous parameter ϵ ∈ [0, λ) labelling eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Such huge added degeneracy in the spectrum of the shift operators is a general feature of the polymer representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' We will show that this degeneracy can show up in Dirac observables of central physical importance such as the mass operator in Section III G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Quantum dynamics The dynamics is dictated by the quantization and imposition of the constraint (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' As the operator corresponding to pa does not exist in our kinematical Hilbert space we introduce a polymerized version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Traditionally, this is achieved by replacing the infinitesimal translation operator λ�pa −→ traditional � sin λpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='12) The rule consists of making some ‘minimal’ substitution of pa by a periodic regularization satisfying that in the limit λ → 0 the functional choice will approximate the original function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Such rule is intrinsically ambiguous and, it opens in general the door for an infinite set of possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Such choices are to be interpreted as quantization ambiguities of the Hamiltonian constraint with potential quantitative dynamical consequences (for a general discussion see [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Dynamical implications of these ambiguities can be analysed in detail in simple models of quantum cosmology [38] and black holes [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In the full theory a new perspective on the regularization issue has been introduced motivated by the novel math- ematical notion of generalized gauge covariant Lie derivatives [41] and their geometric interpretation allowing for the introduction of a natural regularization (and subsequent) anomaly free quantization of the Hamiltonian constraint [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Even when the procedure does not eliminate all ambiguities of quantization (choices are available in the part of the quantum constraint responsible for propagation [43]), the new technique reduces drastically some of them in the part of the Hamiltonian that is more stringently constrained by the quantum algebra of surface deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' What we want to emphasize here is that the analogous procedure in the case of our symmetry reduced Hamiltoinian has a similar effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Indeed, because our classical Hamiltonian constraint is linear in the variable pa (whose associated Hamiltonian vector field has a crystal-clear geometric interpretation of infinitesimal translations in a), one has an 13 unambiguous choice of quantization: the obvious choice to replace infinitesimal translations (which do not exist in the polymer representation) by finite translations or shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' From this geometric perspective the right polymerization is the regularization that make the replacement λ�pa −→ � eiλpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='13) In other words the differential time evolution in the Schrodinger equation must be represented in the polymer Hilbert space by a finite translation with a polymerization scale λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' However, as such an action is associated with a clear geometric meaning, the geometric compatibility with the Schrodinger equation can be preserved if the second term in the classical Hamiltonian (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='20) is exponentiated too in order to produce the well known unitary evolution operator that produces finite area evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Disregarding for the moment quantum corrections that will have to be included near the a = 0 region (see Section III E), the quantum constraint is taken to be exp(iλpa) � �� � finite areatime translation |ψ⟩ − finite areatime unitary evolution operator � �� � exp � i 2 log � a + λℓ2 p a � � m + p2 φ 16πℓ2pm �� |ψ⟩ = 0, (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='14) whose action is well defined in the polymer representation and whose solutions are easily found (by acting on the left with ⟨m, pφ, a|) to be wave functions satisfying the discrete dynamics given by ψ(m, pφ, a + λℓ2 p) = e i 2 log � a+λℓ2 p a �� m+ p2 φ 16πℓ2pm � ψ(m, pφ, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='15) The physical Hilbert space is defined via the usual inner product at fixed (discrete) time a via ⟨ψ1(a), ψ2(a)⟩phys = � +∞ −∞ � +∞ −∞ ψ1(m, pφ, a)ψ2(m, pφ, a)dmdpφ, (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='16) which independent of the lattice sites as required (a property that we could identify with the intrinsic unitarity of the quantum constraint kernel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' More precisely the physical inner product is a constant of the quantum motion associated to the full history represented by the lattice Γϵ,λ as implied by unitarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Explicitly one has ⟨ψ1(a), ψ2(a)⟩phys = ⟨ψ1(a + λ), ψ2(a + λ)⟩phys ∀ a ∈ Γϵ,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='17) Ambiguities of regularization that are usually associated with the polymerization procedure are thus completely absent in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The reason is the linear dependence of the Hamiltonian constraint in the polymerized variable which allows for a regularization fixed by the geometric interpretation of the classical Hamiltonian vector field associated to the corresponding variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' However, ambiguities remain when one studies the evolution across the would-be- singularity of the Kantowski-Sachs model at a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' We will study this in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Now we would like to concentrate on the evolution when we are away from the a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In such regime the one step evolution (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='15) can be composed to produce the arbitrary initial to final area evolution ψ(m, pφ, ϵ + nλ) = �ϵ + nλ ϵ + qλ � im 2 � 1+ p2 φ 16πℓ2pm2 � ψ(m, pφ, ϵ + qλ), (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='18) for arbitrary integers n, q > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Evolving across the a = 0 point will be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The continuum versus the polymer dynamics The polymer dynamics that arises from the geometric action of the quantum constraint (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='14) enjoys of the appealing feature of being closely related to the dynamics that one would obtain in the continuum Schroedinger representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This statement can be made precise as follows: any solution of the Schroedinger equation (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='4) induces on any given lattice Γϵ,λ a solution of (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Conversely, physical states of the polymer theory represent a discrete sampling of the continuum solutions of (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' However, the Schroedinger evolution is ill defined at 14 the singularity a = 0 due to the divergence of the 1/a factor in front of the second term of (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The polymer representation allows for a well defined evolution across the singularity thanks to the deviations from the 1/a behaviour introduced by the analog of the ‘inverse-volume’ corrections (see next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Nevertheless, with the appropriate modification of the 1/a factor in the Schroedinger equation the correspondence between the discrete and continuum continues to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Quantum evolution across the classical singularity Quantum evolution in a for all values of a including the singularity is dictated by the quantum corrected version of the constraint (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='14) given by ψ(m, pφ, a) = e i 2 � a� a0[ 1 a]qda �� m+ p2 φ 16πℓ2pm � ψ(m, pφ, a0) (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='19) = e i 2 [τ(a)−τ(a0)] � m+ p2 φ 16πℓ2pm � ψ(m, pφ, a0), (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='20) where �1 a � q (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='21) denotes the quantum corrected expression for the operator a−1 (the analog of inverse volume correction in cosmology) that can be implemented in various ways due to inherent ambiguities associated to the polymer quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In general this will give deviations of the a−1 behaviour in the region a ∼ ℓ2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This modifies the integral of a−1 in a way charaterized by the (to a large extend arbitrary [38]) function τ(a) introduced in the second line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' One, among the many possibilities, is the one that follows from the so-called Thiemann’s trick whose most elementary form is (see [11] for its application in cosmology, see [38] for a discussion of the multiplicity of variants) �1 a � Thm ≡ sign(a) � � � |a + ℓ2p| − � |a − ℓ2p| ℓ2p � � 2 = 1 a + O � a−3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='22) Integration leads to the following τ(a) function in (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='19) τ(a) = � � � |a| � |a| − � |a|2 − 1 � + log �� |a|2 − 1 + |a| � − π 2 + 1 1 ≤ |a| −|a| �� 1 − |a|2 − 2 � − sin−1(|a|) |a| ≤ 1 , (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='23) whose graph is presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The classical theory does not fix the evolution uniquely in this high curvature regime where quantum geometry effects cannot be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Quantum geometry effects regularize the dynamics near the a = 0 singularity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' one way of seeing this is that the factor log(a) in the quantum evolution away from the singularity in (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='14) receives inverse volume quantum geometry corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' As discussed in [38] these corrections are ambiguous (as fact that should not be surprising given the expectation that the classical theory cannot guide us all the way to the deep UV in QFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Instead of proposing one particular UV extension, as in the example shown where Thiemann regularization was used, one might simply keep all possibilities open and assume that the corresponding operator is regularized in the relevant region by some arbitrary function log(a) → τ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In regions where τ(a) = log(a) the quantum evolution leads to semiclassical equations that match exactly Einstein’s equations in the KS sector (more details in Section III F) F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Dynamics of semiclassical states and tunelling across the singularity Let us first consider the vacuum pφ = 0 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This case is important because it should correspond to the Schrwarzschild interior in the region r ≪ M (or a ≪ M 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The dynamical evolution (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='19) becomes ψ(m, a) = e i 2 m(τ(a)−τ(a0))ψ(m, a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='24) 15 a ⌧(a) 10 5 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='0 Figure 3: The dynamics across the singularity is regular due to inverse volume corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In this picture we show the function τ(a), defining the dynamical evolution across the singularity, in the case of Thiemann’s regularization of the inverse-area- operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In the pm representation the previous equation simply reads ψ(pm, a) = ψ(pm + τ(a) − τ(a0) 2 , a0), (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='25) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=', a simple translation in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Such dynamical evolution in a implies in an obvious manner that semiclassical states peaked at classical values (m, pm) at area a0 with some given fluctuations will be simply evolved into the translated state with the very same fluctuation properties peaked at (m, pm + (τ(a) − τ(a0))/2) at area a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This is a remarkable property of the quantum system: independently of the quantum gravity effects encoded in the precise form of τ(a), expectation values satisfy well defined effective dynamical equations which are exact (not an approximation), and semiclassical states are not spread by the dynamical evolution (as in the simple case of the harmonic oscillator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Notice that the validity of effective dynamical equations in the context of black hole models have remained a conjecture in other formulations [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In regions where τ(a) = log a the previous evolution gives pm(a) = log(r) + pm(a0), m(a) = m(a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='26) From the definitions (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='17) we have that fpf = m(a) and f = − exp[−pm(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The constraint (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='12), gives us h(r) and the metric becomes ds2 = e−pm(a)dt2 − ℓ2 0 (4ℓp)4π2 e−pm(a) m(a)2 da2 + a 4π dΩ2 (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='27) which corresponds to the Schrwarzschild solution with the two Dirac observables M and pM given in terms of the initial conditions by M = 2 √ 4πℓ4 p ℓ2 0 √a0 m(a0)2epm(a0), pM = ℓ0 √ 4π 2ℓp√a0 e−pm(a0) m(a0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='28) When quantum inverse volume corrections are taken into account then the quantum evolution is perfectly well defined across the classical singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The evolution of the mean values of a semiclassical state is also well defined and given by pm(a) = pm(a0) + 1 2(τ(a) − τ(a0)), m(a) = m(a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='29) 16 Such solutions can also be obtained from the Hamiltonian constraint given that one replaces the operator a−1 by its quantum regularization (effective equations are in this sense exact).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The metric for all values of |a| ≪ M 2 but otherwise arbitrary becomes ds2 = exp � −pm(a0) − 1 2(τ(a) − τ(a0)) � � dt2 − ℓ2 0 (4ℓp)4π2 da2 m(a0)2 � + a 4π dΩ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='30) Since the Thiemann regularization produces a τ(a) ∼ a2 ∼ r4 near a = 0 we see that the previous metric is just given by a two dimensional Minkowski metric fibrated with two dimensional sphere with time dependent radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The shrinking of the spheres leads to a singularity at a = 0 where the spheres collapse and the spacetime geometry (as described by the effective line element) becomes a two dimensional flat one at the singularity in the a-t plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Despite the singular nature of the effective metric the fundamental quantum evolution is well defined across the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In the presence of matter the situation is a bit more involved due to the factor pφ/m appearing in matter contribution to the Hamiltonian constraint (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' However, in the spirit of applying this analysis to macroscopic black holes and modelling the dynamics of a weak scalar excitation (a Hawking particle) falling into the singularity, it is natural to focus on semiclassical states (Gaussian) peaked on values such that m ≫ pφ with fluctuations σm ≪ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' As in the vacuum case m = m0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=', m is a constant of motion as well as its spread σm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The dynamics of the conjugate variable (the mean value pm of the variable pm) can be evaluated using stationary phase methods and the result is pm(a) = pm(a0) + 1 2(τ(a) − τ(a0)) � 1 + p2 φ m(a0)2 � 1 + 3 4 σ2 m m(a0)2 �� + O � σ3 m m(a0)3 � , (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='31) which can be seen to correspond to the classical solutions found in Section II C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Notice that, as expected from the form of the matter coupling, there are here quantum corrections characterized by terms proportional to σ2 m/m(a0)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The spread in the variable pm is not time independent if we take into account higher order corrections, namely σ2 pm(a) = 1 σ2m + σ2 mp4 φ 4m(a0)4 (τ(a) − τ(a0))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='32) The previous equations are derived assuming that the scalar field is in an eigenstate of the momentum pφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This is an idealization that simplifies the analysis of the dynamical evolution of the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Similarly, if we assume that the geometry state was in an eigenstate of m then we can easily analyze the dynamics of the scalar field assuming that it is initially in a gaussian semiclassical state picked about pφ(a0) and φ(a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In accordance with the classical solutions we get pφ(a) = pφ(a0) φ(a) = φ(a0) + (τ(a) − τ(a0)) pφ(a0) 16πℓ2pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='33) One way to quickly derive these equations by inspection is to realize that the Hamiltonian constraint (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='20) is that of a non relativistic point particle with mass proportional to our geometric variable m evolving in dτ = [1/a]qda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Note that (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='31) implies that the back-reaction of the scalar field enters only through a simple modification of the exponential conformal factor in front of the 2-metric in the a-t ‘plane’ in equation (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Unlike the geometry degrees of freedom, the fluctuations in the scalar field grow as one approaches the would-be- singularity: for a given geometry semiclassical state picked around the mass M, the spread of the scalar field σφ in an initially eigenstate of φ at area a grows to a maximum value close to the would-be-singularity such that Mσφ < � M log � a/ℓ2p � 16πℓ0 , σpφ < � 16πM ℓ0 log � a/ℓ2p � (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='34) which are small in the interior if we take ℓ0 ≫ M as expected from appearance of ℓ0 in the fundamental commutation relations [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Note that, if we take into account inverse volume corrections of the type suggested by LQG (see Figure 3), the scalar field reaches a critical point at a = 0 where its area velocity vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The mass operator (in the vacuum case) In this section we concentrate in the vacuum case for simplicity as the mass can be directly read off the form of the metric in this case via a simple comparison with the classical Schwarzschild solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In this case the result is M = 2 √ 4πℓ4 p ℓ2 0 √a m2epm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='35) 17 where we used the vacuum solution (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='3) (in its r ≪ M approximation) and (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' It is easy to verify that the previous is indeed a Dirac observable by showing that it commutes with Hamiltonian constraint (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Its non linear dependence on the basic variables anticipates factor ordering ambiguities when it comes to promoting the mass to a quantum operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Here we focus on the choice � M = α(�a) � �me�pm �m � , (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='36) where α(a) ≡ 2 √ 4πℓ4 p/(ℓ2 0 √a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The eigenstates equation � M|φM⟩ = M|φM⟩, (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='37) turns into the differential equation α(a)epm ∂φM(pm, a) ∂pm + α(a)epm ∂2φM(pm, a) ∂p2m + MφM(pm, a) = 0, (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='38) if we expand the eigenstate in the pm, a basis, namely |M⟩ = � a∈Γϵ,λ � φM(pm, a)|pm⟩|a⟩dpm, (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='39) where the sum runs over the discrete lattice Γϵ,λ defined in (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='10) when introducing the dynamical constraint (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This differential eigenvalue equation is solved by φM(pm, a) ≡ ⟨pm, a| M⟩ = � 2 √ M α(a) e−pm/2J1 � 2 � M α(a)e−pm/2 � , (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='40) where J1 is a Bessel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' One can explicitly verify that the quantum dynamics (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='14) preserves the eigenstates by explicitly showing that the evolution between arbitrary lattice points a1, a2 ∈ Γλ,ϵ sends the wave function of the eigenstate at the a1 lattice point to the a2 lattice point (as expected for a Dirac observable);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' or equivalently, the eigenstates of the mass are physical states solving (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Explicitly, � e i 2 (log(a2)−log(a1))mφM(pm, a1) = φM � pm + 1 2(log(a2) − log(a1)), a0 � = φM (p, a2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='41) Now the evolution across a = 0 requires inverse volume corrections which modifies the previous dynamical law by replacing log(a) → τ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The mass Dirac observable still exists once inverse volume corrections are turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' It corresponds to the modification of (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='36) via the substitution �a → exp(�τ(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Eigenstates are also obtained by the same substitution in (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='40) and satisfy the expected Dirac observable condition (which now holds for lattice points at different sides across the singularity) � e i 2 (τ(a2)−τ(a1))mφM(pm, a1) = φM � pm + 1 2(τ(a2) − τ(a1)), a0 � = φM (p, a2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='42) When supported on the same lattice, one can show that they satisfy the orthogonality relation ⟨M|M ′⟩phys = δ(M, M ′), (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='43) where the inner product is computed with the physical inner product (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Thus the spectrum of the mass operator is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' It was argued in the context of the full LQG theory in [31, 46, 47] that the eigenspaces of the mass should be infinite degenerate due to the underlying discrete structure of the fundamental theory and the existence of defects that would not be registered in the ADM mass operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Interestingly, the conjectured property is illustrated explicitly in our simple toy model as the eigenvectors (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='39) for a given eigenvalue M there are infinitely many and labelled by a continuum parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' More precisely they are associated with wave functions of the form (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='40) supported on lattices with different values of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Thus eigenstates of the mass should then be denoted |M, ϵ⟩ with orthogonality relation ⟨M, ϵ|M ′, ϵ′⟩phys = δ(M, M ′)δϵ,ϵ′, (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content='44) 18 where δϵ,ϵ′ is the Kronecker delta symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The existence of such a large degeneracy is a generic feature of the polymer representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Even when this is a toy model of quantum gravity, this feature is likely to reflect a basic property of the representation of the algebra of observables in the full LQG context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Here we are showing that the mass operator is hugely degenerate suggesting that the usual assumption of the uniqueness of the vacuum in background dependent treatments of quantum field theory might fail in a full loop quantum gravity context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Alternative factor orderings of the quantum operator M could be treated similarly (some simple choices lead to slightly different eigenvectors written also in terms of Bessel functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Such an ambiguity is not relevant for our purposes (and it does not change the key fact that the spectrum of M is infinitely degenerate) as the aim of the model is not to construct any quantitative physical prediction but rather to use it as a toy model to investigate possibly sufficiently generic features that could actually survive in the full theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The large degeneracy of the mass spectrum is, in our view, an interesting example of one such feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' DISCUSSION We have shown that test field solutions of the Klein-Gordon equation with zero angular momentum behave like solutions of the KS symmetry reduced model in the deep interior region r ≪ M defined in terms of the background Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This implies that, spherically symmetric scalar matter falling into a spherical black hole can be modelled by the KS solutions near the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Despite the expected limitations of symmetry reduced models in capturing the full physics in the UV regime, the model includes back-reaction of the scalar matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Focusing in the deep interior region and using perturbation theory in pφ/M we show that it is possible to interpret the solutions of KS with matter as Schwarzschild solutions with matter excitations falling towards the r = 0 singularity (this interpretation is not global but it shown to be correct in the deep interior region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The Hamiltonian dynamics simplifies considerably in that regime becoming tractable both at the classical as well as the quantum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Perturbation theory applies (we have shown) to the situation involving Hawking particles falling into the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In close analogy to LQG, we define a quantization of the system describing the deep interior region where the area of the r =constant spheres has a discrete spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This leads to the polymer representation of the area of the orbits of the rotation group and its conjugate momentum that allows for a well defined quantum evolution across the singularity if one introduces customary ‘inverse-volume’ corrections to the quantum scalar constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The Hamiltonian constraint admits a simple geometric interpretation in the to-be-polymerized sector due to the linearity of the Hamiltonian constraint in the momentum variable conjugated to the area of the r =constant spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The geometric nature of the action of the classical constraint allows for the introduction of a unique polymerization prescription respecting this action at the quantum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This reduces the ambiguity usually associated to the procedure of quantization of the dynamical constraints for reasons that resonate with the ones that lead to similar advantages in the full theory [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Remarkably, the dynamics is exactly solvable at the quantum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In the vacuum case, the mass operator is a Dirac observable that we quantize and whose spectrum is given explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Semiclassical states dynamics leads to effective evolution equations that can be characterized exactly in the vacuum case and using suitable stationary phase approximations in the case where matter is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' These effective equations coincide with Einsteins equations in regions where the inverse volume corrections can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' An important formal aspect of the model is that it presents a concrete example of violation of the ‘unicity of the vacuum’ assumption that permeates discussions of Hawking’s information puzzle for over 40 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' In loop quantum gravity the discrete structure of the theory at the Planck scale suggests that a given (macroscopic) ADM mass configuration need not correspond to a unique quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' High degeneracy due to the contribution of microscopic degrees of freedom is expected but hard to prove at the present stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' This leads to a certain degree of disagreement on the status of such statements in the field at large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Although, a key instance where such degeneracy is accepted with little controversy is in the loop quantum gravity models designed to calculate black hole entropy (for reviews and references see [48, 49]) where the statistical origin of the entropy lies precisely in the large multiplicity of underlying microscopic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Our simple model might still be too simple to represent definite evidence in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' Nevertheless, the results of Section III G do provide a toy model to eventually study the implications of the large degeneracy of the mass spectrum in discussion of the fate of information in black hole evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The model we introduce here is simple and workable, we hope it could provide potentially useful insights in dealing with qualitative questions concerning black hole evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' The investigation of these interesting possibilities is left for the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE2T4oBgHgl3EQfhwde/content/2301.03951v1.pdf'} +page_content=' 19 V.' 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